Arithmetical Properties of Commutative Rings and Monoids

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Arithmetical Properties of Commutative Rings and Monoids

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universität Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

N. Jacobson, Exceptional Lie Algebras L.-Å. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis I. Satake, Classification Theory of Semi-Simple Algebraic Groups F. Hirzebruch et al., Differentiable Manifolds and Quadratic Forms I. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Burckel, Characterization of C(X) Among Its Subalgebras B. R. McDonald et al., Ring Theory Y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus et al., Calkin Algebras and Algebras of Operators on Banach Spaces E. O. Roxin et al., Differential Games and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thomier, Topology and Its Applications J. M. Lopez and K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups G. B. Seligman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis L. Cesari et al., Nonlinear Functional Analysis and Differential Equations J. J. Schäffer, Geometry of Spheres in Normed Spaces K. Yano and M. Kon, Anti-Invariant Submanifolds W. V. Vasconcelos, The Rings of Dimension Two R. E. Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology S. K. Jain, Ring Theory B. R. McDonald and R. A. Morris, Ring Theory II R. B. Mura and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynamical Systems H.-C. Wang, Homogeneous Branch Algebras E. O. Roxin et al., Differential Games and Control Theory II R. D. Porter, Introduction to Fibre Bundles M. Altman, Contractors and Contractor Directions Theory and Applications J. S. Golan, Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis R. Gordon, Representation Theory of Algebras M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. I. Arruda et al., Mathematical Logic F. Van Oystaeyen, Ring Theory F. Van Oystaeyen and A. Verschoren, Reflectors and Localization M. Satyanarayana, Positively Ordered Semigroups D. L Russell, Mathematics of Finite-Dimensional Control Systems P.-T. Liu and E. Roxin, Differential Games and Control Theory III A. Geramita and J. Seberry, Orthogonal Designs

46. J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces 47. P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics 48. C. Byrnes, Partial Differential Equations and Geometry 49. G. Klambauer, Problems and Propositions in Analysis 50. J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields 51. F. Van Oystaeyen, Ring Theory 52. B. Kadem, Binary Time Series 53. J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems 54. R. L. Sternberg et al., Nonlinear Partial Differential Equations in Engineering and Applied Science 55. B. R. McDonald, Ring Theory and Algebra III 56. J. S. Golan, Structure Sheaves Over a Noncommutative Ring 57. T. V. Narayana et al., Combinatorics, Representation Theory and Statistical Methods in Groups 58. T. A. Burton, Modeling and Differential Equations in Biology 59. K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory 60. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces 61. O. A. Nielson, Direct Integral Theory 62. J. E. Smith et al., Ordered Groups 63. J. Cronin, Mathematics of Cell Electrophysiology 64. J. W. Brewer, Power Series Over Commutative Rings 65. P. K. Kamthan and M. Gupta, Sequence Spaces and Series 66. T. G. McLaughlin, Regressive Sets and the Theory of Isols 67. T. L. Herdman et al., Integral and Functional Differential Equations 68. R. Draper, Commutative Algebra 69. W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras 70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems 71. J. Van Geel, Places and Valuations in Noncommutative Ring Theory 72. C. Faith, Injective Modules and Injective Quotient Rings 73. A. Fiacco, Mathematical Programming with Data Perturbations I 74. P. Schultz et al., Algebraic Structures and Applications 75. L Bican et al., Rings, Modules, and Preradicals 76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry 77. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces 78. C.-C. Yang, Factorization Theory of Meromorphic Functions 79. O. Taussky, Ternary Quadratic Forms and Norms 80. S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications 81. K. B. Hannsgen et al., Volterra and Functional Differential Equations 82. N. L. Johnson et al., Finite Geometries 83. G. I. Zapata, Functional Analysis, Holomorphy, and Approximation Theory 84. S. Greco and G. Valla, Commutative Algebra 85. A. V. Fiacco, Mathematical Programming with Data Perturbations II 86. J.-B. Hiriart-Urruty et al., Optimization 87. A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups 88. M. Harada, Factor Categories with Applications to Direct Decomposition of Modules 89. V. I. Istra’tescu, Strict Convexity and Complex Strict Convexity 90. V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations

91. H. L. Manocha and J. B. Srivastava, Algebra and Its Applications 92. D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems 93. J. W. Longley, Least Squares Computations Using Orthogonalization Methods 94. L. P. de Alcantara, Mathematical Logic and Formal Systems 95. C. E. Aull, Rings of Continuous Functions 96. R. Chuaqui, Analysis, Geometry, and Probability 97. L. Fuchs and L. Salce, Modules Over Valuation Domains 98. P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics 99. W. B. Powell and C. Tsinakis, Ordered Algebraic Structures 100. G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications 101. R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications 102. J. H. Lightbourne III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential Equations 103. C. A. Baker and L. M. Batten, Finite Geometrics 104. J. W. Brewer et al., Linear Systems Over Commutative Rings 105. C. McCrory and T. Shifrin, Geometry and Topology 106. D. W. Kueke et al., Mathematical Logic and Theoretical Computer Science 107. B.-L. Lin and S. Simons, Nonlinear and Convex Analysis 108. S. J. Lee, Operator Methods for Optimal Control Problems 109. V. Lakshmikantham, Nonlinear Analysis and Applications 110. S. F. McCormick, Multigrid Methods 111. M. C. Tangora, Computers in Algebra 112. D. V. Chudnovsky and G. V. Chudnovsky, Search Theory 113. D. V. Chudnovsky and R. D. Jenks, Computer Algebra 114. M. C. Tangora, Computers in Geometry and Topology 115. P. Nelson et al., Transport Theory, Invariant Imbedding, and Integral Equations 116. P. Clément et al., Semigroup Theory and Applications 117. J. Vinuesa, Orthogonal Polynomials and Their Applications 118. C. M. Dafermos et al., Differential Equations 119. E. O. Roxin, Modern Optimal Control 120. J. C. Díaz, Mathematics for Large Scale Computing 121. P. S. Milojevic, Nonlinear Functional Analysis 122. C. Sadosky, Analysis and Partial Differential Equations 123. R. M. Shortt, General Topology and Applications 124. R. Wong, Asymptotic and Computational Analysis 125. D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics 126. W. D. Wallis et al., Combinatorial Designs and Applications 127. S. Elaydi, Differential Equations 128. G. Chen et al., Distributed Parameter Control Systems 129. W. N. Everitt, Inequalities 130. H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differential Equations 131. O. Arino et al., Mathematical Population Dynamics 132. S. Coen, Geometry and Complex Variables 133. J. A. Goldstein et al., Differential Equations with Applications in Biology, Physics, and Engineering 134. S. J. Andima et al., General Topology and Applications 135. P Clément et al., Semigroup Theory and Evolution Equations

136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178.

K. Jarosz, Function Spaces J. M. Bayod et al., p-adic Functional Analysis G. A. Anastassiou, Approximation Theory R. S. Rees, Graphs, Matrices, and Designs G. Abrams et al., Methods in Module Theory G. L. Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing M. C. Joshi and A. V. Balakrishnan, Mathematical Theory of Control G. Komatsu and Y. Sakane, Complex Geometry I. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations T. Mabuchi and S. Mukai, Einstein Metrics and Yang–Mills Connections L. Fuchs and R. Göbel, Abelian Groups A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum G. Dore et al., Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedt et al., Functional Analysis K. G. Fischer et al., Computational Algebra K. D. Elworthy et al., Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, et al., Commutative Ring Theory S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions P. Clément and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research W. O. Bray et al., Fourier Analysis J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pavel, Optimal Control of Differential Equations M. Ikawa, Spectral and Scattering Theory X. Liu and D. Siegel, Comparison Methods and Stability Theory J.-P. Zolésio, Boundary Control and Variation M. Kr’íz’’ek et al., Finite Element Methods G. Da Prato and L. Tubaro, Control of Partial Differential Equations E. Ballico, Projective Geometry with Applications M. Costabel et al., Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twistor Theory H. Cook et al., Continua D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et al., Complex Analysis and Geometry E. Casas, Control of Partial Differential Equations and Applications N. Kalton et al., Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et al., Differential Equations and Control Theory P. Marcellini et al. Partial Differential Equations and Applications A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type

179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222.

M. Maruyama, Moduli of Vector Bundles A. Ursini and P. Aglianò, Logic and Algebra X. H. Cao et al., Rings, Groups, and Algebras D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models J. E. Andersen et al., Geometry and Physics P.-J. Cahen et al., Commutative Ring Theory J. A. Goldstein et al., Stochastic Processes and Functional Analysis A. Sorbi, Complexity, Logic, and Recursion Theory G. Da Prato and J.-P. Zolésio, Partial Differential Equation Methods in Control and Shape Analysis D. D. Anderson, Factorization in Integral Domains N. L. Johnson, Mostly Finite Geometries D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm–Liouville Problems W. H. Schikhof et al., p-adic Functional Analysis S. Sertöz, Algebraic Geometry G. Caristi and E. Mitidieri, Reaction Diffusion Systems A. V. Fiacco, Mathematical Programming with Data Perturbations M. Kr’íz’cek et al., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups V. Drensky et al., Methods in Ring Theory W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions P. E. Newstead, Algebraic Geometry D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology Z. Chen et al., Advances in Computational Mathematics X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs C. Y. Yédérém and S. A. Stepanov, Number Theory and Its Applications D. E. Dobbs et al., Advances in Commutative Ring Theory F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry J. Kakol et al., p-adic Functional Analysis M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of Algebras R. Costa et al., Nonassociative Algebra and Its Applications T.-X. He, Wavelet Analysis and Multiresolution Methods H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnol et al., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. Ali Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra Á. Granja et al., Ring Theory and Algebraic Geometry A. K. Katsaras et al., p-adic Functional Analysis

223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241.

R. Salvi, The Navier-Stokes Equations F. U. Coelho and H. A. Merklen, Representations of Algebras S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory G. Lyubeznik, Local Cohomology and Its Applications G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications W. A. Carnielli et al., Paraconsistency A. Benkirane and A. Touzani, Partial Differential Equations A. Illanes et al., Continuum Theory M. Fontana et al., Commutative Ring Theory and Applications D. Mond and M. J. Saia, Real and Complex Singularities V. Ancona and J. Vaillant, Hyperbolic Differential Operators and Related Problems G. R. Goldstein et al., Evolution Equations A. Giambruno et al., Polynomial Identities and Combinatorial Methods A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups J. Bergen et al., Hopf Algebras A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao S. Caenepeel and F. van Oystaeyen, Hopf Algebras in Noncommutative Geometry and Physics J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis S. T. Chapman, Arithmetical Properties of Commutative Rings and Monoids

Arithmetical Properties of Commutative Rings and Monoids

Scott T. Chapman Trinity University San Antonio, Texas, U.S.A.

Boca Raton London New York Singapore

Published in 2005 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2327-9 (Hardcover) International Standard Book Number-13: 978-0-8247-2327-9 (Hardcover) Library of Congress Card Number 2004061830 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Arithmetic properties of commutative rings and monoids / edited by Scott T. Chapman. p. cm. — (Lecture notes in pure and applied mathematics v. 240) Includes bibliographical references and index. ISBN 0-8247-2327-9 (alk. paper) 1. Commutative rings—Congresses. 2. Monoids—Congresses. I. Chapman, Scott T. II. Series. QA251.3.A72 2005 512'.44--dc22

2004061830

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Preface This volume contains the Proceedings of two related meetings which took place in October 2003. The ﬁrst event was a one day Mini-Conference on Factorization Properties of Commutative Rings and Monoids hosted by the Department of Mathematics at the University of North Carolina at Chapel Hill. The Mini-Conference was followed by a regular Special Session on Commutative Rings and Monoids at the Fall 2003 Southeast Sectional Meeting of the American Mathematical Society. A major focus of these two meetings was the study of properties related to nonunique factorizations of elements into irreducible elements in commutative rings and monoids. Such problems have emerged as an independent area of research only over the past 30 years. Almost every result obtained in this area since 1970 can be traced to the following Theorem from a short two page paper by Leonard Carlitz (Proc. Amer. Math. Soc. 11(1960), 391–392). Theorem: The algebraic number ﬁeld Z has class number ≤ 2 if and only if for every nonzero integer α ∈ Z the number of primes πj in every factorization α = π1 π2 · · · πk only depends on α. While Carlitz’s result has been referenced well over 50 times, it did not initially gain much attention. None the less, according to Math Science Net, since the early 1970’s well over 200 papers dedicated to the study of factorization properties of integral domains and monoids, and related topics, have appeared in refereed mathematical journals or conference proceedings. During this same period, over 14 Doctoral Dissertations which touch on factorization properties have been completed. The institutions where such degrees were awarded include The University of Iowa, The University of Tennessee at Knoxville, The University of Nebraska at Lincoln, The University of North Carolina at Chapel Hill, North Dakota State University, Karl-Franzens-Universit¨at-Graz and Universite D’Aix-Marseille III. That this has become a highly active area of mathematical research is demonstrated by Marcel Dekker’s publication in 1997 of the monograph Factorization in Integral Domains, edited by D.D. Anderson. This monograph contains the Proceedings of a 1996 Mini-Conference held at the University of Iowa on factorization problems followed by a Special Session on Commutative Algebra at the Midwestern Regional Meeting of the American Mathematical Society. The meetings in Iowa were followed by a ﬂurry of research activity and the 2003 Meetings in Chapel Hill xi

xii

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were organized with the intent of allowing some of the leading researchers in this ﬁeld to review the major results of this period. I invited D. D. Anderson (University of Iowa), D. F. Anderson (University of Tennessee at Knoxville), Jim Coykendall (North Dakota State University), Alfred Geroldinger (Karl-Franzens-Universit¨at, Graz), Franz Halter-Koch (Karl-FranzensUniversit¨ at, Graz) and Ulrich Krause (Universit¨at Bremen) to deliver 45 minute addresses at the Mini-Conference. These authors have produced a series of 7 papers based on these lectures which make up the introductory chapters of this volume. These chapters demonstrate the diverse approach that many authors have taken in studying nonunique factorizations. Several are written in a purely monoidal setting, while several others demonstrate that ring theoretic properties play a key role in how elements factor. Moreover, these chapters exhibit the broad range of mathematical techniques necessary to obtain results in this area. Such techniques can range from purely algebraic and combinatorial, to number theoretic. It is hoped that these chapters not only oﬀer mathematicians new to this area the opportunity to survey its current trends and major results, but also oﬀer an up to date introduction to factorization theory for beginning graduate students. The remaining chapters in this volume reﬂect research which is motivated by arithmetical properties of commutative rings and monoids. This is not restricted to factorization problems, as aspects of the following topics are all considered: multiplicative ideal theory, the factorization of ideals and ideal generation problems, integer-valued polynomials, Pr¨ ufer rings and Domains, block monoids and their combinatorial invariants, and numerical monoids. The two sessions in Chapel Hill and subsequent Proceedings Volume have generated a signiﬁcant amount of interest. The sessions attracted 30 speakers from 6 diﬀerent countries. This volume features 26 papers from 33 diﬀerent contributors from 7 diﬀerent countries. I wish to take this opportunity to thank all those who participated in the sessions, and to all those who submitted papers for consideration to this volume. I am indebted to a large number of referees, who as always will remain nameless. A special thanks goes to the Department of Mathematics at the University of North Carolina at Chapel Hill, and in particular to Professor William W. Smith who was instrumental in planning and organizing this event. I also wish to thank the Department of Mathematics at Trinity University for their continued support and help in preparing this manuscript. The organization and preparation of the ﬁnished work would not have been possible without the help and patience of Denise Wilson, our Departmental Secretary and technical typist. I am of course indebted to Maria Allegra and her staﬀ at Marcel Dekker for giving me the support and opportunity to complete this manuscript. I hope the readers of this volume enjoy its contents as much as I enjoyed putting it together. Scott Chapman San Antonio, Texas January 14, 2005

Contents Preface

xi

Contributors

xv

1 Non-Atomic Unique Factorization in Integral Domains Daniel D. Anderson

1

2 Divisibility Properties in Graded Integral Domains David F. Anderson

22

3 Extensions of Half-Factorial Domains: A Survey Jim Coykendall

46

4 C-Monoids and Congruence Monoids in Krull Domains Franz Halter-Koch

71

5 Monotone Chains of Factorizations in C-Monoids Andreas Foroutan and Alfred Geroldinger

99

6 Transfer Principles in the Theory of Non-unique Factorizations Alfred Geroldinger and Franz Halter-Koch

114

7 Cale Monoids, Cale Domains, and Cale Varieties Scott T. Chapman and Ulrich Krause

142

8 Weakly Krull Inside Factorial Domains Daniel D. Anderson, Muhammed Zafrullah, and Gyu Whan Chang

172

9 The m-Complement of a Multiplicative Set David F. Anderson and Gyu Whan Chang

180

10 Some Remarks on Infinite Products Jim Coykendall

188

11 Rings with Prime Nilradical Ayman Badawi and Thomas G. Lucas

198

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12 On the Ideal Generated by the Values of a Polynomial Jean-Luc Chabert and Sabine Evrard

213

13 Using Factorizations to Prove a Partition Identity David E. Dobbs and Timothy P. Kilbourn

226

14 On Inside Factorial Integral Domains David E. Dobbs, Gabriel Picavet, and Martine Picavet-L’Hermitte

233

15 Polynomial Separation of Points in Algebras Sophie Frisch

253

16 k-Factorized Elements in Telescopic Numerical Semigroups Jose C. Rosales, Pedro A. Garc´ıa-S´ anchez, and Juan I. Garc´ıa-Garc´ıa

260

17 Pr¨ ufer Conditions in Rings with Zero-Divisors Sarah Glaz

272

18 Unmixedness and the Generalized Principal Ideal Theorem Tracy Dawn Hamilton

282

19 A Note on Sets of Lengths of Powers of Elements of Finitely Generated Monoids Wolfgang Hassler

293

20 UMV-Domains Evan Houston and Muhammad Zafrullah

304

21 On Local Half-Factorial Orders Florian Kainrath

316

22 On Factorization in Krull Domains with Divisor Class Group Z2k Karl M. Kattchee

325

23 Integral Morphisms Jack Maney

337

24 A Special Type of Invertible Ideal Stephen McAdam and Richard G. Swan

356

25 Factorization into Radical Ideals Bruce Olberding

363

26 Strongly Primary Ideals Gyu Whan Chang, Hoyoung Nam, and Jeanam Park

378

Index

389

Contributors Daniel D. Anderson, Department of Mathematics, University of Iowa, Iowa City, Iowa, 52242, [email protected] David F. Anderson, Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, [email protected] Ayman Badawi, Department of Mathematics and Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates, [email protected] Jean-Luc Chabert, Laboratoire Ami´enois de Math´ematiques Fondamentale et Appliqu´ee, CNRS-UMR 6140, Universit´e de Picardie, 33 rue Saint Leu, 80039 Amiens, France, [email protected] Gyu Whan Chang, Department of Mathematics, University of Incheon, Incheon, 402-748 Korea, [email protected] Scott T. Chapman, Department of Mathematics, Trinity University, One Trinity Place, San Antonio, Texas 78212-7200, [email protected] Jim Coykendall, Department of Mathematics, North Dakota State University,Fargo, North Dakota 58105-5075, [email protected] David E. Dobbs, Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, [email protected] Sabine Evrard, Laboratoire Ami´enois de Math´ematiques Fondamentale et Appliqu´ee, CNRS-UMR 6140, Universit´e de Picardie, 33 rue Saint Leu, 80039 Amiens, France, [email protected] Andreas Foroutan, Institut f¨ ur Mathematik, Karl-Franzens-Universit¨at, Heinrichstrasse 36, 8010 Graz, Austria, a− [email protected] Sophie Frisch, Institut f¨ ur Mathematik, Technische Universit¨at Graz, A-8010 Graz, Austria, [email protected] ´ Pedro A. Garc´ıa-S´anchez, Departamento de Algebra, Universidad de Granada, Facultad de Ciencias, Campus Fuentenueva s/n, 18071 Granada, Spain, [email protected] xv

xvi

Contributors

´ Juan I. Garc´ıa-Garc´ıa, Departamento de Algebra, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain, [email protected] Alfred Geroldinger, Institut f¨ ur Mathematik, Karl-Franzens-Universit¨at, Heinrichstrasse 36, 8010 Graz, Austria, [email protected] Sarah Glaz, Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269, [email protected] Franz Halter-Koch, Institut f¨ ur Mathematik, Karl-Franzens-Universit¨at, Heinrichstrasse 36, 8010 Graz, Austria, [email protected] Tracy Dawn Hamilton, Department of Mathematics and Statistics, California State University Sacramento, 6000 J Street, Sacramento, California 95819, [email protected] Wolfgang Hassler, Institut f¨ ur Mathematik, Karl-Franzens Universit¨ at Graz, Heinrichstraße 36/IV, A-8010 Graz, Austria, [email protected] Evan Houston, University of North Carolina at Charlotte, Department of Mathematics, Charlotte, North Carolina 28223-0001, [email protected] Florian Kainrath, Institut f¨ ur Mathematik, Karl-Franzens-Universit¨at Graz, Heinrichstraße 36, A-8010 Graz, Austria, ﬂ[email protected] Karl M. Kattchee, Mathematics Department, University of Wisconsin-La Crosse, 1725 State Street, La Crosse, Wisconsin 54601, [email protected] Ulrich Krause, Fachbereich Mathematik/Informatik, Universit¨at Bremen, 28334 Bremen, Germany, [email protected] Timothy P. Kilbourn, Department of Mathematics, University of Illinois at UrbanaChampaign, Urbana, Illinois 61801-2975, [email protected] Thomas G. Lucas, Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, North Carolina 28223, [email protected] Jack Maney, Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105-5075, [email protected] Stephen McAdam, Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712-0257, [email protected] Hoyoung Nam, Department of Mathematics, Inha University, Incheon, 402-751, Korea Bruce Olberding, Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001, [email protected]

Contributors

xvii

Jeanam Park, Department of Mathematics, Inha University, Incheon, 402-751, Korea Gabriel Picavet, Laboratoire de Math´ematiques Pures, Universit´e Blaise Pascal, 63177 Aubi`ere Cedex, France, [email protected] Martine Picavet-L’Hermitte, Laboratoire de Math´ematiques Pures, Universit´e Blaise Pascal, 63177 Aubi`ere Cedex, France, [email protected] ´ Jose C. Rosales, Departamento de Algebra, Universidad de Granada, Facultad de Ciencias, Campus Fuentenueva s/n, 18071 Granada, Spain, [email protected] Richard G. Swan, Department of Mathematics, The University of Chicago, Chicago, Illinois 60637, [email protected] Muhammed Zafrullah, Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085, [email protected]

Chapter 1

Non-Atomic Unique Factorization in Integral Domains by

Daniel D. Anderson Abstract UFDs can be characterized by the property that every nonzero nonunit is a product of principal prime elements or equivalently that every nonzero nonunit x can be written in the form x = upa1 1 · · · pann where u is a unit, p1 , . . . , pn are nonassociate principal primes, and each ai ≥ 1. Each pai i , in addition to being a power of a prime, has a number of other properties, each of which is subject to generalization. We survey various generalizations of (unique) factorization into prime powers in integral domains.

1

Introduction

Unique factorization domains are of course the integral domains in which every nonzero nonunit element has a unique factorization (up to order and associates) into irreducible elements or atoms. Now UFDs can also be characterized by the property that every nonzero nonunit is a product of principal primes or equivalently that every nonzero nonunit has the form upa1 1 · · · pann where u is a unit, p1 , . . . , pn are nonassociate principal primes, and each ai ≥ 1. Each of the pai i , in addition to being a power of a prime, has other properties, each of which is subject to generalization. For example, each pai i is primary, each is contained in a unique maximal t-ideal, and the pai i are pairwise coprime. The goal of this chapter is to survey various generalizations of (unique) factorization into prime powers in integral domains. This follows the thesis of M. Zafrullah that the pai i are the building blocks in a UFD. The author would like to thank M. Zafrullah for a number of discussions of these topics over the past several years. 1

2

Anderson

This chapter consists of four sections besides the introduction. Section 2 covers integral domains whose elements are products of primary elements, Section 3 relates locally ﬁnite intersections of localizations to factorizations, Section 4 surveys factorizations into pairwise comaximal elements, and Section 5 summarizes the various generalizations of prime powers and the integral domains whose elements are products of these various generalizations. A fairly complete list of references is also given. The next four paragraphs outline in more detail these four sections. Then a brief review of some of the terms and notation used throughout the chapter is given. In particular, star-operations, the t-operation, the group of divisibility, and splitting sets are discussed. In Section 2 we consider weakly factorial domains, integral domains with the property that every nonzero nonunit is a product of primary elements. While factorization into primary elements need not be unique, it is unique (up to order and associates) once we combine primaries with the same radical, that is, it is unique when the primary elements involved are pairwise v-coprime. Now UFDs are also characterized as being the Krull domains D with trivial divisor class group Cl(D) or the integral domains D whose group of divisibility G(D) is a cardinal sum of copies of Z (with the usual order). In a similar manner, an integral domain D is weakly factorial if and only if D is weakly Krull, that is, D = ∩P ∈X (1) (D) DP (here X (1) (D) is the set of height-one primes of D) where the intersection is locally ﬁnite and D has t-class group Clt (D) = 0 or if and only if the natural map G(D) → ⊕P ∈X (1) (D) G(DP ) is an order isomorphism (or equivalently, is just surjective). Also, D is weakly factorial if and only if every saturated multiplicatively closed subset of D is a splitting set. In the case of a weakly factorial domain D, the (unique) factorization of an element into v-coprime primary elements is given by the intersection D = ∩P ∈X (1) (D) DP in the following sense. Let x be a nonzero nonunit of D that is contained in the height-one primes P1 , . . . , Pn . Then each xDPi ∩ D = (xi ) is a principal Pi primary ideal and (x) = ∩P ∈X (1) (D) xDP = (x1 ) ∩ · · · ∩ (xn ) = (x1 ) · · · (xn ) where (x1 ) ∩ · · · ∩ (xn ) is also the reduced primary decomposition for (x). In Section 3 we consider factorizations induced by other locally ﬁnite intersections of localizations D = ∩P ∈S DP . For example, call a nonzero nonunit element t-pure if it is contained in a unique maximal t-ideal (as is the case for a nonzero primary element). Then every nonzero nonunit of D is a product of t-pure elements if and only if the intersection D = ∩P ∈t-Max(D) DP (where t-M ax(D) is the set of maximal t-ideals of D) is locally ﬁnite, is independent (distinct maximal t-ideals contain no common nonzero prime ideal) and Clt (D) = 0. Again, this factorization into t-pure elements is unique once elements contained in the same maximal t-ideal are combined, or equivalently, when the elements in the product are pairwise v-coprime. We also consider factorization into homogeneous elements and into rigid elements. In Section 4 we consider the comaximal factorizations recently introduced by McAdam and Swan [27]. They deﬁned a nonzero nonunit element d of an integral domain D to be pseudo-irreducible (pseudo-prime) if d = ab (d|ab) with a and b comaximal implies that a or b is a unit (d|a or d|b). A factorization d = d1 · · · dn is a (complete) comaximal factorization if each di is a nonzero nonunit (pseudoirreducible) and the di ’s are pairwise comaximal. The integral domain D is a comaximal factorization domain (CFD) if each nonzero nonunit has a complete comaximal factorization and a unique comaximal factorization domain (UCFD) is

Non-Atomic Unique Factorization in Integral Domains

3

a CFD in which complete comaximal factorizations of an element are unique up to order and associates. The main result is that for a CFD D the following are equivalent: (1) every two-generated invertible ideal of D is principal, (2) every nonzero nonunit of D has a comaximal factorization into pseudo-prime elements, and (3) D is a UCFD. We give a star-operation generalization, and relate these comaximal factorizations to the factorizations given in Sections 2 and 3. It is interesting to note that a one-dimensional domain is a UCFD if and only if it is weakly factorial. In Section 5 we summarize the various prime power generalizations given in Sections 2–4 and give several additional ones. Then the integral domains whose nonzero nonunits are products of these prime power generalizations are considered. Diagrams are given to show the various implications among the prime power generalizations and the integral domains deﬁned. We next review some of the terms and notation used in this chapter. Throughout D is an integral domain with quotient ﬁeld K. Let F (D) denote the set of nonzero fractional ideals of D and let f (D) be the set of ﬁnitely generated members of F (D). A star-operation ∗ on D is a closure operation on F (D) that further satisﬁes D∗ = D and (xA)∗ = xA∗ for all 0 = x ∈ K and A ∈ F (D). Here A ∈ F (D) is called a ∗-ideal if A∗ = A and A has ﬁnite type if A∗ = B ∗ for some B ∈ f (D). The star-operation ∗ has ﬁnite character if for each A ∈ F (D), A∗ = ∪{B ∗ |B ⊆ A with B ∈ f (D)}. Suppose that ∗ has ﬁnite character. Then every proper ∗-ideal is contained in a maximal ∗-ideal and a maximal ∗-ideal is prime. A prime ideal minimal over a ∗-ideal is a ∗-ideal. Hence a prime ideal minimal over a principal ideal is a ∗-ideal. We use ∗-Max(D) to denote the set of maximal ∗-ideals. Here D = ∩P ∈∗-Max(D) DP . With any star-operation ∗ is associated the ﬁnite character star-operation ∗s where A∗s = ∪{B ∗ |B ⊆ A where B ∈ f (D)}. Evidently ∗ has ﬁnite character precisely when ∗ = ∗s . Examples of star-operations include (1) the d-operation Ad = A, (2) the voperation Av = (A−1 )−1 = ∩{Dx|Dx ⊇ A where 0 = x ∈ K}, (3) the t-operation At = Avs = ∪{Bv |B ⊆ A with B ∈ f (D)}, and (4) for a set {Dα } of overrings of D with D = ∩Dα , the star-operation induced by ∩Dα A∗ = ∩ADα . Here (1) and (3) have ﬁnite character while (2) generally does not. An intersection D = ∩Dα has ﬁnite character or is locally ﬁnite if each 0 = x ∈ D is a unit in almost all Dα . If the intersection D = ∩Dα is locally ﬁnite, then the induced star-operation has ﬁnite character, but not conversely [1, Theorem 1]. An ideal A ∈ F (D) is ∗-invertible if there is a B ∈ F (D) with (AB)∗ = D. In this case we can take B = A−1 . The set Inv∗ (D) of ∗-invertible ∗-ideals forms a group under the ∗-product A ∗ B = (AB)∗ . The ∗-class group Cl∗ (D) of D is Inv∗ (D)/ Princ(D) where Princ(D) is the subgroup of nonzero principal fractional ideals. Of particular importance is the t-class group Clt (D) (or just the class group) of D. For D a Krull domain Clt (D) is the usual divisor class group while for a Pr¨ ufer domain D or one-dimensional domain D Clt (D) is the Picard group Pic(D) = Inv(D)/ Princ(D) = Cld (D). If ∗ is a ﬁnite character star-operation, then a ∗-invertible ideal has ﬁnite type, in fact A ∈ F (D) is ∗-invertible if and only if A has ﬁnite type and AP is principal for each P ∈ ∗-Max(D). Two ideals A and B of D are ∗-comaximal if (A, B)∗ = D. If A and B are ∗-comaximal, then it is easily proved that A∗ ∩ B ∗ = (A ∩ B)∗ = (AB)∗ . In the case where ∗ = d, we just say comaximal. Two elements a, b ∈ D are ∗-coprime if

4

Anderson

(a, b)∗ = ((a), (b))∗ = D and a and b are coprime if [a, b] = 1 (here [a, b] denotes the GCD of a and b). Thus a, b ∈ D are v-coprime if (a, b)v = D, or equivalently, a and b are not contained in any maximal t-ideal. Of particular interest are star-operations induced by intersections of localizations D = ∩P ∈S DP : A → A∗S = ∩P ∈S ADP . For any star-operation ∗ we have the associated ﬁnite character star-operation ∗w deﬁned by A∗w = {x ∈ K|xI ⊆ A for some I ∈ f (D) with I ∗ = D} = ∩P ∈∗s -Max(D) ADP . Here ∗s -Max(D) = ∗w -Max(D), A is ∗s -invertible if and only if A is ∗w -invertible and Cl∗s (D) = Cl∗w (D). See [7] for details. General references for star-operations include [21], [23], and [24]. For results on ∗-invertibility, class groups, and t-ideals see [18] and [35]. For star-operations induced by overrings, see [1] and [7]. For an integral domain D with quotient ﬁeld K let K ∗ = K − {0} and U (D) be the group of units of D. Then the multiplicative group G(D) = K ∗ /U (D) is called the group of divisibility of D. Now G(D) is partially ordered by aU (D) ≤ bU (D) ⇐⇒ a|b in D ⇐⇒ Da ⊇ Db. Note that the map G(D) → Princ(D) given by aU (D) → Da is an order isomorphism where Princ(D) is ordered by reverse inclusion. Given a family {(Gλ , ≤λ˙ )} of partially ordered abelian groups, the cardinal product is ΠGλ with the order (aλ ) ≤ (bλ ) ⇐⇒ each aλ ≤ bλ . The cardinal sum is deﬁned in a similar manner. Suppose that D = ∩P ∈S DP . Then we have an order preserving monomorphism ϕ: G(D) → ΠP ∈S G(DP ) (cardinal product) given by ϕ(xU (D)) = (xU (DP )). Note that if D = ∩P ∈S DP has ﬁnite character, then im ϕ ⊆ ⊕P ∈S G(DP ). A subgroup H of a partially ordered abelian group (G, ≤) is convex if whenever 0 ≤ a ≤ h with h ∈ H, then a ∈ H and H is directed (or ﬁltered ) if given a, b ∈ H, there exists c ∈ H with a ≤ c and b ≤ c or equivalently H+ = {h ∈ H|h ≥ 0} generates H as a group. Let S be a saturated multiplicatively closed subset of D. Then < S >= {s1 s−1 2 U (D)|s1 , s2 ∈ S} is a convex directed subgroup of G(D). The converse is also true, see [28]. A convex directed subgroup H of G(D) is a cardinal summand of G(D) if there is a convex directed subgroup K of G(D) with G(D) = H ⊕ K where H ⊕ K is the cardinal sum of H and K. A saturated multiplicatively closed subset S of D is a splitting set if every nonzero element x of D can be written in the form x = st where s ∈ S and t is v-coprime to each element of S. It is not hard to show that a saturated multiplicatively closed set S is a splitting set if and only if < S > is a cardinal summand of G(D) ([29, Proposition 4.1] and [4, Theorem 22]). Perhaps the main example of a splitting set is a multiplicatively closed subset generated by a set {pα } of height-one principal primes satisfying pαn D = 0 for each countable subcollection {pαn } of {pα }. Splitting sets were introduced in [29] and studied in [4] and [17]. There are of course many other generalizations of UFDs. A UFD is characterized by the property that every nonzero prime ideal contains a nonzero principal prime (ideal). Thus for each of the generalizations of a prime power, we can ask what domains have the property that each nonzero prime ideal contains such an element. While we will not pursue that theme here, the reader is referred to [14] for various “Kaplansky-like” theorems. Recall that an integral domain D is atomic if every nonzero nonunit of D is a product of irreducible elements or atoms. The theme of studying integral domains with various good atomic factorization properties is put

Non-Atomic Unique Factorization in Integral Domains

5

forth in [3]. For a survey of extensions of unique factorization to integral domains, rings with zerodivisors, rings without identity, or to modules, the reader is referred to [2]. In fact, this chapter is an elaboration of Section 2 of that chapter. Portions of this chapter were presented at a mini-conference on factorization organized by Professor Scott Chapman held October 23, 2003 preceding the regional American Mathematical Society meeting at the University of North Carolina.

2

Weakly Factorial Domains

Throughout D will denote an integral domain with quotient ﬁeld K. Recall that D is weakly factorial if every (nonzero) nonunit of D is a product of primary elements or equivalently every proper principal ideal of D is a product of principal primary ideals. Weakly factorial domains were introduced by D. D. Anderson and L. A. Mahaney [9] and further studied by Anderson and Zafrullah [13]. We begin by looking at factorizations into primary elements in an integral domain. Suppose that (q) is a nonzero P -primary ideal of D. Then P is the unique prime t-ideal containing (q) [5, Lemma 1]. Thus nonzero primary elements with distinct radicals are v-coprime. If (q1 ) and (q2 ) are P -primary, then so is (q1 ) (q2 ) = (q1 q2 ) [9, Corollary 2]. Thus if a nonzero nonunit x ∈ D is a product of primary elements, then by combining primary elements with the same radical, we can write x = q1 · · · qn where (qi ) is Pi -primary and the Pi ’s are distinct, or equivalently, the qi ’s are pairwise v-coprime. In this case, (x) = (q1 ) · · · (qn ) = (q1 ) ∩ · · · ∩ (qn ) where (q1 ) ∩ · · · ∩ (qn ) is the unique normal decomposition for (x) [9, Corollary 5]. It follows that the factorization of an element into a product of pairwise v-coprime primary elements is unique up to order and associates. We next give a number of conditions equivalent to D being weakly factorial and then sketch the proof of several of the implications. Theorem 2.1. For an integral domain D the following conditions are equivalent. (1 ) D is weakly factorial. (2 ) Every nonzero nonunit of D has a unique factorization (up to order and associates) into a product of pairwise v-coprime primary elements. (3 ) D = ∩P ∈X (1) (D) DP is locally ﬁnite and Clt (D) = 0. (4 ) For every nonzero nonunit x ∈ D, if P is a prime ideal minimal over (x), then ht P = 1 and xDP ∩ D is principal. (5 ) D = ∩P ∈X (1) (D) DP is locally ﬁnite and the natural map G(D) → ⊕P ∈X (1) (D) G(DP ) is surjective (and hence an order isomorphism). (6 ) Every convex directed subgroup of G(D) is a cardinal summand of G(D). (7 ) For (nonzero) prime ideal P of D, D − P = every U(D)|s is a cardinal summand of G(D). s1 s−1 1 , s2 ∈ D − P 2 (8 ) Every saturated multiplicatively closed subset of D is a splitting set. (9 ) For every (nonzero) prime ideal P of D, D − P is a splitting set. Proof. We have already remarked the equivalence of (1) and (2). The equivalence of (1) and (3)-(7) is [13, convex directed subgroup of G(D) Theorem]. Since every U (D) |s , s ∈ S for some saturated multiplicatively is of the form S = s1 s−1 1 2 2

6

Anderson

closed subset S of D and conversely and since S is a cardinal summand of G(D) if and only if S is a splitting set, we get the equivalence of (6) and (8) and of (7) and (9).

We next sketch the proof of several of the implications of the previous theorem. Suppose that D is weakly factorial. Since each nonzero prime ideal contains a nonzero primary element, each nonzero prime ideal contains the radical of a nonzero principal primary ideal which is a maximal t-ideal. Hence X (1) (D) = t-Max (D). As we always have D = ∩P ∈t-Max(D) DP , we have D = ∩P ∈X (1) (D) DP . Moreover, each proper principal ideal having a primary decomposition involving only heightone primes gives that each nonzero principal ideal is contained in only ﬁnitely many height-one primes and hence the intersection is locally ﬁnite. (In fact, D = ∩P ∈X (1) (D) DP with the intersection being locally ﬁnite if and only if every nonzero principal ideal has a primary decomposition involving only height-one primes [9, Theorem 13].) So D is weakly Krull. Let A be an integral t-invertible ideal of D. Let P1 , · · · , Pn be the maximal t-ideals (=height-one primes) containing A. Now APi is principal, so APi = qi DPi where qi D is Pi -primary. Then A ⊆ (AP1 ∩ D) ∩ · · · ∩ (APn ∩ D) = q1 D ∩ · · · ∩ qn D = q1 · · · qn D. So A = q1 · · · qn A where A is contained in no height-one prime. Hence At = D, so At = q1 · · · qn D is principal. Thus Clt (D) = 0. Conversely, suppose that D = ∩P ∈X (1) (D) DP where the intersection is locally ﬁnite and that Clt (D) = 0. Let x be a nonzero nonunit of D. Then xD = (xDP1 ∩ D) ∩ · · · ∩ (xDPn ∩ D) where P1 , · · · , Pn are the height-one primes containing xD. But each xDPi ∩D is Pi -primary and xDP1 ∩D, · · · , xDPn ∩D are v-coprime. Hence xD = (xDP1 ∩ D) ∩ · · · ∩ (xDPn ∩ D) = ((xDP1 ∩ D) · · · (xDPn ∩ D))t ; so each xDPi ∩ D is t-invertible. Hence Clt (D) = 0 gives (xDPi ∩ D)t = qi D where qi D is Pi -primary. So xD = q1 D · · · qn D. Thus D is weakly factorial. We next show that (1) and (4) are equivalent. Suppose that D is weakly factorial. If P is minimal over (x), then ht P = 1 and (x)P ∩D is principal being the P -primary component in the reduced primary decomposition of (x). Conversely, suppose that (4) holds. Let S be the multiplicatively closed subset of D consisting of all products of primary elements and units. Then S is saturated. Let y be a nonunit factor of an element x ∈ S. Then (x) being a product of principal primary ideals has only ﬁnitely many minimal primes each of which by hypothesis has height-one. Since y is a factor of x, it too is contained in only ﬁnitely many height-one primes P1 , · · · , Pn . Now y ∈ (y)P1 ∩ D = (q1 ) where (q1 ) is P1 -primary and y = q1 t1 where t1 ∈ P2 ∩ · · · ∩ Pn − P1 . Continuing in this manner we get y = q1 · · · qn tn where (qi ) is Pi -primary and tn is contained in no height-one prime, hence a unit. So y ∈ S. Suppose that some nonzero nonunit of D is not in S. Then (a) ∩ S = ∅ since S is saturated. Hence (a) can be enlarged to a prime ideal P ⊇ (a) maximal with respect to P ∩ S = ∅. Shrinking P to a prime ideal Q minimal over (a), we have a height-one prime ideal Q containing no primary elements. But this is absurd, for (a)Q ∩ D is itself a principal primary ideal contained in Q. We next give some examples of weakly factorial domains.

Non-Atomic Unique Factorization in Integral Domains

7

Example 2.2. (1) A one-dimensional quasilocal domain has every nonunit element primary and hence is weakly factorial. Conversely, a domain in which every nonunit is primary is either a ﬁeld or one-dimensional quasilocal. (2) [9, page 149] A one-dimensional domain D is weakly factorial if and only if each nonunit of D is contained in only ﬁnitely many maximal ideals and P ic(D) = 0. (3) [9, Theorem 21] Given any collection {Hλ } of rank-one totally ordered abelian groups, there is a weakly factorial Bezout domain D with G(D) order-isomorphic to the cardinal direct sum ⊕Hλ . (4) [13, Example] Let K ⊆ L be ﬁelds and D = K +XL[X] = {f (X) ∈ L[X]|f (0) ∈ K}. Then D is an atomic weakly factorial domain. It is easily seen that every nonzero nonunit of D may be written in one of the following two forms: p1 · · · pn or p1 · · · pn (aX r ) where p1 , · · · , pn are principal primes and aX r (r ≥ 1, a ∈ L) is primary. (5) [5, Lemma 3] An atomic domain in which almost all atoms are prime is weakly factorial. (6) [5, Corollary 2] An atomic domain is weakly factorial ⇐⇒ every atom is primary ⇐⇒ each atom is contained in a unique maximal t-ideal. While D[X] is a UFD whenever D is, this need not be the case for a weakly factorial domain. We have the following result. Recall that D is a generalized Krull domain if D is weakly Krull and for each P ∈ X (1) (D), DP is a valuation domain. Theorem 2.3. For an integral domain D the following conditions are equivalent. (1 ) D[X] is weakly factorial. (2 ) D is a weakly factorial GCD domain. (3 ) D is a weakly factorial generalized Krull domain. (4 ) D is a generalized Krull domain and a GCD domain. (5 ) D is weakly factorial and if p and q are non v-coprime primary elements then p|q or q|p. Proof. (1)⇐⇒(2) [9, Theorem 17]. The equivalence of (2)–(4) is [9, Theorem 20]. (2)⇐⇒(5) [9, Theorem 18] and its proof. It is interesting to observe that D[X] is an atomic weakly factorial domain if and only if D[X] and hence D is a UFD. This follows from the previous theorem and the well-known fact that an atomic GCD domain is a UFD.

3

Independent Locally-Finite Intersections of Localizations

In Section 2 we saw that factorization into primary elements was intimately connected to the locally ﬁnite representation D = ∩DP ∈X (1) (D) DP . Now of course an integral domain need not have such a representation. In this section we extend the results of Section 2 to more general intersections of localizations. Much of this comes from Anderson and Zafrullah [16]. We begin with some deﬁnitions given there.

8

Anderson

Let D be an integral domain with quotient ﬁeld K. A family F = {Pi }i∈I of nonzero prime ideals of D is called a deﬁning family of primes for D if D = ∩i∈I DPi . If, further, every nonzero nonunit of D belongs to at most ﬁnitely many members of F , F is of ﬁnite character, and if no two members of F contain a common nonzero prime ideal, F is independent. An integral domain is independent of ﬁnite character F (or an F -IFC domain) if it has a deﬁning family F of prime ideals that is independent and of ﬁnite character. Let us note that the h-local domains of Matlis [26] (where F = Max(D)), Noetherian domains whose grade-one primes are of height one, Krull domains, generalized Krull domains of Ribenboim [30], independent rings of Krull type [22], and weakly Krull domains [10] are all F -IFC domains with well-deﬁned family of primes. Let F be a deﬁning family of primes for D. Then F induces the star-operation ∗F on D by A∗F = ∩P ∈F ADP for A ∈ F (D). Moreover, if F has ﬁnite character, so does ∗F [1, Theorem 1]. We call an integral ideal A of D unidirectional if A is contained in a unique member of F . The following four theorems come from [16]. Theorem 3.1. ([16, Theorem 2.1] ) Let D be an integral domain and let F be a deﬁning family of primes for D. Then the following are equivalent. (1 ) D is an F -IFC domain. (2 ) For every nonzero nonunit x of D, (x) is a ∗F -product of ﬁnitely many unidirectional ∗F -ideals of D. (3 ) Every nonzero prime ideal of D contains a nonzero element x such that (x) is a ∗F -product of ﬁnitely many unidirectional ∗F -ideals. If P1 , · · · , Pn are the members of F containing (x) in (2), then the factorization alluded to is (x) = (xDP1 ∩ D) ∩ · · · ∩ (xDPn ∩ D) = ((xDP1 ∩ D) · · · (xDPn ∩ D))

∗F

.

Theorem 3.2. ([16, Theorem 3.3] ) Let F be a deﬁning family of mutually incomparable primes for D such that ∗F is of ﬁnite character. Then the following are equivalent. (1 ) F is independent of ﬁnite character. (2 ) Every nonzero prime ideal of D contains an element x such that (x) is a ∗F product of unidirectional ∗F -ideals. (3 ) Every nonzero prime ideal of D contains a unidirectional ∗F -invertible ∗F -ideal. (4 ) For P ∈ F and 0 = x ∈ P , xDP ∩ D is ∗F -invertible and unidirectional. (5 ) F is independent and for any nonzero ideal A of D, A∗F is of ﬁnite type whenever ADP is ﬁnitely generated for all P in F . Theorem 3.3. ([16, Corollary 3.5]) Let F be a deﬁning family for D consisting of incomparable primes. Then the following are equivalent. (1 ) F is independent of ﬁnite character and every ∗F -invertible ∗F -ideal is principal. (2 ) Every nonzero nonunit x of D may be written in the form x = x1 · · · xn where each xi generates a unidirectional ideal. (3 ) For each nonzero nonunit x in D and for each P in F containing x, xDP ∩ D is principal and unidirectional. (4 ) The natural map G(D) → ΠP ∈F G(DP ) has image ⊕P ∈F G(DP ).

Non-Atomic Unique Factorization in Integral Domains

9

The factorization in (2) becomes unique (up to order and associates) once the xi ’s belonging to the same unidirectional ideal are combined. If P1 , · · · , Pn are the primes of F containing x and if xDPi ∩ D = (xi ), then this factorization is x = ux1 · · · xn for some unit u. Now suppose that ∗ is a star-operation on D. Now D = ∩P ∈∗s -Max(D) DP , so F = ∗s -Max(D) is a deﬁning family for D. Here the star-operation ∗sF is given by A∗sF = ∩P ∈F ADP . But by [7], ∩P ∈F ADP = A∗w = {x ∈ K|xI ⊆ A for some I ∈ f (D) with I ∗ = D}. Also, ∗sF = ∗w has ﬁnite character, A is ∗sF -invertible if and only if A is ∗s -invertible and Cl∗sF (D) = Cl∗s (D). So we have the following result. Theorem 3.4. Let ∗ be a ﬁnite character star-operation on the integral domain D. Then the following conditions are equivalent. (1 ) ∗-Max(D) is independent of ﬁnite character and Cl∗ (D) = 0. (2 ) Every nonzero nonunit x of D may be written in form x = x1 · · · xn where each xi is contained in a unique member of ∗-Max(D). (3 ) Every nonzero nonunit x of D may be written uniquely (up to order and associates) in the form x = x1 · · · xn where each xi is contained in a unique Pi ∈ ∗Max(D) and for i = j, Pi = Pj , or equivalently, xi and xj are ∗-coprime. (4 ) For each nonzero nonunit x in D and for each P ∈ ∗-Max(D), xDP ∩ D is principal and unidirectional. (5 ) The natural map G(D) → ΠP ∈∗- Max(D) G(DP ) has image ⊕P ∈∗- Max(D) G(DP ) (and hence is an order isomorphism where ⊕G(DP ) has the cardinal order ). (6 ) ∗-Max(D) is independent, Cl∗ (D) = 0, and for each nonzero ideal A of D, A∗w is of ﬁnite type whenever ADP is ﬁnitely generated for all P ∈ ∗-Max(D). Of special interest is the case where ∗ is the t-operation. Here ∗w = tw is the w-operation introduced by Wang and McCasland [20]: Aw = {x ∈ K|xI ⊆ D for some ﬁnitely generated ideal I with I −1 = D}. Let P be a prime ideal of a domain D. Then an ideal A of D is P -pure if AP ∩ D = A (we usually assume A ⊆ P ) and A is t-pure if A is P -pure for some P ∈ t-Max(D) [10]. For A t-invertible (with At = D), At is t-pure if and only if A is contained in a unique maximal t-ideal [11, Lemma 2.1]. A nonzero nonunit x ∈ D is t-pure if (x) is t-pure, or equivalently, if x is contained in a unique maximal t-ideal. Following [11] we call an integral domain D t-pure if every nonzero nonunit of D is t-pure and we call D semi-t-pure if every nonzero nonunit of D is a ﬁnite product of t-pure elements. Evidently D is t-pure if and only if D is quasilocal with maximal ideal a t-ideal. Observe that if 0 = (q) is P -primary, then (q) is P -pure and t-pure and q is t-pure. We have the following t-version of Theorem 3.4. Theorem 3.5. For an integral domain D, the following conditions are equivalent. (1 ) t-Max(D) is independent of ﬁnite character and Clt (D) = 0. (2 ) D is semi-t-pure, i.e., every nonzero nonunit of D is a ﬁnite product of t-pure elements. (3 ) Every nonzero nonunit x of D may be written uniquely in the form x = x1 · · · xn where each xi is t-pure and for i = j, xi and xj are v-coprime. (4 ) For 0 = x ∈ D and x ∈ P ∈ t-Max(D), xDP ∩ D is principal and t-pure. (5 ) The natural map G(D) → ΠP ∈t- Max(D) G(DP ) has image ⊕P ∈t- Max(D) G(DP ) (and hence is an order ismorphism where ⊕G(DP ) has the cardinal order ).

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Anderson

(6 ) t-Max(D) is independent, Clt (D) = 0, and for each nonzero ideal A of D, Aw is of ﬁnite type whenever ADP is ﬁnitely generated for all P ∈ t-Max(D). (7 ) For each P ∈ t-Max(D), D − P is a splitting set, or equivalently, < D − P >= {s1 s−1 2 U (D)|s1 s2 ∈ D − P } is a cardinal summand of G(D). (8 ) For each nonempty subset F ⊆ t-Max(D), S = D − UP ∈F P is a splitting set, or equivalently < S > is a cardinal summand of G(D). (9 ) D is an F -IFC domain, for each Q ∈ F, DQ is t-pure, and Clt (D) = 0. Proof. The equivalence (1)–(6) follows from Theorem 3.4. The equivalence of (1), (7) and (8) follows from [16, Theorem 2.14]. And the equivalence of (1)–(3) and (8) is given in [11, Theorem 3.4]. Observe that Theorem 2.1 (D weakly factorial) is just the case X (1) (D) = tMax(D). We next give three more generalizations of prime powers that turn out to be equivalent to being t-pure in GCD domains. Let h be a nonzero nonunit of an integral domain D. Then h is rigid [31] (homogeneous, strongly homogeneous [11]) if for x, y ∈ D, x, y|h (with [x, y] = 1, (x, y)v = D) implies x|y or y|x (x or y is a unit). And D is (semi-) rigid if every nonzero nonunit of D is (a product of) rigid (elements). Similar deﬁnitions for (semi-) homogeneous domains and (semi-) strongly homogeneous domains may be given. It is easy to see that an atom or prime power is rigid, that a rigid element is strongly homogeneous, and that a strongly homogeneous or t-pure (and hence primary) element is homogeneous. Evidently an integral domain is rigid if and only if it is a valuation domain. Thus any one-dimensional quasilocal domain that is not a valuation domain will have a nonzero primary element that is not rigid. To study factorization in a nonatomic setting, P.M. Cohn [19] deﬁned a nonzero nonunit h ∈ D to be primal if h|xy implies h = h1 h2 where h1 |x and h2 |y and to be completely primal if each of its nonunit factors is primal. (See [15] for a survey of completely primal elements.) Thus an atom is (completely) primal if and only if it is prime. An integral domain in which every nonzero nonunit is (completely) primal is called a pre-Schreier domain [34] and an integrally closed pre-Schreier domain is a Schreier domain. A GCD domain is a Schreier domain [19]. We have the following result. Theorem 3.6. ([11, Theorem 2.3, Corollary 2.4]) Let h be a completely primal element of an integral domain D. Then the following are equivalent. (1 ) h is homogeneous. (2 ) h is strongly homogeneous. (3 ) h is t-pure. If further D is a GCD domain, we have (1)–(3) equivalent to (4 ) h is rigid. It is well-known that Clt (D) = 0 for a pre-Schreier domain D [34, Corollary 3.7]. Combining this with Theorem 3.5 and Theorem 3.6 we get our next result. Theorem 3.7. ([11, Corollary 3.7]) For an integral domain D, the following are equivalent. (1 ) D is a pre-Schreier semi-homogeneous integral domain.

Non-Atomic Unique Factorization in Integral Domains

11

(2 ) Every nonzero nonunit of D may be written uniquely as a product of mutually coprime completely primal homogeneous elements. (3 ) D is pre-Schreier and t-Max(D) is independent of ﬁnite character. (4 ) D is a F -IFC domain, for each Q ∈ F, DQ is a homogeneous pre-Schreier domain, and Clt (D) = 0. Combining Theorem 3.5 and the second part of Theorem 3.6 we get the following result which should be compared to Theorem 2.3. Recall that an integral domain D is an independent ring of Krull type if t-Max(D) is independent of ﬁnite character and DP is a valuation domain for each P ∈ t-Max(D). The implication (1)=⇒(3) was ﬁrst given in [31] while the implication (3)=⇒(1) was given in [32]. Theorem 3.8. ([11, Corollary 3.8]) For an integral domain D, the following conditions are equivalent. (1 ) D is a semi-rigid GCD domain. (2 ) D is a semi-t-pure GCD domain. (3 ) D is a GCD domain that is an independent ring of Krull type. (4 ) D is an independent ring of Krull type with Clt (D) = 0. We end this section with the following result. Recall that two ideals I and J of a ring are condensed if IJ = {ij|i ∈ I, j ∈ J}. Theorem 3.9. ([8, Theorem 2.1] ) Let D be an atomic domain. Then the following statements are equivalent. (1 ) Every pair of comaximal ideals of D is condensed. (2 ) Every pair of distinct maximal ideals of D is condensed. (3 ) Each atom of D is unidirectional. (4 ) D is h-local with Pic(D) = 0. (5 ) Every nonzero nonunit of D is a product of unidirectional elements. (6 ) For each nonzero nonunit x ∈ D and each maximal ideal M containing x, xDM ∩ D is a principal unidirectional ideal. As before, the factorization into unidirectional elements given in (5) is unique up to units and associates once elements contained in the same maximal ideal are combined. Note that (4)–(6) are still equivalent if we drop the hypothesis that D is atomic. Indeed these three statements are just three of the equivalences of Theorem 3.4 where ∗ is the d-operation.

4

Comaximal Factorizations and Generalizations

Recently McAdam and Swan [27] studied comaximal factorizations of elements and ideals in integral domains and rings. In this section we discuss their comaximal factorizations of elements in integral domains, give a generalization to star-operations, and compare it to results from Sections 2 and 3. We begin with some deﬁnitions from their chapter. Let D be an integral domain. A nonzero nonunit b of D is pseudo-irreducible (resp. pseudo-prime) if b = cd with c and d comaximal (i.e., (c, d) = D) implies c or d is a unit (resp. b|cd with c and d comaximal implies that b|c or b|d). Clearly b pseudo-prime implies that b is pseudo-irreducible. Note that b is pseudo-prime

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if and only if D/ (b) is indecomposable [27, Lemma 3.1]. For a nonzero nonunit b of D we call b = b1 · · · bm a (complete) comaximal factorization of b if the bi are pairwise comaximal nonunit (pseudo-irreducible) elements. Evidently a comaximal factorization b = b1 · · · bm is complete if and only if it has no proper reﬁnements that are also comaximal factorizations of b. Then D is a comaximal factorization domain (CFD) if every nonzero nonunit of D has a complete comaximal factorization and D is a unique comaximal factorization domain (UCFD) if D is a CFD in which complete comaximal factorizations are unique up to order and associates. They showed [27, Lemma 1.1] that an integral domain D is a CFD if either (i) each nonzero nonunit of D has only ﬁnitely many minimal primes or (ii) each nonzero nonunit of D is contained in only ﬁnitely many maximal ideals. Here (i) insures that a Noetherian domain is a CFD. To this list of CFDs we can add: D has a deﬁning family of primes of ﬁnite character. The proof is similar to the proof given for (i). A key concept in their work is the notion of an S-ideal. A nonzero ideal I of D is called an S-ideal if I = (a, c) = a2 , c for some a, c2 ∈ I. The relation to comaximal factorizations is the observation that (a, c) = a , c if and only if there is an element b ∈ D with (a, b) = D and c|ab [27, Lemma 1.2]. They proved that S-ideals are invertible and that any two-generated invertible ideal is isomorphic to an S-ideal [27, Lemma 1.5]. They gave the following characterization of UCFDs (we have added (3)). Theorem 4.1. ([27, Theorem 1.7] ) For an integral domain D the following conditions are equivalent. (1 ) D is a UCFD. (2 ) D is a CFD and every pseudo-irreducible element of D is pseudo-prime. (3 ) Every nonzero nonunit of D has a comaximal factorization into pseudo-prime elements. (4 ) Every two-generated invertible ideal of D is principal. (5 ) Every S-ideal of D is principal. It should be noted that in the previous theorem we can not add P ic(D) = 0 as the following example from Section 4 of [27] shows (but see Theorem 4.2 below). Let An be the subring of Bn = R[x0 , · · · , xn ]/ x20 + · · · + x2n − 1 , the ring of realvalued polynomial functions on the n-sphere S n , consisting of all even functions. Then for n ≥ 2, An is a regular domain that is a UCFD, but P ic(An ) = 0. Now a UFD D is a UCFD, indeed every nonzero nonunit of D is contained in only ﬁnitely many minimal prime ideals and P ic(D) = 0. For exactly the same reason, a weakly factorial domain is a UCFD. Note that the domains given by Theorem 3.4 for the case where ∗ is the d-operation are UCFDs. Indeed, these are precisely the UCFDs in which every pseudo-irreducible element is contained in a unique maximal ideal. In the case of a one-dimensional UFD (that is, a PID) a pseudo-prime element is just a primary element and a complete comaximal factorization is the same thing as a factorization into v-coprime prime powers or primary elements. Actually this result extends to any one-dimensional domain. Theorem 4.2. For a one-dimensional integral domain D the following are equivalent. (1 ) D is weakly factorial.

Non-Atomic Unique Factorization in Integral Domains

13

(2 ) Every nonzero nonunit of D is contained in only ﬁnitely many maximal ideals and P ic(D) = 0. (3 ) D is a UCFD. Proof. (1)⇐⇒(2) Example 2.2 (2). (2)=⇒(3) Since every nonzero nonunit is contained in only ﬁnitely many minimal primes, D is a CFD. By Theorem 4.1 D is a UCFD. (3)=⇒(1) Let d ∈ D be pseudo-prime. Hence D/ (d) is indecomposable and (d) is indecomposable. Since D/ (d) is zero-dimensional and reduced, thus D/ (d) is von Neumann regular and hence a ﬁeld since it is indecomposable. Thus D/ (d) is a maximal ideal and hence (d) is a primary ideal. Thus every nonzero nonunit of D is a product of pseudo-prime elements or equivalently a product of primary elements. So D is weakly factorial. Also, a quasilocal domain (D, M ) is a UCFD. Indeed, for 0 = a ∈ M , a = a is the unique comaximal factorization of a ! One should note that while a UFD is a UCFD, the complete comaximal factorization of an element need not be the prime-power factorization. Indeed in K[X, Y ], K a ﬁeld, XY is pseudo-irreducible. We next generalize comaximal factorizations to coprime and ∗-comaximal factorizations where ∗ is a ﬁnite character star operation on D. A nonzero nonunit a ∈ D is ∗-pseudo-irreducible (resp. ∗-pseudo-prime) if for b, c ∈ D with (b, c)∗ = D, a = bc (resp. a|bc) implies b or c is a unit (resp. a|b or a|c). Certainly a ∗-pseudoprime element is ∗-pseudo-irreducible. A factorization b = b1 · · · bm into nonunits is a (complete) ∗-comaximal factorization if for i = j (bi , bj )∗ = D (and each bi is ∗-pseudo-irreducible). Evidently b = b1 · · · bm is a complete ∗-comaximal factorization if and only if it has no proper reﬁnements that are ∗-comaximal factorizations. Finally, D is a ∗-CFD if each nonzero nonunit of D has a complete ∗-comaximal factorization and a ∗-CFD is a ∗-UCFD if each complete ∗-comaximal factorization is unique up to units and associates. Note that if we take ∗ to be the d-operation (Ad = A) we get the comaximal factorizations of McAdam and Swan. We can also deﬁne coprime factorizations by replacing (a, b)∗ = D by [a, b] = 1, but we must be careful (see next paragraph). Deﬁne a nonzero nonunit a ∈ D to be [ ] -pseudo-irreducible (resp. [ ]-pseudoprime) if a = a1 · · · an (resp. a|a1 · · · an ) where [ai , aj ] = 1 for i = j implies some ai is a unit (a|ai for some i). A (complete) coprime factorization is a factorization b = b1 · · · bm where each bi is a nonunit and [bi , bj ] = 1 for i = j (with each bi [ ]-pseudo-irreducible). Evidently a coprime factorization is complete if and only if it has no proper reﬁnement which is again a coprime factorization. The domain D is a [ ]-CFD if each nonzero nonunit has a complete coprime factorization and a [ ]-CFD is a [ ]-UCFD if the complete coprime factorization of an element is unique up to units and associates. Note that if a is ∗-pseudo-irreducible and a = a1 · · · an where (ai , aj )∗ = D for i = j, then some ai is a unit. Indeed, this follows from the fact that if (a, b)∗ = (a, c)∗ = D, then (a, bc)∗ = D. For we can write a = a1 (a2 · · · an ) where (a1 , a2 · · · an )∗ = D. However, [a, b] = [a, c] = 1 does not imply [a, bc] = 1. (In fact, a domain D satisﬁes this property (called the P P -property) if and only if D satisﬁes GL: the product of two primitive polynomials is primitive. And it is well-known that an atomic domain satisfying GL is a UFD. See [12] for details.) Consider the element X 3 of Q+XR[X]. Then X 3 is not [ ]-pseudo-irreducible since

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Anderson

√ 1 3 X 3 = X( 3 2X)( √ does not have a 3 X) is a coprime factorization. However, X 2 3 coprime factorization into two elements. For if X = ab, then we can take a = uX and b = u−1 X 2 where u ∈ R. But [uX, u−1X 2 ] = uX. Note that if ∗1 and ∗2 are ﬁnite character star-operations on D with ∗1 ≤ ∗2 (i.e., A∗1 ⊆ A∗2 for each A ∈ F (D)), then a ∈ D ∗2 -pseudo-irreducible implies a is ∗1 -pseudo-irreducible and that a [ ]-pseudo-irreducible implies that a is t-pseudoirreducible. Similar statements hold for the corresponding types of pseudo-prime elements. However, for comaximal factorizations we have the reverse situation. If ∗1 ≤ ∗2 , then a ∗1 -comaximal factorization is a ∗2 -comaximal factorization. Thus the relationship between ∗1 -UCFDs and ∗2 -UCFDs is somewhat murky. As in the case of CFDs, a domain D is a ∗-CFD if each nonzero nonunit is either (i) contained in only ﬁnitely many minimal primes or (ii) contained in only ﬁnitely many maximal ∗-ideals. Also, it is not hard to show that a domain satisfying ACCP is a ∗-CFD. We have the following partial analog of Theorem 4.1. Theorem 4.3. Let ∗ be a ﬁnite character star-operation on an integral domain D. Consider the following four conditions. (1 ) D is a ∗-CFD and each two-generated ∗-invertible ∗-ideal is principal. (2 ) D is a ∗-CFD and each ∗-pseudo-irreducible element is ∗-pseudo-prime. (3 ) Every nonzero nonunit of D has a ∗-comaximal factorization into ∗-pseudoprime elements. (4 ) D is a ∗-UCFD. Then (1 )=⇒(2 )⇐⇒(3 )=⇒(4 ). Proof. The proof of (1)=⇒(2)=⇒(4) is the star operation version of [27, Lemma 1.3]. We give the details for the readers not familiar with star-operations. (1)=⇒(2) Let a be ∗-pseudo-irreducible. Suppose that a|bc where (b, c)∗ = D. Then ((a, b), (a, c))∗ = D, so (a, b)∗ ∩ (a, c)∗ = ((a, b)(a, c))∗ . Hence (a) ⊆ (a, b)∗ ∩ (a, c)∗ = ((a, b)(a, c))∗ ⊆ (a); so (a) = ((a, b)(a, c))∗ . So (a, b) and (a, c) are ∗-invertible. Thus (a, b)∗ = (d) and (a, c)∗ = (e) for some d, e ∈ D. So (a) = (d)(e) and hence a = ude for some unit u. But then a ∗-pseudo-irreducible and (d, c)∗ = D gives ud or e is a unit. If ud is a unit, (a) = (e) and c ∈ (a, c)∗ = (e) = (a). Likewise e a unit gives c ∈ (a). So a is ∗-pseudo-prime. (2)=⇒(3) Clear. (3)=⇒(2) Let a be ∗-pseudo-irreducible. Then a has a ∗-comaximal factorization a = a1 · · · an where each ai is ∗-pseudo-prime. Since a is ∗-pseudo-irreducible, n = 1, that is, a = a1 is ∗-pseudo-prime. (2)=⇒(4) Suppose that x1 · · · xn = y1 · · · ym are two complete ∗-comaximal factorizations. By hypothesis, each xi , yj is ∗-pseudo-prime. Now x1 ∗-pseudo-prime and x1 |y1 · · · ym implies x1 |yi for some i which we can take to be 1. Likewise yi |xj for some j. Then x1 |xj and since (x1 , xj )∗ = D for j = 1, we must have j = 1. Thus x1 and y1 are associates. Cancelling x1 gives x2 · · · xn = (uy2 ) · · · ym for some unit u. By induction n = m and after re-ordering xi and yi are associates. Note that if D is a domain satisfying the equivalent conditions of Theorem 3.4, then D is a ∗-UCFD. Indeed, these are precisely the ∗-UCFDs in which each ∗-pseudo-irreducible element is contained in a unique maximal ∗-ideal. We do not know if (4)=⇒(1), even in the case of the t-operation. However, in the onedimensional case we have the following result.

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Theorem 4.4. For a one-dimensional domain D and a ﬁnite character staroperation ∗ on D the following are equivalent. (1 ) D is weakly factorial. (2 ) D is a ∗-UCFD. Proof. This follows from Theorem 4.2 and the fact that for D one-dimensional, ∗-Max(D) = Max(D). However, we can not add D is a [ ]-UCFD to Theorem 4.4. For let D = K + XL[[X]] where K ⊆ L are ﬁelds. Now D is always a [ ]-CFD, but D is a [ ]-UCFD if and only if K = L, i.e., D is a UFD. Now up to associates the nonunits (atoms) of D have the form uX n where u ∈ L∗ and n ≥ 1 (n = 1). Let u, v ∈ L∗ . Then uX and vX are associates ⇐⇒ uv −1 ∈ K and for n > 1, uX n = (vX)(uv −1 X n−1 ). Hence [uX n1 , vX n2 ] = 1 ⇐⇒ n1 = n2 = 1 and uv −1 ∈ / K. Let a be a nonzero nonunit of D. Then either a is [ ]-pseudo-irreducible or a = a1 · · · an where each ai is a nonunit and [ai , aj ] = 1 for i = j. In the latter case [ai , aj ] = 1 gives that each ai has order one and hence is irreducible and thus [ ]-pseudo-irreducible. Thus D is a [ ]-CFD. Suppose that K L and let u ∈ L − K. Then uX 2 = uX · X = u2 X · u−1 X = u(u + 1)X · (u + 1)−1 X. Now the ﬁrst two complete coprime factorizations are distinct unless u2 ∈ K while the ﬁrst and the third complete coprime factorizations are distinct unless u(u + 1) ∈ K. But then uX 2 having a unique complete coprime factorization gives that u = u(u + 1) − u2 ∈ K, a contradiction.

5

Generalizations of Prime Powers

In this section we summarize the various generalizations of prime powers given in Sections 2–4 and give several new ones (see Figure 1). For each of these prime power generalizations, we consider the domains whose nonzero nonunits are a product of such elements (see Figure 2). The pattern runs as follows. Suppose that D is an integral domain and that P is a property generalizing prime power. Then D could be called a P-domain if each nonzero nonunit of D has a property P and a semi-P-domain if each nonzero nonunit of D is a ﬁnite product of elements with property P. In several cases we have seen that D is a semi-P-domain if and only if D is an F -IFC domain where for each P ∈ F, DP is a P-domain. For example, this is the case where property P is “prime power”, “primary”, or “t-pure”. In [6] a nonzero nonunit q was deﬁned to be a prime quantum if q satisﬁes Q1 : For every nonunit r|q, there exists a natural number n with q|rn , Q2 : For every natural number n, if r|q n and s|q n , then r|s or s|r, and Q3 : For every natural number n, each element t with t|q has the property that if t|ab, then t = t1 t2 where t1 |a and t2 |b. Thus Q2 says that each power of q is rigid and Q3 says that q is completely primal (since a product of completly primal elements is completely primal). A prime power pn is a prime quantum and a prime quantum q is primary. An integral domain D is a generalized unique factorization domain (GUFD) if every nonzero nonunit of D is a product of prime quanta. Clearly a UFD is a GUFD and a GUFD is weakly factorial and a GCD domain [6]. M. Zafrullah [33] considered yet another generalization of a prime power. He deﬁned a nonzero nonunit x ∈ D to be a packet if (x) is prime, that is, there is a

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unique minimal prime over (x). He studied GCD domains, called unique representations domains (URD ), with the property that every nonzero nonunit is a product of packets. In a URD, each nonzero nonunit has only ﬁnitely many primes minimal over it. Thus by Theorem 4.1, a URD is a UCFD. For a study of domains with this and related properties, see [25]. The various generalizations of a prime element are indicated in Figure 1.

Non-Atomic Unique Factorization in Integral Domains

17

prime = irreducible B B

prime power

B BN rigid

Z Z Z ~ Z

prime quantum A

A AA U

homogeneous A

completely primal A

A AAU

strongly homogeneous A

-

primary Q Q Q s Q

primal

packet

t-pure

A A U A

t-pseudo-irreducible

-

Figure 1.

A AAU

pseudo-irreducible

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Anderson

Corresponding to Figure 1, we have Figure 2 showing the various generalizations of UFDs. Again, the only implications are the obvious ones.

atomic domain

? semi-rigid domain semi-strongly-homogeneous domain A

A AA U

semi-homogeneous domain

UFD

?

GCD domain : GUFD @ @ R @ A A Schreier domain AAU weakly factorial domain

?

pre-Schreier domain

semi-t-pure domain H HH HH HH j H ? t-UCFD

Figure 2.

UCFD

Bibliography [1] D.D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988), 2535–2553. [2] D.D. Anderson, Extensions of unique factorization: A survey, Advances in Commutative Ring Theory (Fez, 1997), 31-53, Lecture Notes in Pure and Appl. Math., 205, Dekker, New York, 1999. [3] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19. [4] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), 17–37. [5] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Atomic domains in which almost all atoms are prime, Comm. Algebra 20 (1992), 1447–1462. [6] D.D. Anderson, D.F. Anderson, and M. Zafrullah, A generalization of unique factorization, Boll. Un. Mat. Ital. A(7) 9 (1995), 401–413. [7] D.D. Anderson and S.J. Cook, Two star-operations and their induced lattices, Comm. Algebra 28 (2000), 2461–2475. [8] D.D. Anderson and T. Dumitrescu, Condensed domains, Canad. Math. Bull. 46 (2003), 3–13. [9] D. D. Anderson and L.A. Mahaney, On primary factorizations, J. Pure Appl. Algebra 54 (1988), 141–154. [10] D.D. Anderson, J. L. Mott, and M. Zafrullah, Finite character representations for integral domains, Boll. Un. Mat. Ital. B(7) 6 (1992), 613–630. [11] D.D. Anderson, J.L. Mott, and M. Zafrullah, Unique factorization in nonatomic integral domains, Boll. Unione Mat. Ital. Sez B Artic Ric. Mat. (8) 2 (1999), 341–352. [12] D.D. Anderson and R.O. Quinterro, Some generalizations of GCD-domains, Factorization in Integral Domains, 189–195, Lecture Notes in Pure and Appl. Math., 189, Dekker, New York, 1997. [13] D.D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), 907–913. 19

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[14] D.D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A(7) 8 (1994), 397–402. [15] D.D. Anderson and M. Zafrullah, P.M. Cohn’s completely primal elements, Zero Dimensional Commutative Rings, 115–123, Lecture Notes in Pure and Appl. Math., 171, Dekker, New York, 1995. [16] D.D. Anderson and M. Zafrullah, Independent locally ﬁnite intersections of localizations, Houston J. Math. 25 (1999), 433–452. [17] D.D. Anderson and M. Zafrullah, Splitting sets in integral domains, Proc. Amer. Math. Soc. 129 (2001), 2209–2217. [18] D.F. Anderson, The class group and local class group of an integral domain, 33-55, Non-Noetherian Commutative Ring Theory, Mathematics and Its Applications, vol. 520, Kluwer Academic Publishers, Dordrect/Boston/London, 2000. [19] P.M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 251–264. [20] Wang Fanggui and R.L. McCasland, On w-modules over strong Mori domains, Comm. Algebra 25 (1997), 1285–1306. [21] R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Appl. Math. 90, Queen’s University, Kingston, Ontario, 1992. [22] M. Griﬃn, Rings of Krull type, J. Reine Angew. Math. 229 (1968), 1–17. [23] F. Halter-Koch, Ideal Systems - An Introduction to Multiplicative Ideal Theory, Pure and Applied Mathematics, vol. 211, Dekker, New York, 1998. [24] P. Jaﬀard, Les Syst`emes d’Ideaux, Travaux et Recherches Math´ematiques, vol IV, Dunod, Paris, 1960. [25] P. Malcolmson and F. Okoh, Minimal prime ideals and generalizations of factorial domains, preprint. [26] E. Matlis, Torsion-free Modules, The University of Chicago Press, ChicagoLondon, 1972. [27] S. McAdam and R.G. Swan, Unique comaximal factorization, J. Algebra (to appear). [28] J.L. Mott, Convex directed subgroups of the group of divisibility, Canad. J. Math. 26 (1974), 532–542. [29] J.L Mott and M. Schexnayder, Exact sequences of semi-value groups, J. Reine Angew. Math. 283/284 (1976), 388–401. [30] P. Ribenboim, Anneaux normaux, r´eels `a caract`ere ﬁni, Summa Brasil Math. 3 (1956), 213–253. [31] M. Zafrullah, Semirigid GCD domains, Manuscripta Math. 17 (1975), 55–66.

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[32] M. Zafrullah, Rigid elements in GCD domains, J. Natur. Sci. Math. 17 (1977), 7–14. [33] M. Zafrullah, Unique representation domains, J. Natur. Sci. Math. 18 (1978), 19–29. [34] M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895–1920. [35] M. Zafrullah, Putting t-invertibility to use, 429-457, Non-Noetherian Commutative Ring Theory, Mathematics and Its Applications, vol. 520, Kluwer Academic Publishers, Dordrect/Boston/London, 2000.

Chapter 2

Divisibility Properties in Graded Integral Domains by

David F. Anderson Abstract Let R be a graded integral domain. In this chapter, we study the extent to which conditions on the homogeneous elements or the homogeneous ideals of R determine divisibility properties of R.

1

Introduction

There are many divisibility properties one can consider on an integral domain R. They range from chain conditions and closedness properties to lengths of factorizations and class groups. If, in addition, R is a graded ring, then in some cases we need only consider homogeneous information. Here we investigate to what extent conditions on the monoid S of nonzero homogeneous elements of R or the homogeneous ideals of R determine divisibility properties of R. It is somewhat surprising that in some cases divisibility properties of R are completely determined by S; this depends on the divisibility property considered and how R is graded. Things behave best when R is a Z+ -graded integral domain or a semigroup ring. Many standard ring-theoretic constructions such as polynomial rings, semigroup rings, and the A + XB[X] construction yield graded integral domains. These constructions illustrate the diversity in graded integral domains, and they have been very useful in constructing examples. We are also interested in determining when these various classes of graded integral domains satisfy certain divisibility conditions. The literature on these constructions is vast; so we will not needlessly duplicate it. For more on semigroup rings, we refer the reader to [54], and for the A + XB[X] and the related D + XDS [X] constructions, we refer the reader to [74], [44], and [89]. This chapter has six sections including the introduction. In the second section, we review some basic facts about graded integral domains and the homogeneous quotient ﬁeld of a graded integral domain. In the third section, we discuss several 22

Divisibility Properties in Graded Integral Domains

23

divisibility properties on a graded integral domain and investigate the extent to which conditions on the homogeneous elements or homogeneous ideals of R determine divisibility properties of R. In the fourth section, we look at some factorization properties in several special classes of graded integral domains, with emphasis on semigroup rings and graded subrings of polynomial rings. In the ﬁfth section, we discuss the Picard group and (t-)class group of a graded integral domain. In the ﬁnal section, we give several examples using graded integral domains, usually semigroup rings. Of course, there are many divisibility properties we do not discuss. We have chosen those divisibility properties we have worked on and are most interested in. Hopefully, we have supplied enough additional references for the interested reader to pursue other properties. Our notation is standard. We will usually denote an integral domain by either D or R, a ﬁeld by K, a monoid by Γ, and a torsionfree abelian group by G. Given an integral domain R, we will denote its quotient ﬁeld and group of units by qf (R) and U (R), respectively. By the dimension of a ring, we will always mean Krull dimension. Usually, X, Y, Z, W , or {Xα } will denote indeterminates. The nonzero elements of a subset A of a ring R will be denoted by A∗ . Given a ring D and an additive semigroup Γ, the semigroup ring will be denoted by D[Γ] = { as X s | as ∈ D, s ∈ Γ } with X s X t = X s+t . As usual, N, Z, Q, R, C, and Z/nZ will denote the positive integers, integers, rational numbers, real numbers, complex numbers, and the integers modulo n, respectively. For a partially ordered abelian group G, let G+ = { x ∈ G | x ≥ 0 }. References for any undeﬁned concepts or notation are [53], [54], [49], [80], [64].

2

Graded Domains and the Homogeneous Quotient Field

In this section, we review some basic properties of graded integral domains which will be used throughout this article. Let R = R0 ⊕ R1 ⊕ · · · be a Z+ -graded integral domain. That is, each Rn is an additive subgroup of R and Ri Rj ⊆ Ri+j for all i, j ∈ Z+ . We will say that a nonzero x ∈ Rn is homogeneous of degree n, and deg(0) = 0. Thus each x ∈ R may be written uniquely as x = x1 + · · · + xn with deg(xi ) = mi and m1 < . . . < mn . In this case, R0 is a subring of R, and (R0 )∗ is a saturated multiplicatively closed subset of R. Thus U (R) = U (R0 ). Also, the set S = { 0 = x ∈ R | x is homogeneous } = ∪n∈Z+ (Rn )∗ of nonzero homogeneous elements of R is a saturated multiplicatively closed subset of R. Then RS is a Z-graded integral domain via deg(a/s) = deg(a) − deg(s) for all a, s ∈ S. We will call RS the homogeneous quotient ﬁeld of R; each nonzero homogeneous element of R is a unit. Note that (RS )0 is a ﬁeld, and moreover, RS = (RS )0 [t, t−1 ] for any homogeneous t ∈ RS of smallest positive degree (such a t is transcendental over (RS )0 ), and hence RS is a PID. The simplest example of a Z+ -graded integral domain is the polynomial ring R = D[X] over an integral domain D with each Rn = DX n . More generally, let {Dn }n∈Z+ be a family of subrings of an integral domain D with Dn ⊆ Dn+1 for all n ∈ Z+ . Then R = ⊕n∈Z+ Dn X n is a Z+ -graded integral domain with each Rn = Dn X n . Special cases of this construction which have received considerable

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attention, often for constructing examples, are the A + XB[X] construction and the D + XDS [X] construction. Let Γ be a commutative, cancellative monoid, written additively, with the property that nx = ny for x, y ∈ Γ and n a positive integer implies that x = y. We will call such a monoid a torsionless grading monoid. In this case, Γ is a submonoid of its quotient group, the torsionfree abelian group Γ = {a − b | a, b ∈ Γ}. (Thus a torsionless grading monoid is just a submonoid of a torsionfree abelian group.) The group of units of Γ is U (Γ) = Γ ∩ −Γ. Note that a cancellative monoid can be totally ordered if and only if it is torsionless. Also, the semigroup ring D[Γ] is an integral domain if and only if D is an integral domain and Γ is a torsionless grading monoid. General references for torsionless grading monoids are [54] and [80]. Let Γ be a torsionless grading monoid (which we will always assume to be totally ordered). In this article, we will study Γ-graded integral domains. That is, R = ⊕α∈Γ Rα , where each Rα is an additive subgroup of R and Rα Rβ ⊆ Rα+β for all α, β ∈ Γ. We say that a nonzero x ∈ Rα is homogeneous of deg(x) = α, and we will usually make the harmless assumption that each Rα is nonzero. In this case, R0 is a subring of R, and thus (R0 )∗ is multiplicatively closed, but (R0 )∗ is saturated if and only if Γ ∩ −Γ = 0. Let S = ∪α∈Γ (Rα )∗ be the saturated multiplicatively closed set of nonzero homogeneous elements of R. Note that U (R) ⊆ S. The homogeneous quotient ﬁeld RS of R is Γ-graded with (RS )α = { a/s | a, s ∈ S with deg(a) − deg(s) = α } ∪ {0} for each α ∈ Γ. The best example of a Γ-graded integral domain is the semigroup ring R = D[Γ] over an integral domain D, graded by deg(aX α ) = α for all a ∈ D∗ and α ∈ Γ. For R = D[Γ], its set of nonzero homogeneous elements is S = { aX α | a ∈ D∗ , α ∈ Γ }, and its homogeneous quotient ﬁeld RS is just the group ring K[G], where K = qf (D) and G = Γ. So, when we say a “graded integral domain R”, we mean an integral domain R graded by a torsionless grading monoid Γ. A general reference for graded integral domains is [80]. We have seen that RS = (RS )0 [Z] when R is a Z+ - or Z-graded integral domain, and in this case, RS is a UFD (in fact, a PID). However, in general RS need be neither a UFD nor a group ring over Γ. The UFD case is easy. Just let R = K[Q] for any ﬁeld K; then RS = R is not a UFD because ACCP fails since (1 − X) ⊆ (1 − X 1/2 ) ⊆ (1 − X 1/4 ) ⊆ . . . is a properly ascending chain of principal ideals of R (see Theorem 2.3(a) below). We next show that RS need not be a group ring. Example 2.1. Let D be an integral domain, Γ a torsionless grading monoid, and T ⊆ Γ such that [T ] = Γ, where [T ] is the submonoid generated by T . Deﬁne R = D[{Xα }α∈T ]. Then R is Γ-graded with deg(Xαn11 · · · Xαnrr ) = n1 α1 + · · · + nr αr . In fact, R = D[Γ ], where Γ is the free abelian monoid on T. Let D be a UFD and T = Γ = Q+ . Then R, and hence RS , is a UFD. If RS were a group ring over Q, then necessarily RS ∼ = (RS )0 [Q]. But the group ring (RS )0 [Q] is not a UFD since ACCP fails. So R is not isomorphic to a group ring over Q. However, the homogeneous quotient ﬁeld RS is a twisted group ring. Let K = (RS )0 and G = Γ. For each α ∈ G, pick a nonzero Yα ∈ (RS )α . Then (RS )α = KYα and Yα Yβ = γ(α, β)Yα+β for some nonzero γ(α, β) ∈ K. Thus RS = (RS )γ0 [G] is a twisted group ring (see [34] for more details). But, we may not always be able to choose the Yα ’s so that each γ(α, β) = 1. Some cases when RS is a group ring or

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a UFD are given in [12]. For example, RS is a UFD if Γ satisﬁes ACC on cyclic subgroups [12, Proposition 3.5] (note that the converse fails by Example 2.1). Although RS need not be a UFD, it is “close” to being a UFD. In fact, by our next result, RS is a UFD if and only if it is a Krull domain, if and only if it satisﬁes ACCP. Theorem 2.2. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then RS is a completely integrally closed GCD-domain. Proof. The ring RS is completely integrally closed by [12, Proposition 3.2] and is a GCD-domain by [12, Proposition 3.3]. For future reference, we record the following result for group rings. The semigroup ring case will be discussed in the next section. The condition that G has ACC on cyclic subgroups is equivalent to each rank-one subgroup of G being free (cyclic) [12, Lemma 2.5] or each nonzero element of G having type (0, 0, 0, . . . ) (see [51, Section 85]). We will always use the ACC on cyclic subgroups condition. Theorem 2.3. Let D be an integral domain and G a torsionfree abelian group. (a) D[G] is a UFD if and only if D is a UFD and G satisﬁes ACC on cyclic subgroups. (b) D[G] is a Krull domain if and only if D is a Krull domain and G satisﬁes ACC on cyclic subgroups. Proof. Part (a) is due to Gilmer and Parker [57, Theorem 7.13], and part (b) is due to Matsuda [75, Proposition 3.3]. They each used the type (0, 0, 0, . . .) condition on G. Following Cohn [42], we say that an extension A ⊆ B of integral domains is inert if whenever xy ∈ A for x, y ∈ B, then x = ru and y = su−1 for some r, s ∈ A and u ∈ U (B). Clearly R0 ⊆ R = ⊕α∈Γ Rα is inert if and only if Rα contains a unit for each α ∈ Γ ∩ −Γ. For example, this happens if Γ ∩ −Γ = 0, R0 is a ﬁeld, or R = R0 [Γ]. In particular, if R is Z+ -graded, then R0 ⊆ R is always an inert extension. We will see that things usually behave better when R0 ⊆ R is an inert extension. An integral ideal I of R = ⊕α∈Γ Rα is homogeneous if I = ⊕α∈Γ (I ∩ Rα ); equivalently, if I is generated by homogeneous elements. A fractional ideal I of R is homogeneous if sI is an integral homogeneous ideal of R for some s ∈ S (thus I ⊆ RS ). For x = x1 + · · · + xn ∈ RS with each xi ∈ (RS )αi and α1 < . . . < αn , we deﬁne the content of x to be the homogeneous ideal C(x) = (x1 , . . . , xn ). Thus a fractional ideal I ⊆ RS is homogeneous if and only if C(x) ⊆ I for each x ∈ I. For x, y ∈ R, we have C(x)n C(xy) = C(x)n+1 C(y) for some integer n ≥ 0 [79]. A subring A of R is a graded subring if A = ⊕α∈Γ (A ∩ Rα ). In a similar manner, we deﬁne a graded overring of R to be a graded subring of RS containing R.

3

Divisibility Properties of Graded Domains

Divisibility properties of an integral domain R are often equivalent to the corresponding divisibility properties in the multiplicative monoid R∗ . Some of these equivalences are obvious (R is a UFD iﬀ R∗ is a factorial monoid); others are more

26

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diﬃcult (R is a Krull domain iﬀ R∗ is a Krull monoid). In fact, nowadays one often studies divisibility properties in the more general context of a commutative, cancellative monoid (for example, see [64] or [40], and the next paragraph). In this section, we investigate to what extent divisibility properties of R are actually determined by the corresponding divisibility properties on the monoid S of nonzero homogeneous elements of R, or by conditions on the homogeneous ideals of R. One can also ask how divisibility properties of R0 relate to those of R. In other words, how do divisibility properties on the submonoids (R0 )∗ and S of R∗ relate to divisibility properties of R (or R∗ )? Let H be a commutative, cancellative (multiplicative) monoid with quotient group H = { a/b | a, b ∈ H }. Ring-theoretic properties can often be translated to monoid properties. This is obvious if it is a purely multiplicative property, and we usually won’t give the explicit deﬁnition in this case. For example, we say that H is integrally closed (resp., completely integrally closed) if xn ∈ H (resp., hxn ∈ H for some h ∈ H) for x ∈ H and some (resp., all) n ∈ N implies x ∈ H. Also, H is a Krull monoid if H is completely integrally closed and H satisﬁes ACC on integral v-ideals (for other equivalent conditions, see [41], [40], or [64]). Recall that a torsionless grading monoid Γ is written additively; so here integrally closed would mean that nx ∈ Γ for some n ∈ N and x ∈ Γ implies x ∈ Γ. Let R be an integral domain with quotient ﬁeld K. Then its group of divisibility is the abelian group G(R) = K ∗ /U (R) partially ordered by xU (R) ≤ yU (R) if and only if y/x ∈ R. Divisibility properties of R are often reﬂected in its group of divisibility. For example, R is a GCD-domain (resp., valuation domain) if and only if G(R) is lattice ordered (resp., totally ordered). In a similar manner, for a graded integral domain R, we deﬁne its homogeneous group of divisibility to be HG(R) = U (RS )/U (R). It is a partially ordered subgroup of G(R). We ﬁrst give two examples to show that divisibility properties of R need not be determined by S. We call a graded integral domain R a graded valuation domain if for x, y ∈ S, then either x|y or y|x (cf. [66]); equivalently, HG(R) is a totally ordered abelian group. Certainly RS is always a graded valuation domain, and any polynomial ring over a ﬁeld is a graded valuation domain, but neither is a valuation domain. So a condition on the nonzero homogeneous elements of R need not force that given condition on all the nonzero elements of R. Secondly, for the group ring R = K[G] over a ﬁeld K, its set of nonzero homogeneous elements S = { aX α | a ∈ K ∗ , α ∈ G } ∼ = K ∗ × G is certainly a very nice monoid and satisﬁes almost any conceivable divisibility property. For example, S is always a factorial monoid, but R need not be a UFD (cf. Theorem 2.3(a) and Theorem 3.3). A given integral domain R may be graded in many diﬀerent ways (cf. [3]); so R0 and S depend on the given grading of R. Our next example illustrates this problem. Example 3.1. Let R = Q[X, Y, Z, W ]. Then R is a 4-dimensional Noetherian UFD. We consider three very diﬀerent gradings on R. (a) R is Z+ -graded in the usual way with degX = degY = degZ = degW = 1. In this case, R0 = Q is a UFD and R0 ⊆ R is an inert extension. (b) R is (Z+ )4 -graded with degX = (1, 0, 0, 0), degY = (0, 1, 0, 0), degZ = (0, 0, 1, 0), and degW = (0, 0, 0, 1). In this case, R = Q[(Z+ )4 ] is a semigroup ring, R0 = Q is a UFD, and R0 ⊆ R is an inert extension. (c) R is Z-graded with degX = degY = 1 and degZ = degW = −1. In this

Divisibility Properties in Graded Integral Domains

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case, R0 = Q[XZ, XW, Y Z, Y W ] is not a UFD. In fact, R0 is a Krull domain with Cl(R0 ) = Z [49, Proposition 14.8]. Note that R0 ⊆ R is not an inert extension. As in [1], we deﬁne a graded integral domain R to be a graded GCD-domain if each pair of nonzero homogeneous elements of R has a (necessarily homogeneous) GCD; equivalently, S is a GCD-monoid, or HG(R) is a lattice ordered abelian group. Example 3.1(c) shows that R a graded GCD-domain does not imply that R0 is a GCD-domain. However, if R0 ⊆ R is an inert extension and R is a graded GCD-domain, then it is easily proved that R0 is a GCD-domain. The key fact that gives the equivalence in the next theorem is that RS is a GCD-domain. Theorem 3.2. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then R is a GCD-domain if and only if S is a GCD-monoid. Thus R is a GCD-domain if and only if R is a graded GCD-domain. Proof. This is proved in [1, Theorem 3.4] using [57, Theorem 3.1]. In a similar manner, we deﬁne a graded integral domain R to be a graded UFD if each homogeneous nonzero nonunit of R is a product of (necessarily homogeneous) prime elements of R; equivalently, S is a factorial monoid, or HG(R) is orderisomorphic to a free abelian group with the usual product order. We have already seen that a graded UFD need not be a UFD since RS , which is trivially a graded UFD, need not be a UFD. Example 3.1(c) also shows that R0 need not be a UFD when R is a graded UFD. However, R0 is a UFD when R is a graded UFD and R0 ⊆ R is an inert extension. We next give the analog of Theorem 3.2 for UFDs. But in this case, and the next, the graded and nongraded conditions are not equivalent. Theorem 3.3. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then R is a UFD if and only if S is a factorial monoid and RS is a UFD. In particular, if R is Z+ - or Z-graded, then R is a UFD if and only if R is a graded UFD. Proof. This is proved in [1, Theorem 4.4]. The “in particular” statement follows since RS ∼ = (RS )0 [Z] is a UFD (PID) when R is Z+ - or Z-graded. This case was ﬁrst proved in [10, Theorem 5]. We next consider the Krull domain case. We deﬁne a graded integral domain R to be a graded Krull domain if R is completely intregrally closed with respect to homogeneous elements of RS (i.e., a/b almost integral over R for a, b ∈ S implies a/b ∈ R) and R satisﬁes ACC on homogeneous integral v-ideals; equivalently, S is a Krull monoid. However, unlike the previous two cases, if R is a graded Krull domain, then R0 is also a Krull domain [1, Corollary 5.9]. Theorem 3.4. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then R is a Krull domain if and only if S is a Krull monoid and RS is a Krull domain (UFD). In particular, if R is Z+ - or Z-graded, then R is a Krull domain if and only if R is a graded Krull domain. Proof. This is proved in [1, Theorem 5.8]. We have already observed that RS is a Krull domain if and only if it is a UFD. The “in particular” statement follows since RS is a UFD (PID) in this case.

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We next recall several closedness properties on an integral domain R. For a positive integer n, we say that R is n-root closed if xn ∈ R for x ∈ qf (R) implies x ∈ R, and that R is root closed if R is n-root closed for each positive integer n. Also, R is seminormal if x2 , x3 ∈ R for x ∈ qf (R) implies x ∈ R. An integrally closed domain is root closed, and an n-root closed (for some n ≥ 2) domain is seminormal, but neither converse holds (see Example 6.7). The seminormal property will be useful in Section 5 when we discuss Picard groups. By a result of Swan [84], R is seminormal if and only if whenever x2 = y 3 for x, y ∈ R, then x = z 3 and y = z 2 for some z ∈ R. In the obvious manner, we can deﬁne a (n-)root closed or seminormal monoid (for seminormality, we can use either equivalent condition). Note that for monoids, the deﬁnitions of integrally closed and root closed are the same. The n-root closed version of the next theorem is left to the reader. Theorem 3.5. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then the following statements are equivalent. (a) R is root closed (resp., seminormal). (b) R is root closed (resp., seminormal) with respect to homogeneous elements of RS , i.e., if xn ∈ R for some positive integer n (resp., x2 , x3 ∈ R) for a homogeneous x ∈ RS , then x ∈ R. (c) S is a root closed (resp., seminormal) monoid. Proof. The root closed (resp., seminormal) part is given in [16, Theorem 2.4] (resp., [2, Theorem 6.1]). Theorem 3.6. Let R be a graded integral domain and S its set of nonzero homogeneous elements. (a) R is integrally closed (resp., completely integrally closed) if and only if R is integrally closed (resp., completely integrally closed) with respect to homogeneous elements of RS , i.e., if a homogeneous x ∈ RS is integral (resp., almost integral) over R, then x ∈ R. (b) R is completely integrally closed if and only if S is a completely integrally closed monoid. Proof. (a) The integrally closed (resp., completely integrally closed) case is given in [1, Theorem 5.4] (resp., [1, Theorem 5.2]). The integrally closed case also follows from [66, Theorem 2.10]. (b) This follows directly from part (a). The above seven properties P all satisfy: the graded integral domain R is a Pdomain if and only if R is a “graded P-domain” and RS is a P-domain. Note that RS is always a completely integrally closed GCD-domain, and hence is integrally closed, root closed, and seminormal. Also, as noted above, RS is a UFD if and only if it is a Krull domain. This equivalence also holds for π-domains, PVMDs, and G-GCD domains (see [1] and [2]). Note that RS is a valuation domain if and only if R is trivially graded, i.e., R = R0 , so RS = qf (R). Thus this property also has the above form. We next give the semigroup ring analogs of Theorem 2.3 which were promised in the previous section. These are all fairly immediate consequences of the previous theorems.

Divisibility Properties in Graded Integral Domains

29

Theorem 3.7. Let D be an integral domain and Γ a torsionless grading monoid with group of units U (Γ) = Γ ∩ −Γ. (a) D[Γ] is a GCD-domain if and only if D is a GCD-domain and Γ is a GCDmonoid. (b) D[Γ] is a UFD if and only if D is a UFD, Γ is a factorial monoid, and U (Γ) satisﬁes ACC on cyclic subgroups. (c) D[Γ] is a Krull domain if and only if D is a Krull domain, Γ is a Krull monoid, and U (Γ) satisﬁes ACC on cyclic subgroups. (d ) D[Γ] is integrally closed if and only if D is integrally closed and Γ is a root closed monoid. (e) D[Γ] is completely integrally closed if and only if D is completely integrally closed and Γ is a completely integrally closed monoid. (f ) D[Γ] is (n-)root closed if and only if D is (n-)root closed and Γ is a (n-)root closed monoid. (g) D[Γ] is seminormal if and only if D is seminormal and Γ is a seminormal monoid. Proof. We prove parts (f) and (g); the proofs of the others may all be found in [54]. If D[Γ] is root closed (resp., seminormal), then clearly D and Γ are each root closed (resp., seminormal). Conversely, to show that D[Γ] is root closed (resp., seminormal), by Theorem 3.5 we need only check that (aX α )n ∈ D[Γ] (resp., (aX α )2 , (aX α )3 ∈ D[Γ]) for aX α ∈ qf (D)[Γ] implies aX α ∈ D[Γ]. But this is clear if D and Γ are each root closed (resp., seminormal). Remark 3.8. (a) By Theorem 3.7, If D is an integrally closed domain, then R = D[Γ] is integrally closed if and only if it is root closed. (b) We give a more concrete description of factorial and Krull monoids. A factorial monoid has the form G ⊕ F+ and a Krull monoid has the form G ⊕ T , where G is a torsionfree abelian group, F = ⊕Zα is a free abelian group with the usual product order, and T is a submonoid of F such that T = T ∩ F+ . Thus a UFD semigroup ring is just a polynomial ring over a UFD group ring, and a Krull domain semigroup ring is just a subring of a polynomial ring over a Krull group ring generated by monomials (cf. [1, pages 204 and 209], [41, Proposition 1], and [57, Lemma 7.15]). (c) By Theorem 3.7 (with Γ = Z+ ), we have that D[X] is a P-domain if and only if D is a P-domain. Thus it seems natural to ask if a graded integral domain R satisﬁes property P if and only if R0 satisﬁes property P and RS satisﬁes property P. However, this fails for all seven properties in Theorem 3.7. Let R = Z + ZX + X 2 Q[X] be considered as a graded subring of Q[X]. Here R0 = Z and RS = Q[X, X −1 ] satisfy all seven properties, but R does not satisfy any of them.

4

Factorization Properties in Graded Domains

In this section, we consider several divisibility properties related to factorizations for an integral domain R (they all translate in the obvious way to monoids). We say that R is atomic if each nonzero nonunit of R is a product of a ﬁnite number of irreducible elements (atoms) of R. We say that R is a bounded factorization domain (BFD) if R is atomic and for each nonzero nonunit x of R there is a bound

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on the length of factorizations of x into irreducible elements. We say that R is a half-factorial domain (HFD) if R is atomic and each factorization of a nonzero nonunit of R into a product of irreducible elements has the same length. Finally, R is a ﬁnite factorization domain (FFD) if each nonzero nonunit of R has only a ﬁnite number of factorizations up to order and associates. These properties, and several others, have been studied systematically in [6], [7], and [9]. We have the following implications (no implication is reversible, see [6] and Examples 6.1-6.4). UFD ⇓ FFD

⇒ HF D ⇓ ⇒ BF D

⇒ ACCP

⇒ atomic

It is well-known that R[X] satisﬁes ACCP if and only if R satisﬁes ACCP. Also, R[X] is a BFD (resp., FFD) if and only if R is a BFD (resp., FFD) [6, Proposition 2.5 (resp., Proposition 5.3)]. Clearly R is atomic (resp., an HFD) if R[X] is atomic (resp., an HFD). However, R an atomic integral domain does not imply that R[X] is atomic [83]. For R a Krull domain, R[X] is an HFD if and only if |Cl(R)| ≤ 2 [90, Theorem 2.4]. Also, R = R + Y C[Y ] is an HFD (see Theorem 4.3(a)), but R[X] is not an HFD [7, Example 5.4]. In fact, if R[X] is an HFD, then R is necessarily integrally closed [45, Theorem 2.2]. For more on HFDs, see the survey article [39]. Several divisibility properties are studied for group rings and semigroup rings in [54] and [70]. We mention three for group rings. Theorem 4.1. Let D be an integral domain and G a torsionfree abelian group. Then D[G] satisﬁes ACCP (resp., is a BFD, FFD) if and only if D satisﬁes ACCP (resp., is a BFD, FFD) and G satisﬁes ACC on cyclic subgroups. Proof. The ACCP case is proved in [54, Theorem 14.17], and the BFD and FFD cases are from [70, Theorem 3.1] and [70, Theorem 4.1], respectively. We next consider Z+ -graded integral domains of the form R = ⊕n∈Z+ Dn X n , where {Dn }n∈Z+ is a family of subrings of an integral domain D with Dn ⊆ Dn+1 for all n ∈ Z+ . Various divisibility and factorization properties of R have been studied in [65], [58], [68], and [35]. For example, if D0 is a ﬁeld, then R is atomic, satisﬁes ACCP, and is a BFD; but R need not be an HFD or an FFD (see Theorems 4.2 and 4.3). However, R is completely integrally closed if and only if Dn = D0 for all n ∈ Z+ and D0 is completely integrally closed. Thus R is a UFD (resp., Krull domain) only in the trivial case when R = D0 [X] and D0 is a UFD (resp., Krull domain). The other closedness properties we have studied all force Dn = D1 for all n ∈ N. Thus we will concentrate on the special case of the A + XB[X] construction. There has also been considerable work on the related A + XB[[X]] construction. In fact, most papers treat both the A + XB[X] and A + XB[[X]] constructions simultaneously. We will just give a few results for the A + XB[X] construction; for many more see [7], [27], [32], [33], [38], [44], [47], [48], [58], [59], [60], [61], [65] [68] [69], [74], [87], [88],and [89]. For the deﬁnition of splitting set in part (d), see the next section. Theorem 4.2. Let A ⊆ B be an extension of integral domains and R = A+XB[X]. (a) R is completely integrally closed (resp., a UFD, Krull domain) if and only if A = B and A is completely integrally closed (resp., a UFD, Krull domain).

Divisibility Properties in Graded Integral Domains

31

(b) R is integrally closed (resp., root closed, seminormal) if and only if B is integrally closed (resp., root closed, seminormal) and A is integrally closed in qf (B) (resp., root closed in qf(B), seminormal). (c) R satisﬁes ACCP if and only if U (B) ∩ A = U (A) and every ascending chain b1 B ⊆ b2 B ⊆ . . . with each bn ∈ B nonzero and bn /bn+1 ∈ A terminates. (d ) R is a GCD-domain if and only if A is a GCD-domain and B = AS for S a splitting multiplicative set of A. (e) R is a BFD if and only if U (B) ∩ A = U (A) and for each nonzero nonunit b ∈ B, there is an N (b) ∈ N such that whenever b = b1 · · · bn with each bi ∈ B a nonunit, then at most N (b) of the bi ’s are in A. (f ) R is an FFD if and only if B is an FFD and U (B)/U (A) is ﬁnite. Proof. We have already mentioned part (a). Part (b) is clear by comments in the previous section. Part (c) is from [32, Proposition 1.1]; part (d) is from [33, Theorem 2.10]; part (e) is from [32, Proposition 2.1]; and part (f) is from [32, Proposition 3.1] and uses facts from [11]. The most popular problem seems to be to determine conditions for A + XB[X] to be an HFD. Partial results have appeared in many of the papers listed above. Probably the simplest example of an HFD which is not a UFD, is k + XK[X] for any proper extension k K of ﬁelds [7, Theorem 5.3]. Below we give three conditions for A + XB[X] to be an HFD. A surprising consequence of part (b) is that A + XB[X] can be an HFD without B[X] being an HFD. (Note that A is an HFD if A + XB[X] is an HFD.) Theorem 4.3. Let A ⊆ B be an extension of integral domains and R = A+XB[X]. (a) If B is a ﬁeld, then R is an HFD if and only if A is a ﬁeld. (b) If A is a ﬁeld, then R is an HFD if and only if B is integrally closed. (c) If B is a UFD, U (B) ∩ A = U (A), and each irreducible element of A is also irreducible in B, then R is an HFD. Proof. Part (a) is from [7, Theorem 5.3], and part (b) is from [47, Theorem 2.1]. For part (c), see [59, Proposition 1.8] and [32, Proposition 5.4]. For an atomic integral domain R, the elasticity of R is ρ(R) = sup{ m/n | x1 · · · xm = y1 · · · yn for irreducible xi , yj ∈ R }. (In a similar manner, one can deﬁne the elasticity ρ(H) of an atomic monoid H.) Then 1 ≤ ρ(R) ≤ ∞, and ρ(R) = 1 if and only if R is an HFD. Not much more can be said since for each real number r ≥ 1 or r = ∞, there is a Dedekind domain R with torsion class group and ρ(R) = r [4, Theorem 3.2]. This concept is due to Valenza [85]. For a recent survey article on the elasticity, see [20]. Let R be a graded integral domain with S its monoid of nonzero homogeneous elements. It is clear that ρ(R0 ) ≤ ρ(R) if R0 ⊆ R is an inert extension and that ρ(S) ≤ ρ(R). The next example shows that a “graded HFD” need not be an HFD and that we may have ρ(R0 ) > ρ(R). Example 4.4. Let R = Q[X, Y, Z, W ] be Z-graded with degX = degY = 1 and degZ = degW = −1 as in Example 3.1(c). Here R is a UFD, and hence an HFD. In this case, R0 = Q[XZ, XW, Y Z, Y W ] = Q[Γ], where Γ = (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 0, 1) ⊆ (Z+ )4 , is not a UFD. We observed in Example 3.1(c) that

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R0 is a Krull domain with Cl(R0 ) = Z. Thus ρ(R0 ) = ∞ since each nonzero divisor class of Cl(Z[Γ]), and hence Cl(R0 ), contains a height-one prime ideal [71, Theorem 7](also see Theorem 4.5(c)). Hence 1 = ρ(R) < ρ(R0 ) = ∞. Note that R0 ⊆ R is not an inert extension. Let S0 be the set of nonzero homogeneous elements of R0 . Then S0 ∼ = Q∗ × Γ. Clearly Γ is an HFD monoid, and thus so is S0 ; but 1 = ρ(S0 ) < ρ(R0 ) = ∞. Hence R0 is a “graded HFD”, but not an HFD. The elasticity of certain families of graded integral domains has been studied extensively. We will just mention two cases for graded subrings of a polynomial ring, and one for Krull semigroup rings. For a ﬁnite abelian group G, we deﬁne the Davenport constant of G, denoted by D(G), to be the least positive integer d such that for each sequence S ⊆ G with |S| = d, some nonempty subsequence of S has sum 0 (D(G) = ∞ if G is inﬁnite). Then D(G) ≤ |G| and D(Z/nZ) = n. A special case of our next result is given in Example 6.4. For other related results, see [28], [31], [36], [63], and [72] for semigroup rings, and [35], [58], [59], [60], [61], and [72] for the A + XB[X] and related constructions. Theorem 4.5. (a) Let K be a ﬁeld and R = K[X n , X n+1 , . . . , X 2n−1 ]. Then ρ(R) = (nD(P ic(R)) + 3n − 2)/2n. (b) For n ≥ 2, let K0 ⊆ K1 ⊆ . . . ⊆ Kn−1 be an ascending sequence of subﬁelds of a ﬁeld K with Kn−1 K, and let R = K0 + K1 X + · · · + Kn−1 X n−1 + X n K[X]. Then ρ(R) = (2n + D(P ic(R)) − 1)/2. (c) Let D be a UFD with proper quotient ﬁeld K and D[Γ] a Krull domain which is not a UFD. Then ρ(D[Γ]) = D(Cl(K[Γ]))/2. Proof. Part (a) follows from [31, Theorem 2.4] and [35, Remark 4.6(e)]; part (b) is proved in [35, Theorem 4.4 and Remark 4.6(e)]; and part (c) is proved in [72, Corollary 2.3].

5

The Picard Group and Class Group of a Graded Domain

In this section, we investigate the Picard group and (t-)class group of a graded integral domain. We ﬁrst recall a few deﬁnitions. Let R be an (not necessarily graded) integral domain with quotient ﬁeld K. For a nonzero fractional ideal I of R, let I −1 = { x ∈ K | xI ⊆ R }, Iv = (I −1 )−1 , and It = ∪{ Jv | 0 = J ⊆ I is ﬁnitely generated }. A nonzero fractional ideal I of R is a v-ideal (resp., t-ideal ) if Iv = I (resp., It = I), and I has ﬁnite type if I = Jv with J ⊆ I ﬁnitely generated. A fractional ideal I of R is t-invertible if (II −1 )t = R. A t-invertible t-ideal necessarily has ﬁnite type. Let T (R) be the group of t-invertible t-ideals of R under the t-multiplication I ∗ J = (IJ)t , and let P rin(R) be its subgroup of nonzero principal fractional ideals. Then the (t-)class group of R is the abelian group Cl(R) = T (R)/P rin(R). Let Inv(R) ⊆ T (R) be the subgroup of invertible ideals of R. Then the Picard group or ideal class group of R, P ic(R) = Inv(R)/P rin(R), is a subgroup of Cl(R). If R is a Krull domain, then Cl(R) is just the usual divisor class group of R; while if R is a Pr¨ ufer domain or a one-dimensional integral domain, then Cl(R) = P ic(R). The class group is important because ring-theoretic properties of R are often reﬂected

Divisibility Properties in Graded Integral Domains

33

in group-theoretic properties of Cl(R). For example, if R is a Krull domain (resp., Pr¨ ufer domain), then Cl(R) = 0 if and only if R is a UFD (resp., B´ezout domain). For more on the divisor class group of a Krull domain, see [49]; for a recent survey article on the t-class group, see [22]. Given an extension of integral domains A ⊆ B, we write P ic(A) = P ic(B) (resp., Cl(A) = Cl(B)) to mean that the inclusion map induces an isomorphism P ic(A) −→ P ic(B) (resp., Cl(A) −→ Cl(B)) given by [I] → [IB], where [I] denotes the class of I in P ic(A) or Cl(A). Note that this map is always deﬁned for P ic and is deﬁned for Cl when B is a ﬂat A-module. A ﬁrst natural question for polynomial rings is: when does Cl(D) = Cl(D[X]) or P ic(D) = P ic(D[X])? In this case, we have nice answers. Theorem 5.1. Let D be an integral domain. (a) P ic(D) = P ic(D[X]) if and only if D is seminormal. (b) Cl(D) = Cl(D[X]) if and only if D is integrally closed. Proof. Part (a) is due to Gilmer and Heitmann [56, Theorem 1.6], and part (b) is due to Gabelli [52, Theorem 3.6]. We next give the semigroup ring analog of Theorem 5.1 for the Picard group; the class group case will be discussed later. Recall that an integral domain D is called quasinormal if P ic(D) = P ic(D[X, X −1 ]) ( = P ic(D[Z])). An integrally closed domain is always quasinormal, and a quasinormal integral domain is seminormal, but neither converse holds. An interesting consequence of the next theorem is that P ic(D) = P ic(D[Γ]) for all seminormal torsionless grading monoids Γ if and only if P ic(D) = P ic(D[Z]). Theorem 5.2. Let D be an intergal domain and Γ a torsionless grading monoid with group of units U (Γ) = Γ ∩ −Γ. (a) P ic(D) = P ic(D[Γ]) if and only if D[Γ] is seminormal and P ic(D) = P ic(D[U (Γ)]). (b) If U (Γ) = 0, then P ic(D) = P ic(D[Γ]) if and only if D is seminormal and Γ is seminormal. (c) If U (Γ) = 0, then P ic(D) = P ic(D[Γ]) if and only if D is quasinormal and Γ is seminormal. Proof. This is from [18, Corollary]. Part (b) follows directly from part (a) and Theorem 3.7(g). The key fact needed for part (c) is that D is quasinormal if and only if P ic(D) = P ic(D[Zn ]) for all positive integers n (see [86]), and hence if and only if P ic(D) = P ic(D[G]) for all torsionfree abelian groups G. Let R be a graded integral domain. If I is a nonzero homogeneous fractional ideal of R, then I −1 , Iv , and It are also homogeneous fractional ideals of R. Thus the abelian groups HP rin(R) ⊆ HInv(R) ⊆ HT (R) can be deﬁned in the natural way, and we can deﬁne the homogeneous analogs of P ic(R) and Cl(R) by HP ic(R) = HInv(R)/HP rin(R) and HCl(R) = HT (R)/HP rin(R), the homogeneous Picard group and homogeneous (t-)class group, respectively. Thus HCl(R) = Cl(R) if and only if for each I ∈ T (R), I = xJ for some x ∈ RS and J ∈ HT (R). A similar result holds for HP ic(R).

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It is well-known that HCl(R) = Cl(R) if R is a graded Krull domain (see [49, Proposition 10.2] for the Z+ -graded case and [12, Theorem 4.2] for the general Γgraded case) since Cl(R) is generated by the classes of the height-one homogeneous prime ideals of R. We next give some conditions to have HP ic(R) = P ic(R) and HCl(R) = Cl(R). We ﬁrst give some relevant deﬁnitions from [2] and [15]. Recall from Section 3 that a graded integral domain R is integrally closed (resp., seminormal) if and only if R is integrally closed (resp., seminormal) with respect to homogeneous elements of RS . With this motivation, we deﬁne R to be almost normal (resp., almost seminormal) if R is integrally closed (resp., seminormal) with respect to homogeneous elements of RS of nonzero degree. Thus R is integrally closed (resp., seminormal) if and only if R is almost normal (resp., almost seminormal) and R0 is integrally closed (resp., seminormal) in (RS )0 . Moreover, if R contains a (homogeneous) unit of nonzero degree, then R is integrally closed (resp., seminormal) if and only if R is almost normal (resp., almost seminormal) [2, Theorem 3.7(1)]. Also, note that D[Γ] is almost normal (resp., almost seminormal) if and only if it is integrally closed (resp., seminormal). Based on earlier work of Querre [82] for polynomial rings, a careful study of invertible ideals and v-ideals in graded integral domains was initiated in [2]. In [37], these ideas were extended to t-invertible t-ideals. We next give a sampling of results from [2]. Theorem 5.3. Let R be a graded integral domain and S its set of nonzero homogeneous elements. Then the following statements are equivalent. (a) Each integral v-ideal of R which contains a nonzero homogeneous element of R is homogeneous. (b) C(xy)v = (C(x)C(y))v for all nonzero x, y ∈ R. (c) xRS ∩ R = xC(x)−1 for all nonzero x ∈ R. (d ) If I is an integral v-ideal of R of ﬁnite type, then I = xJ for some x ∈ RS and homogeneous v-ideal J of R of ﬁnite type. Proof. This is part of [2, Theorem 3.2]. The ﬁnite type hypothesis may be deleted in part (d) above if IRS is a principal ideal of RS [2, Proposition 3.3]. In particular, this holds if R is Z+ - or Z-graded. However, an example given in [76] shows that it can not be deleted in general. We next examine the content condition in Theorem 5.3(b) more carefully. In [78], an example of a Z-graded integral domain is given which is almost normal, but does not satisfy the content condition. Thus the inert extension hypothesis can not be deleted in Theorem 5.4(c). Theorem 5.4. Let R be a graded integral domain. (a) If R is integrally closed, then C(xy)v = (C(x)C(y))v for all nonzero x, y ∈ R. (b) If C(xy)v = (C(x)C(y))v for all nonzero x, y ∈ R, then R is almost normal. (c) If R0 ⊆ R is an inert extension, then C(xy)v = (C(x)C(y))v for all nonzero x, y ∈ R if and only if R is almost normal. Proof. Parts (a) and (b) are proved in [2, Theorem 3.5], and part (c) is proved in [2, Theorem 3.7(2)].

Divisibility Properties in Graded Integral Domains

35

We next discuss the relationship between P ic(R) and HP ic(R). An example given in [77] shows that the inert extension hypothesis in Theorem 5.5(b) can not be eliminated. Theorem 5.5. Let R be a graded integral domain. (a) If HP ic(R) = P ic(R), then R is almost seminormal. Suppose, in addition, that R0 ⊆ R is an inert extension. (b) P ic(R0 ) = HP ic(R). (c) If R is almost normal, then P ic(R0 ) = P ic(R). Proof. Part (a) is [2, Theorem 4.3(3)], part (b) is [2, Theorem 5.4], and part (c) is [2, Theorem 5.5]. Our next goal is to determine when Cl(D) = Cl(D[Γ]). The key fact is the next result, which is the t-analog of Theorems 5.3 and 5.4. Theorem 5.6. Let R be a graded integral domain such that R0 ⊆ R is an inert extension. Then HCl(R) = Cl(R) if and only if R is almost normal. Proof. This is proved in [37, Theorem 1.1]. Corollary 5.7. Let R be a Z+ -graded integral domain. Then HCl(R) = Cl(R) if and only if R is almost normal. Proof. This is [37, Corollary 1.8]. It follows directly from Theorem 5.6 since in this case R0 ⊆ R is an inert extension. An example of a Z-graded almost normal integral domain R with HCl(R) = Cl(R) is given in [37, Example 1.11]. Thus the inert extension hypothesis is needed in Theorem 5.6. However, in Theorem 5.6 it is suﬃcient to assume that R contains a (homogeneous) unit of nonzero degree [37, Remark 1.7]. If D[Γ] is a Krull domain, then Cl(D[Γ]) = Cl(D)⊕Cl(K[Γ]), where K = qf (D), and Cl(K[Γ]) is independent of the ﬁeld K [12, Proposition 7.2]. In fact, one can deﬁne the divisor class group Cl(Γ) of the Krull monoid Γ, and it turns out that Cl(Γ) ∼ = Cl(K[Γ]) (see [41] and [54, Section 16]). So it seems natural to ask what happens in the general semigroup ring case. In [67], it is shown that Cl(D) = Cl(D[G]) when G is a torsionfree abelian group which satisﬁes ACC on cyclic subgroups, and in [73] it is shown that Cl(D[Γ]) = Cl(D) ⊕ Cl(Γ) when D[Γ] is a PVMD and Γ satisﬁes ACC on cyclic subgroups. For integrally closed semigroup rings, a very satisfactory analog of the Krull domain result is given in [37]. Theorem 5.8. Let D be an integral domain with quotient ﬁeld K and Γ a torsionless grading monoid. (a) HCl(D[Γ]) = Cl(D[Γ]) if and only if D[Γ] is integrally closed. (b) HCl(D[Γ]) = Cl(D) ⊕ HCl(K[Γ]). Proof. Part (a) follows from Theorem 5.6; it is in [37, Proposition 2.2]. Part (b) is in [37, Theorem 2.6]. Corollary 5.9. Let D be an integral domain with quotient ﬁeld K and Γ a torsionless grading monoid. If D[Γ] is integrally closed, then Cl(D[Γ]) = Cl(D)⊕Cl(K[Γ]).

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Proof. This is proved in [37, Corollary 2.8]. It is an immediate consequence of Theorems 5.6 and 5.8. In analogy with integral domains, one can deﬁne the (t-)class group Cl(Γ) for any torsionless grading monoid Γ to be the group of t-invertible t-ideals of Γ under t-multiplication modulo its subgroup of principal fractional ideals. By [37, Theorem 2.9], HCl(K[Γ]) ∼ = Cl(Γ) for any ﬁeld K, and thus Cl(K[Γ]) ∼ = Cl(Γ) when Γ is integrally closed [37, Corollary 2.10]. Hence Cl(D[Γ]) = Cl(D) ⊕ Cl(Γ) when D[Γ] is integrally closed [37, Corollary 2.11]. Theorem 5.10. Let D be an integral domain and Γ a torsionless grading monoid. Then Cl(D) = Cl(D[Γ]) if and only if D[Γ] is integrally closed and Cl(Γ) = 0. Proof. This is from [37, Theorem 2.12] and follows immediately from the previous results. Example 5.11. (a) When Γ is a Krull monoid and K is a ﬁeld, then Cl(K[Γ]) = Cl(Γ) can be calculated. If Γ ⊆ F+ with Γ = Γ ∩ F+ (here, F = ⊕Zα is a free abelian group with the usual product order and prα is the natural projection map, cf. Remark 3.8(b)) and the prα |Γ ’s are distinct essential valuations of Γ, then Cl(Γ) ∼ = F/Γ ([41, Theorem 2], [54, Section 16], and [13]). For some speciﬁc calculations, see Example 6.6. (b) If R = D[Γ] is a Krull domain, then Cl(D) = Cl(D[Γ]) if and only if Cl(Γ) = 0, if and only if Γ is a factorial monoid. This need not be the case in general. For example, R = K[Q+ ] is a B´ezout domain for any ﬁeld K [54, Theorem 13.5]; so Cl(R) = 0. Thus Cl(Q+ ) = 0, but Q+ is not a factorial monoid. (c) However, even for Krull domains, we may have Cl(D) ∼ = Cl(D[Γ]) with Cl(Γ) nonzero. For example, let k be any ﬁeld, Γ = (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 0, 1) ⊆ (Z+ )4 , D = k[⊕n∈Z+ Γ], and K = qf (D). Then Cl(D) ∼ = ⊕n∈Z+ Z [49, Proposition 14.8], and thus Cl(D[Γ]) = Cl(D) ⊕ Cl(K[Γ]) = Cl(D) ⊕ Z ∼ = Cl(D). (d) Another consequence of Theorem 5.10 is that Cl(D) = Cl(D[G]) for a torsionfree abelian group G if and only if D is integrally closed. However, we always have Cl(D[F+ ]) = Cl(D[F ]) for any free abelian group F with the usual product order [8, Example 4.5]. Next one should try to compute Cl(D[Γ]) in general. As a step in this direction, we have the following result from [25]. Recall that a numerical semigroup Γ is a submonoid of Z+ with Z+ \Γ ﬁnite. So for Γ a numerical semigroup, D[Γ] is a subring of D[X] generated by monomials over D with quotient ﬁeld K(X) and D[Γ] ⊆ D[X] integral. Theorem 5.12. Let D be an integral domain with quotient ﬁeld K and Γ a numerical semigroup. Then Cl(D[Γ]) = Cl(D[X]) ⊕ P ic(K[Γ]). In particular, if D is integrally closed, then Cl(D[Γ]) = Cl(D) ⊕ P ic(K[Γ]). Proof. Let N = { X α | α ∈ Γ } and T = D∗ ; each is a multiplicatively closed subset of D[Γ]. In [25, Theorem 5], it is shown that the natural homomorphism Cl(D[Γ]) −→ Cl(D[Γ]N ) ⊕ Cl(D[Γ]T ) = Cl(D[X, X −1 ]) ⊕ Cl(K[Γ]), given by [I] → ([IN ], [IT ]), is an isomorphism. Since Cl(D[X]) = Cl(D[X, X −1 ]) for any integral domain D by Example 5.11(d) and Cl(K[Γ]) = P ic(K[Γ]) because K[Γ] is onedimensional, the result follows. For the “in particular” statement, use Theorem 5.1(b).

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37

The above result can be iterated to hold when Γ = Γ1 × · · · × Γn is a ﬁnite direct product of numerical semigroups Γi , 1 ≤ i ≤ n. Also, note that P ic(K[Γ]) can be calculated (for example, by using the Mayer-Viertoris exact sequence for (U, P ic)). It seems reasonable to ask if Cl(D[Γ]) = Cl(D[Γ]) ⊕ Cl(K[Γ]) for any integral domain D and torsionless grading monoid Γ. We next consider the natural homomorphism φ : P ic(R0 ) −→ P ic(R) given by φ([I]) = [IR]. When R is a Z+ -graded integral domain, φ is always injective (in fact, splits since P ic is a functor), and we have a nice answer for when φ is surjective. Theorem 5.13. Let R be a Z+ -graded integral domain. Then P ic(R0 ) = P ic(R) if and only if R is almost seminormal. Proof. This is proved in [15, Theorem 1]. It is easy to see that imφ ⊆ HP ic(R); so we have φ : P ic(R0 ) −→ HP ic(R) ⊆ P ic(R). We have already seen in Theorem 5.5 that φ is injective and imφ = HP ic(R) when R0 ⊆ R is an inert extension, and that P ic(R0 ) = P ic(R) if, in addition, R is almost normal. In [77], an example of a Z-graded integral domain R is given with imφ = HP ic(R). For some other results, mainly in the Z-graded case, see [18], [81], and [2]. However, kerφ is easy to describe. Theorem 5.14. Let R be a Γ-graded integral domain, and let φ : P ic(R0 ) −→ P ic(R) be the natural homomorphism. Deﬁne Γ1 = { α ∈ U (Γ) | Rα R−α = R0 } and Γ2 = { α ∈ Γ | Rα ∩ U (R) = ∅ }. Then Γ2 ⊆ Γ1 are subgroups of U (Γ). (a) kerφ ∼ = Γ1 /Γ2 . (b) Let 0 −→ A −→ B −→ C −→ 0 be a short exact sequence of abelian groups. Then there is a graded integral domain R with P ic(R0 ) ∼ = B, P ic(R) ∼ = C, ∼ and kerφ = A. Proof. These are proved in [19]. The analog of Theorem 5.13 for the class group fails, even for Krull domains. A Z+ -graded Krull domain R may have Cl(R0 ) = 0 and Cl(R) = 0 (see Example 6.6). For other results on the class group of a graded Krull domain, see [12] and [49]. We end this section with some results about P ic(A + XB[X]) and Cl(A + XB[X]). Note that A + XB[X] is almost normal (resp., almost seminormal) if and only if B is integrally closed (resp., seminormal). Also, the natural homomorphism φ : Cl(A) −→ Cl(A + XB[X]) is always injective when B is a ﬂat A-module [29, Lemma 4.2]. However, the overring hypothesis is needed in Theorem 5.15(c) for φ to be surjective [29, Corollary 4.12]. For related results on Cl(A + XB[X]), see [29] and [30]. Theorem 5.15. Let A ⊆ B be an extension of integral domains, R = A + XB[X], and S a multiplicatively closed subset of A. (a) P ic(A) = P ic(R) if and only if B is seminormal. (b) HCl(A) = HCl(R) if and only if B is integrally closed. (c) If B is an integrally closed ﬂat overring of A, then Cl(A) = Cl(R). In particular, Cl(A) = Cl(A + XAS [X]) if A is integrally closed.

38

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Proof. Part(a) is a special case of Theorem 5.13. Part (b) is from [30, Corollary 1.2], and is a special case of Corollary 5.7. Part (c) is from [29, Theorem 4.4]. The “in particular” statement is clear since AS is always a ﬂat overring of A. Let S be a saturated multiplicatively closed subset of an integral domain D. Then N = { 0 = x ∈ D | xD ∩ sD = xsD for all s ∈ S } is a saturated multiplicatively closed subset of D called the m-complement of S. We call S a splitting set if SN = D∗ . If S is a splitting set, then Cl(D) = Cl(DS ) ⊕ Cl(DN ) [8, Corollary 3.8]. For other applications of splitting sets, see [8] and [24]. If S is a splitting set, then we can explicitly calculate Cl(D + XDS [X]) and P ic(D + XDS [X]). Theorem 5.16. Let S be a splitting multiplicative subset of an integral domain D with m-complement N . Then Cl(D + XDS [X]) = Cl(DS [X]) ⊕ Cl(DN ) and P ic(D + XDS [X]) = P ic(DS [X]) ⊕ P ic(DN ). Proof. This is proved in [24, Theorem 4.3].

6

Examples

In this ﬁnal section, we have collected some of our favorite examples illustrating various divisibility properties. In each case, we use a semigroup ring construction. In Examples 6.4, 6.6, and 6.7, the semigroup rings are subrings of a polynomial ring generated by monomials, and hence are also Z+ -graded. Example 6.1. An integral domain with no atoms. Let R = Z/2Z[Q+ ]. Then R is a one-dimensional integral domain with f = X α1 +· · ·+X αn = (X α1 /2 +· · ·+X αn/2 )2 for each f ∈ R; so no element of R is irreducible. Integral domains with no atoms have been investigated in [46]. Example 6.2. An atomic integral domain which does not satisfy ACCP ([62]). Let R = K[Γ]M , where K is a ﬁeld, Γ = 1/3, 1/(2·5), 1/(22·7), . . . , 1/(2k ·pk ), . . . ⊆ Q+ (here p0 = 3, p1 = 5, . . . are the odd primes), and M = { f ∈ K[Γ] | f (0) = 0 }, a maximal ideal of K[Γ]. Then R is a one-dimensional quasilocal integral domain. In [62], it is shown that R is atomic. However, R does not satisfy ACCP since XR ⊂ X 1/2 R ⊂ X 1/4 R ⊂ . . . is a properly ascending chain of principal ideals of R. Example 6.3. An integral domain which satisﬁes ACCP, but is not a BFD ([6, Example 2.1]). Let R = K[Γ], where K is a ﬁeld and Γ = 1/2, 1/3, 1/5, . . . , 1/pk , . . . ⊆ Q+ (here pk is the kth prime). Then R is a one-dimensional integral domain which satisﬁes ACCP. However, X = (X 1/p )p and X 1/p is an atom for each prime p; so R is not a BFD. An integral domain which satisﬁes ACCP, but is not a BFD ([6, Example 2.1]). Let R = K[Γ], where K is a ﬁeld and Γ = (1/2, 1/3, 1/5, . . . , 1/pk , . . .) ⊆ Q+ (here pk is the kth prime). Then R is a one-dimensional integral domain which satisﬁes ACCP. However, X = (X 1/p )p and X 1/p is an atom for each prime p; so R is not a BFD. Example 6.4. An atomic integral domain which is not an HFD. Let R = Z/2Z[X 2 , X 3 ]. Then R is a one-dimensional Noetherian integral domain which is not seminormal. Clearly ρ(R) ≥ 3/2 since (X 2 )3 = (X 3 )2 . In fact, for any ﬁeld K, we have ρ(K[X 2 , X 3 ]) = (D(K) + 2)/2 [28, Theorem 2.4]. Thus ρ(R) = 2. Moreover,

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39

f 2 = g 3 h for irreducible f = X 3 (1 + X + X 2 ), g = X 2 , and h = 1 + X 2 + X 4 ∈ R. As an application of Theorem 5.12, for any integral domain D with quotient ﬁeld K, we have Cl(D[X 2 , X 3 ]) = Cl(D[X]) ⊕ K since P ic(K[X 2 , X 3 ]) = K as additive abelian groups. Example 6.5. A two-dimensional non-Noetherian UFD ([55]). Let R = K[G], where K is a ﬁeld and G is a torsionfree rank-two inﬁnite abelian group which is not free, but each rank-one subgroup is free. Such abelian groups exist (see [51, Section 88]). Then dim(D) = rank(G) = 2, and R is a UFD by Theorem 2.3(a). Clearly R is not Noetherian since G is not ﬁnitely generated. For each integer n ≥ 2, R[X1 , . . . , Xn−2 ] is an n-dimensional non-Noetherian UFD. Note that a one-dimensional UFD is a PID, and thus Noetherian. Example 6.6. Krull domains with speciﬁed class group. Let K be a ﬁeld and G an abelian group. One can construct a Krull domain R = K[Γ] with Cl(R) = G [41, Corollary 2]. In fact, we may localize R to obtain a quasilocal Krull domain A with Cl(A) = G. We show this explicity when G = Z/n1 Z ⊕ · · · ⊕ Z/nr Z ⊕ Zn is ﬁnitely generated. Let R = K[{ Xini , Xi Yi , Yini | 1 ≤ i ≤ r }, { Sj Uj , Sj Vj , Tj Uj , Tj Vj | 1 ≤ j ≤ n }], a subring of T = K[{ Xi , Yi | 1 ≤ i ≤ r }, { Sj , Tj , Uj , Vj | 1 ≤ j ≤ n }] generated by monomials. Then Cl(A) = Cl(R) = G, where A = RM with M = R ∩ ({Xi , Yi }, {Sj , Tj , Uj , Vj })T [13, Example 9 and Theorem 10]. Recall that for any abelian group G, there is a Dedekind domain D with Cl(D) = G [43]. Example 6.7. Independence of n-root closed and seminormal conditions ([16]). For an integral domain R, deﬁne C(R) = { n ∈ N | R is n-root closed }. Then C(R) is a multiplicative submonoid of N such that mn ∈ C(R) if and only if m, n ∈ C(R); so C(R) is generated by primes. We show that for any multiplicative submonoid T of N generated by primes, there is an integral domain R with C(R) = T . If T = N, let R be any root closed integral domain. Otherwise, let Q = {pi }i∈I be the set of primes not in T . Deﬁne R = Q[{ Xipi | i ∈ I }, { XimYjn | m ≥ 0, n ≥ 1, and i, j ∈ I }], a subring of Q[{ Xi , Yi | i ∈ I }] generated by monomials. In [16, Theorem 2.7], it is shown that C(R) = T . Note that if T = {1}, then R is seminormal, but not n-root closed for any integer n ≥ 2. Other constructions for R with C(R) = T are discussed in [21, Section 6].

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Chapter 3

Extensions of Half-Factorial Domains: A Survey by

Jim Coykendall Abstract A half-factorial domain (HFD) is an atomic domain, R, with the property that if one has the irreducible factorizations in R π1 π2 · · · πn = ξ 1 ξ 2 · · · ξ m then n = m. Half-factorial domains were implicitly studied in the case of rings of algebraic integers in a 1960 paper of L. Carlitz [11] and were subsequently abstracted and studied by A. Zaks. Since their inception as a generalization of the classical notion of unique factorization domains, half-factorial domains have been the subject of much interest in commutative algebra. This chapter will give a survey of some recent advances in the study of half-factorial domains with the emphasis on advances in the understanding of ring extension behavior of half-factorial domains.

1

Introduction and A Bit of History

The purpose of this chapter is to give an expository overview of some of the recent work in the study of half-factorial domains and their extension rings. This chapter is written as an expansion on an invited 45-minute address presented in Chapel Hill in October of 2003. The author would like to begin by expressing his gratitude to Professors Scott Chapman and Bill Smith for their kind invitation and hospitality. The concept of “half-factorial domain” was introduced implicitly in 1960 in a striking paper of L. Carlitz [11]. The paper was written as an answer to a challenge of Narkiewicz who asked for a purely arithmetic characterization of rings of integers in terms of the class number. Of course, it was well-known that a ring of algebraic integers, and more generally, a Dedekind domain, has unique factorization if and only if its class number is one. Given this interplay between the concepts of unique 46

Extensions of Half-Factorial Domains: A Survey

47

factorization and class number (and class group), Narkiewicz’ line of questioning was most natural. In 1960, Carlitz’ partial answer to this question appeared in the Proceedings of the American Mathematical Society. In our opinion, this result is one of the most succinct and beautiful results in the study of factorization in rings of integers. It is also, from at least one point of view, the paper that is the ancestor of much of the current study in the general theory of factorization in general integral domains. Additionally, this paper is one of the most cited papers in factorization theory, and the arguments utilized in the proof have made a “factorization industry” possible for large classes of domains. The proofs in the paper underscore the “right” direction in which to proceed in the more general case. The main result of this seminal paper is that if R is a ring of algebraic integers then one has unique length of factorizations if and only if the class number does not exceed 2. Because of the importance of this result and the fundamental nature of its proof, we will state this result (and demonstrate its proof) more formally later. In the interim, we will remind the reader of the “nicest” class of domains from a factorization point of view and give the current deﬁnition of a half-factorial domain as a generalization. This not only gives the historical line of reasoning, but also makes the statement of Carlitz’ result much cleaner. The result of Carlitz’ 1960 paper remained mostly untouched for about 15-20 years until Zaks generalized this “unique length of factorization” property. He published a couple of papers [47, 48] on the subject and coined the terminology “half-factorial domain.” The paper [48] is a longer work with a number of important theorems on the subject of the half-factorial property. It is interesting to note that what seems to be the current accepted deﬁnition of a half-factorial domain is actually diﬀerent from the one in Zaks’ paper. About 10 years later the real gold rush began with a paper by D. D. Anderson, D. F. Anderson, and M. Zafrullah [2]. This paper (and its sequel [3]) is a blueprint for much of the work that is taking place in modern factorization theory, and the scope of the paper far exceeded the scope of the half-factorial property. Historically, this marks the time when much activity began in the factorization theatre. In this chapter, we will concentrate on work that has been done in the theory of half-factorial domains and their extension rings. Because of the large number of results obtained in various aspects of factorization, this chapter will be far from complete. The interested reader would be well served to consult the volume [1] for a very good collection of papers on the half-factorial property, its generalizations, and factorization in general. In this chapter we use the following notation conventions: 1) R is (unless otherwise stated) an atomic domain with quotient ﬁeld K. 2) RS is the localization of R at a multiplicative set S. 3) R is the integral closure of R. 4) R is the complete integral closure of R. 5) U (R) is the unit group of R. 6) Irr(R) is the set of irreducible elements of R.

48

Coykendall 7) Cl(R) is the class group of R. 8) R[x] and R[[x]] are the rings of polynomial and formal power series over R, respectively. 9) D(G) is the Davenport constant of the ﬁnite abelian group G.

10) d(K) is the discriminant of the algebraic number ﬁeld K. 11) Sn is the symmetric group on n letters. Other notations are standard.

2

Some Preliminary Results and Deﬁnitions

Of course, the genesis of factorization theory is motivated by the domains where factorization is extremely well-behaved. For the purposes of this chapter, we will assert some control over the “badness” of the factorization behavior. That is, unless otherwise stated, we will assume that our domains are atomic (i.e., every nonzero, nonunit element of R can be written as a product of irreducible elements). We now recall the deﬁnition of unique factorization domains. Definition 2.1. A unique factorization domain (UFD) is an atomic integral domain, R, such that given the irreducible factorizations π1 π2 · · · πn = ξ1 ξ2 · · · ξm we have that: a) n = m and b) ∃ σ ∈ Sn such that for each 1 ≤ i ≤ n, πi = ui ξσ(i) for some ui ∈ U (R). There are a number of equivalent deﬁnitions of UFDs that are useful in different settings. In particular, a UFD may be more succinctly deﬁned as the class of domains where every nonzero nonunit can be expressed as a product of prime elements. From the ideal theoretic point of view, UFDs can also be characterized as the class of domains where every nonzero prime ideal contains a nonzero principal prime [32]. Unique factorization domains possess a large number of very nice properties and we will compare and contrast some of these with the properties of half-factorial domains. The class of half-factorial domains (HFDs) is, from one point of view, the nicest generalization of the class of UFDs with respect to factorization. Definition 2.2. An (atomic) domain is said to be a half-factorial domain (HFD) if given two irreducible factorizations π1 π2 · · · πn = ξ1 ξ2 · · · ξm then n = m.

Extensions of Half-Factorial Domains: A Survey

49

So, in a certain sense, an HFD is just a UFD with “half the axioms.” It is worth noting that Zaks’ original deﬁnition omitted the “atomic” assumption (and in Zaks’ paper he quickly specialized to atomic cases). Of course, it is natural to ask if the HFD condition without the atomic assumption automatically forces atomicity, but this is not the case as we will see in the next example. Without the “atomic” assumption, there is some pathology that we wish to avoid; for this reason (and since today’s common deﬁnition also includes the assumption “atomic”), we will always assume that HFD implies atomic unless otherwise indicated. In the following example from [25], we will produce a nonatomic domain that has the property that any two equal irreducible factorizations have the same length. Example 2.3. This example will give a nonatomic domain with the property that if π1 π2 · · · πn = ξ1 ξ2 · · · ξm with πi , ξj ∈ Irr(R), 1 ≤ i ≤ n, 1 ≤ j ≤ m, then n = m. The example is a domain with a unique nonprime irreducible element (up to associates) and was constructed in [25]. In this domain, the uniqueness of the irreducible assures that any two irreducible factorizations are of the same length (in fact, unique). The ease of this observation belies the delicate nature of the construction of a domain with a unique nonprime irreducible (for atomic domains with ﬁnitely many irreducibles, the so-called CK domains, see [4]). Speaking very loosely, we will outline the major steps to the construction of such a domain. Begin with a ring R that possesses a nonprime π ∈ Irr(R). One then constructs a new domain, R1 , by indexing all the irreducibles in R that are not associated to π (say our relevant irreducibles are {ξi }i∈I with I some indexing set) and building a larger domain that “forces” each ξi to become reducible. In particular, we let R1 = R[{xi }][

ξi ]i∈I , xi

with each xi a polynomial indeterminate. It is then shown that in R1 , π remains a nonprime irreducible and each ξi is reducible in R1 by construction. Of course, there may be many “new irreducibles”, but one again isolates π and constructs R2 analogously. Continuing in this fashion, we obtain a tower of rings ∞

R1 ⊆ R2 ⊆ R3 ⊆ · · ·

and we deﬁne T := i=1 Ri . It can be shown that in T , π is the unique nonprime irreducible (intuitively, uniqueness comes from the fact that every possible irreducible not associated to π is “destroyed” in the next stage of the construction). This domain has the claimed property, and it is interesting to note that this construction shows that any domain with a nonprime irreducible element can be embedded in such a domain. We remark that the general theme of this construction is very much in the spirit of the construction performed by M. Roitman in [42] where it was shown that a polynomial extension of an atomic domain is not necessarily atomic.

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With these deﬁnitions in hand, we now have the tools to succinctly state the classical result of Carlitz. We give a slightly more general statement in the spirit of the general utility of Carlitz’ proof and in the spirit of much of the research that has taken place concerning the interplay of the class number and factorization. Theorem 2.4. Let D be a Dedekind domain. If |Cl(D)| ≤ 2 then D is an HFD. If Cl(D) is torsion and D has the property that there is a prime ideal in every class, then the converse also holds. Proof. Of course, it is well-known that if Cl(D) is trivial, then D is a UFD (and hence an HFD), so we will concentrate on the case where |Cl(D)| = 2. Assume that we have the irreducible factorizations π1 π2 · · · πn = ξ1 ξ2 · · · ξm . Since a prime factor appears on the left side if and only if it appears on the right, we will reduce to the case where the factorizations above consist entirely of nonprime irreducible elements. Since |Cl(D)| = 2 and each irreducible is a nonprime we can factor (in terms of prime ideals) (i)

(i)

(i)

(i)

πi = P1 P2 and

ξi = Q1 Q2 . We now have the ideal factorization (1)

(1)

(2)

(2)

(n)

(n)

P1 P2 P1 P2 · · · P1 P2

(1)

(1)

(2)

(2)

(m)

(m)

= Q1 Q2 Q1 Q2 · · · Q1 Q2 .

Since prime ideal factorizations are unique, we have that 2n = 2m and hence n = m. On the other hand, assume that D is a Dedekind HFD and Cl(D) has a prime in every class. Since Cl(D) is torsion, every class has ﬁnite order. We ﬁrst assume the existence of a class of order n > 2 and select a prime ideal P in this class. By construction, Pn is principal and generated by some α ∈ Irr(D). We now choose a prime ideal Q in the class of P−1 and note that PQ is principal and generated by some π ∈ Irr(D) and additionally Qn is principal and generated by the irreducible β. We now consider the ideal factorizations (PQ)n = (Pn )(Qn ) which give the elemental factorizations π n = uαβ for some u ∈ U (D). Since n > 2, D is not an HFD. The only case left to consider is the case where Cl(D) is of exponent 2 and |Cl(D)| > 2. In this case, Cl(D) must contain a subgroup isomorphic to the Klein 4-group Z2 ⊕ Z2 . In a fashion similar to the above, we choose prime ideals P in the class corresponding to (0, 1), Q in the class corresponding to (1, 0) and R in the class corresponding to (1,1). It is easy to see that there are irreducibles α, β, γ,

Extensions of Half-Factorial Domains: A Survey

51

and ξ such that P2 = (α), Q2 = (β), R2 = (γ), and PQR = (ξ). The ideal factorizations (PQR)2 = (P2 )(Q2 )(R2 ) give rise to the elemental factorizations ξ 2 = uαβγ for some u ∈ U (R) and once again D is not an HFD. A couple of remarks are worth noting here. First, the original Carlitz result was couched strictly in terms of algebraic number theory. His result was that a ring of algebraic integers, R, is an HFD if and only if the class number of R does not exceed 2. But the seeds of the original proof pointed the way towards much more generality. The second remark is that although the above theorem is slightly more general than the original statement of Carlitz, it is not as general as possible either. There has been much work done in more general cases and the interested reader is encouraged to consult [5, 6, 14].

3

“Good” Properties of UFDs versus HFDs

As a generalization of UFDs, one would like to understand what, if any, “nice” behaviors of UFDs are shared by HFDs. UFDs possess a number of nice properties (especially in the case of localizations and polynomial extensions). Some of these classical results are that if R is a UFD then so is R[x], and the localization RS . Both of these central results for UFDs can be obtained from the following nice ideal-theoretic characterization of UFDs. Theorem 3.1. A domain R is a UFD if and only if every nonzero prime ideal in R contains a (nonzero) prime element. So a UFD is the “multi-dimensional” generalization of a principal ideal domain (PID). Generally speaking, HFDs do not have a readily apparent ideal theoretic characterization. In the case of rings of algebraic integers (and more generally, Dedekind and Krull domains with certain restrictions on the class group), HFDs do have a nice ideal-theoretic characterization (evidenced by Carlitz’ theorem, for one). A sweeping ideal-theoretic characterization for general HFDs seems to be elusive (if indeed such a characterization exists), and we cannot resist the following question. Question 3.2. Find a general ideal-theoretic characterization of HFDs in the spirit of the characterization of UFDs, if it exists. Below we produce a table that records some of the classical results about UFDs and compares them with what is known about the analog property for HFDs. It should be explicitly stated that the entries “no” in the table mean, more precisely, “not always.”

52

Coykendall Table of Inherited Factorization Properties R is a: UFD HFD

R[x] Yes No

R[[x]] No No

R Yes No

R Yes No

RS Yes No

We make a couple of remarks about the table. First it should be noted that if R is a UFD then the (complete) integral closure of R coincides with R, so this result is trivial (but it does indeed invite the question as to whether the (complete) integral closure of an HFD is still an HFD). Also, although it is possible for a power series ring over a UFD to lose unique factorization [43], if R is a one-dimensional UFD (i.e., a PID) then R[[x1 , x2 , · · · , xn ]] is a UFD for all n. If the reader hopes for the half-factorial property to be preserved in standard extensions, then this table will look quite depressing. However, we remark that there are indeed positive results with (in some cases) rather minimal restrictions placed on the HFD. We will look at some of the positive results and some of the pathologies in the next sections.

4

Localizations

In our view, one of the most disturbing results in the table from the previous section is that if R is an HFD, then it is not necessarily true that RS is an HFD (where S is a multiplicatively closed subset of R). The proof that the UFD property holds in localizations is very elementary (the characterization of UFD via every nonzero prime contains a nonzero principal prime is very useful in this regard). Of course, one would not expect general overrings to preserve the half-factorial property since unique factorization is not preserved in general overrings. A quick example of this is the ring K[x, y] where K is a ﬁeld. This ring is a UFD (and hence an HFD), but the overring K[x, xy , xy2 , xy3 , · · · ] is not even atomic. In this section we take a brief look at some examples, positive (and negative) results, and generalizations of HFDs in the case of localizations and more general overrings. We will begin by giving an example of an HFD which possesses a localization which does not have the half-factorial property. This example and other very nice results in this spirit may be found in the papers [7], [8], and [9]. Example 4.1. We begin by noting that one can construct a Dedekind domain, D, with class group Z6 such that the prime ideals of D are restricted to the classes corresponding to {1, 2, 3} ⊆ Z6 (a more general theorem on the construction of such Dedekind domains can be found in [27]). As it turns out, such a Dedekind domain is necessarily an HFD. We now consider the set Γ = {Q|Q is a prime ideal of D with [Q] = 1 or [Q] = 2}.

It is easy to see that if P is a prime ideal such that [P ] = 3 then P ⊆ Q∈Γ Q. Let t ∈ P \ ∪Q∈Γ Q, T = {1, t, t2 , · · · } and form the localization R = DT . We make the following observations. 1. R is a Dedekind domain.

Extensions of Half-Factorial Domains: A Survey

53

2. Cl(R) ∼ = Cl(D)/(ker(τ )) where τ is the natural map from Cl(D) −→ Cl(R) deﬁned by τ : [I] −→ [IR]. It follows ([7]) that Cl(R) ∼ = Z6 /Z2 ∼ = Z3 and the primes of R are in the nontrivial classes of Z3 . With this reduction, the technique of the Carlitz proof can be applied to show that R is not an HFD. This example motivates the following deﬁnitions (once again from [7] and [8]). Definition 4.2. Let R be an integral domain. We say that R is a a) Locally half-factorial domain (LHFD) if every localization of R is an HFD. b) Strong half-factorial domain (SHFD) if every overring of R is an HFD. Of course, an SHFD is an LHFD, but the converse does not hold in general. Here are some nontrivial examples of these types of domains. Example 4.3. Let R be a Dedekind domain with Cl(R) = Zp for a prime integer p, such that all primes of R are contained in the class {1}. Then it can be shown that any (nontrivial) localization RS has trivial class group and hence is a PID. The upshot is that R is a LHFD, and since R has torsion class group, every overring is a localization and hence R is SHFD. We note that this example can be tweaked to the case where Cl(R) is not torsion to give an example of an LHFD that is not an SHFD (the key in this augmentation of the example is that if the class group is not torsion, then there are overrings which are not localizations). Theorem 4.4. Let D be a Dedekind domain satisfying one of the following. a) |Cl(D)| ≤ 5. b) Cl(D) = Zpn . c) Cl(D) = ⊕ki=1 Z2 . Then the following statements are equivalent. 1. D is an HFD. 2. D is a SHFD. 3. D is a LHFD. We conclude this brief section with a couple of observations. The ﬁrst is that the SHFD property is very strong in the sense that if R is an SHFD then the Krull dimension of R is no more than 1 (the introductory example at the beginning of the section of the overring pair K[x, y] ⊆ K[x, xy , xy2 , xy3 , · · · ] illustrates very plainly the hazards of domains where the Krull dimension is 2 or more). Finally, if G is a ﬁnite abelian group NOT of the form a), b), or c) in the theorem, then there exists a Dedekind domain D with Cl(D) = G such that D is not an HFD but every proper overring is an HFD. Some interesting generalizations of HFDs (e.g., “congruence” HFDs and kHFDs) are studied in [15].

54

5

Coykendall

Polynomial Extensions

One of the most famous and useful properties of UFDs is the fact that unique factorization in a coeﬃcient ring R makes a smooth transition to the polynomial ring R[x]. This result, originally due to Gauss, makes polynomial computations (over a well-behaved coeﬃcient ring) quite tractable. Theorem 5.1. If R is a UFD, then R[x] is a UFD. The standard proof of this theorem uses the notion of “content” of a polynomial (recall that the content of a polynomial f ∈ R[x] is the ideal of R generated by its coeﬃcients), but a more streamlined proof can be given using the ideal-theoretic characterization of UFDs (any nonzero prime contains a nonzero prime element). This theorem is a classic and begs the analogous question for HFDs. In [48], Zaks proved the following generalization of this theorem to a special class of HFDs. Theorem 5.2. If R is a Krull domain, then R[x] is an HFD ⇐⇒ |Cl(R)| ≤ 2. Unfortunately, Zaks’ result does not generalize to all HFDs as the following example will show. √ √ √ Example 5.3. Consider the order R := Z[ −3] ⊆ Z[ ω] where ω = 1+ 2 −3 . The √ fact that R is an HFD is easily shown (for example, see [48]). Intuitively, Z[ −3] is an HFD because the integral √ closure, Z[ω], is a UFD and a norm argument shows that every irreducible in Z[ −3] is prime in Z[ω]. Now consider the factorization (2x − 2ω)(2x − 2ω) = (2)(2)(x2 − x + 1) where ω is the complex conjugate of ω. Clearly, the right hand side of the above equation has at least three irreducible factors (in fact, precisely three), and so if both of the factors on the left are ir√ reducible, then Z[ −3][x] is not an HFD. And indeed this is the case, as any irreducible factorization of 2x − 2ω (respectively 2x − 2ω) must consist of a constant √ in Z[ −3] of norm 2 or 4, and a degree one polynomial factor. It is easy to check that there is no such factorization. Although the above example demonstrates the impossibility of a full generalization of Gauss’ theorem to general HFDs, it does provide some insight. A slight tweaking of this example to a slightly more general setting allows us more to glean the following theorem [17]. Theorem 5.4. If R[x] is an HFD, then R is an integrally closed HFD. The proof of this theorem may be found in [17], but the example above holds almost the entire content of this theorem. The interested reader is encouraged to “play” with the√above example; it is easily seen that the failure of the half-factorial property in Z[ −3] is a direct consequence of the fact that the coeﬃcient ring is not integrally closed. As a corollary to the previous theorem (coupled with the results from [48]), one can classify all Noetherian polynomial HFDs.

Extensions of Half-Factorial Domains: A Survey

55

Corollary 5.5. Let R be Noetherian. The following conditions are equivalent. a) R[x] is an HFD. b) R[x1 , x2 , · · · , xn ] is an HFD for all n ≥ 1. c) R is a Krull domain with |Cl(R)| ≤ 2. Proof. It is known from [48] that if R is a Krull domain, then R[x] is an HFD if and only if |Cl(R)| ≤ 2, which gives c) implies a). For a) implies c), we assume that R[x] is an HFD. This means that R is integrally closed, and since R is Noetherian, R is a Krull domain. We apply Zaks’ result again to obtain c), and hence a) and c) are equivalent. Certainly b) implies a) and to get the converse, we merely note that if R is a Krull domain, then so is R[x1 , · · · , xn ] and what is more, the class number is stable upon the adjunction of any ﬁnite number of indeterminates.

6

Power Series Extensions

Power series extensions are oftentimes more problematic than polynomial extensions. Many classical results that hold (in general commutative algebra) sometimes fail wildly in the setting of power series. Passing to a completion (even an x–adic one) is sometimes a bit tricky and some nice properties may be lost. For the sake of perspective, a striking example of this phenomenon is in dimension theory. A classical result for polynomials is that if the (Krull) dimension of a ring (dim(R)) is ﬁnite (say dim(R) = n), then so is the dimension of R[x] (in particular, if dim(R) = n then n + 1 ≤ dim(R[x]) ≤ 2n + 1). This is wildly untrue (and in fact, from the non-Noetherian point of view, usually untrue) in the case of power series rings. In fact, there are 0−dimensional rings whose power series extensions are inﬁnite dimensional. Of course, it should be noted that there are instances in which the behavior of formal power series is at least as nice as the analog behavior in polynomials. One example where this occurs is in the case of the passage of the unit group of a ring to polynomials and power series. It is well-known that U (R) = U (R[x]) and that the set of units in R[[x]] is the set of power series f (x) ∈ R[[x]] such that f (0) ∈ U (R). An even more striking example of good power series behavior involves (semi-) quasi-local rings. It is a central result that R is quasi-local (resp., semi-quasi-local) if and only if R[[x]] is quasi-local (resp. semi-quasi-local). The analogous result for polynomials is not true, since the ring R[x] is never semi-quasi-local. But such nice behavior of power series rings relative to the polynomial case is the exception and not the rule in practice. From a factorization point of view we can ﬁnd bad behavior as well. For example, there are UFDs which have non-UFD power series extensions [43]. As has been pointed out earlier, if R[x] is an HFD, then R is integrally closed. Since for the polynomial case, the coeﬃcient ring being integrally closed is necessary for R[x] to have a chance at the half-factorial property, intuitively one would (perhaps) expect that R[[x]] being an HFD would demand at least this much. In light of this “intuition” let us revisit an earlier example.

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√ √ √ Example 6.1. Consider the order R := Z[ −3] ⊆ Z[ ω] where ω = 1+ 2 −3 . As noted before, R is an HFD with R[x] failing to be an HFD. We again look at the factorization in Z[x] that demonstrated the loss of the half-factorial property:

(2x − 2ω)(2x − 2ω) = (2)(2)(x2 − x + 1). A close inspection of this factorization shows that, in contrast to the Z[x] case, this does not deny the half-factorial property in Z[[x]]. The reason for this is that the element x2 + x + 1, although an irreducible in Z[x], is a unit in Z[[x]]. Hence up to units, the factorizations on both sides of the above equation are of length 2. In fact, in the example above, R[[x]] is, in fact, an HFD. Much more general situations are considered in [22]. We will outline a proof that the above example is a (nonintegrally closed) example of an HFD such that R[[x]] is an HFD. A more thorough treatment of this phenomenon can be found in [22]. Theorem 6.2. Let R be a 1−dimensional domain with integral closure R and conductor I. Also suppose that every nonzero coset of R/I can be written in the form u + I with u ∈ U (R). Then R is a UFD implies that R[[x]] is an HFD. Before beginning this proof, we note that, in particular, the hypotheses apply to √ the ring√Z[ −3]. Indeed, the integral closure of this ring is (the UFD) Z[ω] where √ ω = 1+ 2 −3 and the conductor of Z[ω] to Z[ −3] is the ideal 2Z[ω] = P. Since this conductor ideal is prime, the quotient ring Z[ω]/2Z[ω] is isomorphic to F4 , the ﬁeld of 4 elements. It is easy to see that the nonzero cosets can be written in the form 1 + P, ω + P, and ω 2 + P. We will now give an outline of the proof of the theorem. Proof. We claim that every irreducible element of R[[x]] is again irreducible in R[[x]]. If not, then we factor an irreducible f ∈ R[[x]] as gh with g, h ∈ R[[x]]. First note that if both h and g are in I[[x]], then this is a direct contradiction. We begin by assuming that neither g nor h is an element of I[[x]]; we write g(x) = a0 + a1 x + · · · + ak−1 xk−1 + xk (bk + bk+1 x + · · · ) where bk is the ﬁrst term not in the conductor I. We write bk = u1 + Ig with u1 a unit of R and Ig ∈ I. Collapsing notation we write g = g + xk (ug + Ig ) where ug is a unit power series with constant coeﬃcient u1 ∈ U (R) and g ∈ I[[x]]. In a similar fashion, we write h = h + xm (uh + Ih ) with uh a unit power series with constant coeﬃcient v1 ∈ U (R), h ∈ I[[x]], and Ih ∈ I. Note that gh =gh + gxm (uh + Ih ) + hxk (ug + Ig )+ + xk+m (ug uh + ug Ih + uh Ig + Ig Ih ).

Extensions of Half-Factorial Domains: A Survey

57

Since h, g, Ig , and Ih are in I[[x]] and g and h are in R[[x]], we obtain that ug uh is an element of R[[x]]. As a consequence we note that since g = g + xk (ug + Ig ) we have that uh g = uh g + xk (uh ug + uh Ig ) and so uh g ∈ R[[x]]. Similarly, we obtain that ug h ∈ R[[x]]. We now note that f = gh = (uh g)(ug h)(uh ug )−1 and we have a contradiction. The ﬁnal case to consider is the case when precisely one of the factors, g or h, is an element of I[[x]]. We will assume without loss of generality that g = g is the factor in I[[x]]. In this case, uh g ∈ R[[x]] and we have f = gh = (uh g)(u−1 h h) which is again a contradiction. Now that we have established that every irreducible in R[[x]] is irreducible in R[[x]], we observe that if we have two irreducible factorizations in R[[x]] f1 f2 · · · fn = g1 g2 · · · gm then each fi , gj is irreducible in R[[x]] which is a UFD (since R is a 1 dimensional UFD and hence a PID). Hence n = m. We cannot resist highlighting what we believe to be an interesting implication of this example. Corollary 6.3. There exist HFDs R such that the half-factorial property is lost in R[x] and regained in R[[x]]. √ Proof. If one considers the ring R := Z[ −3], then this is an HFD such that R[x] is not an HFD, since R is not integrally closed. Nonetheless, the above result shows that R[[x]] is indeed an HFD. Here is a ﬁnal observation along these lines. Corollary 6.4. If R[[x]] is a UFD, then R and R[x] are UFDs. If R[[x]] is an HFD, then R is an HFD, but R[x] is not necessarily an HFD. Proof. First we note that any irreducible element of R remains irreducible in R[[x]]. Indeed, if π ∈ Irr(R) and π = f (x)g(x) with f (x), g(x) ∈ R[[x]] then we write ∞ i i f (x) = ∞ i=0 ai x and g(x) = i=0 bi x and factor ∞ ∞ π=( ai xi )( bi xi ). i=0

i=0

It is immediate that we get the factorization

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π = a0 b 0 and since π ∈ Irr(R), either a0 or b0 is a unit in R, forcing either f (x) or g(x) to be a unit in R[[x]]. This establishes our claim. Given this claim, we consider the following factorization in R π1 π2 · · · πn = ξ1 ξ2 · · · ξm with each πi , ξj ∈ Irr(R). If R[[x]] is an HFD, then all of the above irreducible elements remain irreducible in R[[x]] and since R[[x]] is an HFD, we must have n = m and R is an HFD. Additionally, if R[[x]] is a UFD, then every irreducible element of R[[x]] is prime, and hence each πi and ξj is a prime element of R[[x]] (and it is easy to see that the elements are therefore prime as elements of R). Hence R is a UFD. We now have that if R[[x]] is a UFD (respectively HFD) then R is a UFD (respectively HFD). The fact that if R[[x]] is a UFD implies that R[x] is a UFD follows from the previously mentioned result of Gauss, and the absence of the analogous result for HFDs is demonstrated by the above example. We remark here that one might notice that there is a similar result for the case of (semi)quasi-local rings. That is, if R[[x]] is (semi)quasi-local then R is (semi)quasi-local, but R[x] is not. One thing to contrast, however, is that R[x] is never (semi)quasi-local. As was noted before, the above technique is used with some success in [22] in a more general setting. In the previous section, a classiﬁcation of all Noetherian polynomial HFDs was given, but a general classiﬁcation for the power series case (even for Noetherian rings) remains elusive. In hindsight, this might not be too surprising since a good characterization of when the UFD property is preserved in power series extensions is not known. One of the best theorems in this sense is the result that states that if R is a PID then R[[x1 , x2 , · · · , xn ]] is a UFD for all n ≥ 1. The class of PIDs is precisely the class of one-dimensional UFDs and it should be noted that the classical example of a UFD, R, such that R[[x]] is not a UFD is a two-dimensional Noetherian domain. This demonstrates, in a certain sense, that one does not have to travel far from the class of PIDs to ﬁnd examples of bad factorization behavior in power series extensions.

7

Extensions of the Form A + xB[x] and A + xI[x]

A key source of examples in commutative algebra is the so-called “D + M” construction. Recently the D + M construction has shown that its utility extends to the study of factorization (often as a rich source of “bad” factorization behavior). A useful illustration of this is the ring Z + xQ[x]. This ring is one of the simplest examples of the previous statement. It is nonatomic, since the element x is a nonunit that possesses no ﬁnite factorization into irreducibles. To see this, we note that if x = π1 π2 · · · πn is an irreducible factorization of x in Z + xQ[x], then a degree argument, coupled with the fact that Q[x] is a PID, shows that precisely one of the irreducible factors

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(say π1 without loss of generality) must be an associate of x as an element of Q[x]. We write π1 = qx with q ∈ Q and note that π1 = 2( q2 )x, which contradicts the irreducibility of π1 in Z + xQ[x]. The constructions A + xB[x] and A + xI[x], specializations of the standard “D + M” construction, are gold mines for interesting (especially nonunique) factorizations [10] (note that it is assumed that A ⊆ B are domains and in the second case that I ⊆ A is an ideal). Indeed, there is no way that an A + xB[x] construction can ever be a UFD unless A = B and A is a UFD (in fact, unless A = B and A is completely integrally closed, A + xB[x] is not completely integrally closed). Also note that if I is a proper nonzero ideal of A, then x is almost integral over A + xI[x] and hence A + xI[x] is not a UFD either. For the A + xI[x] construction, assuming that A is an UFD is a very natural starting point. With respect to this condition, Gonzalez, Pellerin and Roberts proved the following nice theorem [30]. Theorem 7.1. Let A be a UFD and R := A + xI[x] with I a proper nonzero ideal of A. Then R is an HFD if and only if I is a prime ideal of A. Actually much more is shown in this paper. They show, in fact, that R is atomic and that ρ(R) is ﬁnite if and only if I = P1 · · · Pk with the Pi ’s being noncomparable primes with at most one of them nonprincipal (and if this is the case then ρ(R) = k). This result has also been extended to the case where A is a Dedekind domain. It has been shown that if |Cl(A)| = ∞ then ρ(R) = ∞. In the case where |Cl(A)| < ∞ and I is a product of k distinct principal primes, we have the following from [37]. Theorem 7.2. Let A be a Dedekind domain with ﬁnite class group and let I ⊆ A , be the product of k distinct principal primes. Then ρ(A + xI[x]) = k + D(Cl(A)) 2 where D(Cl(A)) is the Davenport constant of Cl(A). Here we declare that the Davenport constant is 0 if the class group is trivial. We now present some recent advances in the spirit of the work of Gonzalez, Pellerin, and Roberts. This next result can be found in [23] and is an extension of some of the work of Kim [33]. Theorem 7.3. If K is a ﬁeld and B is any domain containing K then K + xB[x] is an HFD if and only if B is integrally closed. We now look at the A + xI[x] construction. Question 7.4. If K ⊆ B is an extension of domains with K a ﬁeld and B a UFD, and P ⊆ B is a prime ideal, then is K + xP[x] an HFD? To help in this, we record the following result [23]. Theorem 7.5. Let A ⊆ B be an extension of domains with A a UFD and let I ⊆ B be a proper ideal. If I contains an irreducible element π ∈ B such that π 2 has no nontrivial factor in A, then A + xI[x] is not an HFD. Proof. (πx2 )(π 2 x) = (πx)(πx)(πx).

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Corollary 7.6. Let K ⊆ B be an extension of domains with K a ﬁeld and let I ⊆ B be a proper ideal. If I contains an irreducible element of B, then K + xI[x] is not an HFD. This answers the question raised above. Once again, we see the usefulness of the ideal theoretic characterization of UFD. This time, the fact that the prime ideal xI[x] ⊆ K + xI[x] contains no prime element helps us to do something “bad”.

8

Integral Closure

As UFDs are (completely) integrally closed, a very natural question is “are HFDs integrally closed?” Earlier examples quickly show that√this is not the case; for example, we have seen the concrete example of the ring Z[ −3] which is a nonintegrally closed HFD. Along this line of thought, a natural next question is, “is the integral closure of an HFD still an HFD?” This question was answered in the negative by the present author in [21]. We will outline this result (as the construction is somewhat intricate). But we will note that, in some sense, this counterexample is a bit unsatisfactory. More speciﬁcally, the reason why the integral closure of the HFD presented in [21] fails to have the half-factorial property is that it fails to be atomic. From the point of view of the current accepted deﬁnition of HFD, this example is a bona ﬁde counterexample, but only because of the loss of atomicity. As one will see from the outline of the construction, it turns out that any two irreducible factorizations in the integral closure of this domain are of the same length. (So from the point of view of the original discarded deﬁnition of Zaks, the integral closure of this HFD is again an HFD.) We will outline this example in a couple of steps and give the briefest idea of how it works. This construction will deliver a speciﬁc example of an HFD whose integral closure does not have the half-factorial property, but it should be noted that this construction can be generalized (for example, the choice of the value group and residue ﬁelds are only for computational convenience). 1. We begin with the semigroup ring F2 [x; Q+ ], and we denote the maximal ideal generated by the “monomials” by M (i.e., M is the ideal generated by the set {xα }α∈Q+ ). 2. We now form the localization R := F2 [x; Q+ ]M . This localization is a 1− dimensional nondiscrete valuation domain with value group Q and residue ﬁeld F2 . 3. Abusing the notation, we let M ⊆ R be the maximal ideal and consider R1 := F2 + tM[t] where t is a polynomial indeterminate. It is worth noting at / R1 . It is this step that the element t ∈ / R1 ; additionally for all α ∈ Q+ , xα ∈ also worth noting that R1 is integrally closed. Indeed, its integral closure is certainly contained in F2 + tR[t] since R[t] is integrally closed and F2 + tR[t] is integrally closed in R[t] (the integral closure of F2 in R is F2 ). It is now easily seen that no element of (F2 + tR[t]) \ (F2 + tM[t]) is integral over F2 + tM[t]. 4. In this step, we simplify by localizing at the ideal tM[t]. The eﬀect of this is to get rid of irreducible “polynomials” that may crop up in the ring R1 and

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clariﬁes factorization behavior. We let R2 := (F2 + tM[t])tM[t] and note that this is an HFD. Indeed, if π ∈ R2 is irreducible, then π ∈ tM[t]R2 , and this observation allows us to conclude that every irreducible in R2 can be written in the form u(xα1 t + 2 xα2 t2 + · · · + n xαn tn ) with each u a unit in R2 , i ∈ U (R2 ) {0}, and each αi ∈ Q+ . At this step it is fairly straightforward to verify that R2 is an HFD. Indeed, a general nonzero nonunit in R2 assumes the form (with notation as above) u(xα0 tm + 1 xα1 tm+1 + · · · + k xαk tm+k ). In this form it is easy to see via a “degree argument” that the above element has at most m irreducible factors (since no rational power of x is an element of R2 ). The fact that if m > 1, the above element actually has a factorization with m elements follows from the fact that the original valuation domain from earlier in this construction was nondiscrete. Note that if α = min(α0 , α1 , · · · , αk ), then we can factor the above element u(xα0 tm + 1 xα1 tm+1 + · · · + k xαk tm+k ) α

= u(x 2(m−1) t)m−1 (xα0 − 2 t + 1 xα1 − 2 t2 + · · · + k xαk − 2 tk+1 ). α

α

α

Of course this element may have many factorizations, but the above shows that each has length m. 5. The problem that we encounter now is that the domain from the previous step, R2 , is integrally closed, as it is a localization of the integrally closed domain, R1 . In particular, the strategy is to make the elements {xα }α∈Q+ integral and to ensure that the integral closure of our target domain is not atomic. To continue our construction we wish to adjoin elements to R2 that make this happen (while not destroying the half-factorial property). With this in mind, we consider R3 := (F2 + tM[t])tM[t] [x + t], obtained by adjoining the element x + t of the quotient ﬁeld of R2 to R2 . In the ring R3 there are two prime ideals worth noting. The ﬁrst is the prime ideal P = (x + t) (it is fairly easy to show that x + t is a prime element of R3 ) and second is the prime ideal Γ which is the extension of the maximal ideal tM[t]R2 to our new domain R3 . 6. We now wish to, once again, simplify for the purposes of factorization. We let R4 := (F2 + tM[t])tM[t] [x + t](P S Γ)c . By construction, the only prime ideals of R4 are (the extensions of) P and Γ. A simple computation shows that the only nonunits of R4 are of the form (x + t)n h where h can be thought of as an element of R2 (up to units). If we consider any irreducible factorization of (x + t)n h, we note ﬁrst that since x + t is a prime element of R4 , we can ignore the eﬀect of the (x + t)n factors. An argument similar to the proof that any nonunit in R2 has the half-factorial property (by counting the multiplicity of t) applies here and so R4 is an HFD.

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Coykendall 7. To see that the integral closure of R4 is not an HFD (more speciﬁcally, not atomic) note that 1 1 tx1+ n xn = tx 1

and so x n is in the quotient ﬁeld of R4 . Also note that 1

1

(x n )2n + (x + t)(x n )n + xt = 0 1

and so x n is a root of the monic polynomial Y 2n + Y n (x + t) + xt ∈ R4 [Y ]. It is easy to see that the existence of the factorizations 1

x = (x n )n for all n deny the atomicity of the integral closure of R4 . Purists (those who would like to follow the original concept of HFDs from Zaks) will be understandably displeased with this example, for, in a certain sense, it is cheating. Along lines similar to the above, it can be shown that the integral closure of R4 contains every (positive) rational power of x and every positive integral power of t and hence is a localization of a polynomial ring over a (nondiscrete) valuation domain. As a consequence, it can be shown that any two irreducible factorizations of the same element are of equal length. The question still remains as to whether or not the integral closure of an HFD is again an HFD if the integral closure remains atomic.

9

Orders in Rings of Integers

Despite the title of this chapter, we take a step back and look at the dual notion of underrings of half-factorial domains (here, by “underring” of a ring T , we mean a ring R ⊆ T such that R and T share the same quotient ﬁeld). Much of the study here is motivated by questions concerning orders in rings of algebraic integers. We brieﬂy recall that an order in a ring of integers is a ring R such that the integral closure R is the full ring of algebraic integers. The study of the factorization behavior is well-motivated by questions in algebraic number theory and is important in a number of applications including representation of integers by quadratic forms. For an interesting example of the ramiﬁcations of the half-factorial property with respect to the representation of integers by quadratic forms applied to ﬁnite geometry, see [24] for an example. Much of the theory of factorization has its roots (and is still motivated by) questions from number theory. In particular, Carlitz’ paper was motivated by a number-theoretic question of Narkiewicz. Carlitz’ result gave an interpretation of the half-factorial property in terms of the class group, which is a subject that has been of interest in algebraic number theory for a very long time. In the spirit of factorization and the class group, here is a very famous open conjecture due to Gauss that has resisted solution for quite some time. Conjecture 9.1. There is an inﬁnite number of real quadratic UFDs.

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Despite the apparent diﬃculty of Gauss’ conjecture, we oﬀer another conjecture along these lines. Conjecture 9.2. There is an inﬁnite number of real quadratic HFDs that are not UFDs. The curious reader may at this juncture wonder about the status of the questions for the imaginary case. In light of the fact that the Carlitz result allows us to completely determine the half-factorial property via the class number (with class number one being a UFD and class number 2 being a non-trivial HFD), we present a couple of results that answer this question. The following result is due to H. Stark [44]. Theorem 9.3. Let R be the ring of integers of an imaginary quadratic ﬁeld. Then |Cl(R)| = 1 if and only if the ﬁeld discriminant is −3, −4, −7, −8, −11, −19, −43, −67, or −163. The proof of the fact that the rings of integers corresponding to the ﬁeld discriminants listed above are all UFDs is fairly straightforward. The converse of the theorem was only completely solved in the late 1960s, when Stark showed that there was no tenth imaginary quadratic ﬁeld. A result from analytic number theory states that there are only ﬁnitely many totally complex abelian extensions of Q of given degree and class number ([35]). This makes the existence of only ﬁnitely many imaginary quadratic HFDs immediate. Due to another result of Stark [45] we can list all imaginary quadratic HFDs that are rings of integers. Theorem 9.4. Let R be the ring of integers of an imaginary quadratic ﬁeld. Then |Cl(R)| = 2 if and only if the ﬁeld discriminant is −15, −20, −24, −35, −40, −51, −52, −88, −91, −115, −123, −148, −187, −232, −235, −267, −403, or −427. It is interesting to note that before this list was complete, an initial upper bound was set for the size of the discriminant of imaginary quadratic ﬁelds K with the class number equal to 2. This bound was |d(K)| < 101030 . Stark’s paper [45] found all of the ﬁelds, and it is interesting to note that, in fact, |d(K)| ≤ 427. There is a striking diﬀerence to consider when looking for HFDs that occur as orders in rings of algebraic integers, for unlike UFDs, HFDs do not have to be integrally closed. UFDs that occur as orders must occur as the full ring of algebraic integers (and in some sense, this may make them easier to ﬁnd, since the tools generally used in this vein become more unwieldy when one leaves the assumption “integrally closed” behind). The upshot of all this is that, in searching for HFDs in rings of integers, we do not have to restrict to the full ring of integers; we can consider proper orders. When considering which orders may contain HFDs, the following theorem is useful. It demonstrates that one can restrict the search for HFDs to orders inside rings of integers which are HFDs. It is worth noting that the following theorem is a positive answer to the question, “Is the integral closure of an HFD again an HFD?” for number theorists. Theorem 9.5. Let R be an order in a ring of algebraic integers with integral closure R. If R is an HFD, then R is an HFD.

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If the search for HFDs in imaginary quadratic ﬁelds is augmented to include the nonintegrally closed case, we are done quickly. A proof of the following theorem using norms can be found in [19]. √ Theorem 9.6. The ring Z[ −3] is the unique nonintegrally closed imaginary quadratic HFD. Since we know that any nonintegrally closed HFD that is an order in an imaginary quadratic ﬁeld must appear as a subring of one of the 27 imaginary HFDs (the 9 UFDs and 18 nontrivial HFDs listed above), we could prove this theorem by restricting to searching for subrings of these 27 rings of integers. It is of interest to note, however, that the above theorem was known before it was known that HFDs as orders must be subrings of integrally closed HFDs (see [19] for a direct result or [31] for at least an implicit one). The proof of this theorem in [19] utilizes the norm, and since in the case of imaginary quadratic ﬁelds, the norm is positive deﬁnite, the proof is rather quick and clean (it basically depends only upon representing certain integers via the norm form). But it should be noted that although perhaps elegant in this case, generalizing this technique to more general ﬁelds (even real quadratic ﬁelds) is fraught with diﬃculties because of the diﬃculty determining integer representations when the form may take on negative values (and it also should be pointed out that even for relatively small degree ﬁelds, the norm form becomes quite large and hard to handle very quickly). Despite the computational diﬃculties inherent in actually applying the norm, the above discussion and results beg the question as to how much information can be obtained by looking at factorizations from the point of view of norms. We begin by recalling a concept known as elasticity. Definition 9.7. Let S be an atomic set (that is, every element of S can be factored into irreducible elements of S). We deﬁne the elasticity of S to be n |α1 α2 · · · αn = β1 β2 · · · βm } m where αi , βj are irreducible elements of S for all i, j. ρ(S) = sup{

Basically, elasticity is a global measure of how long ratios of diﬀering (yet equal) factorizations in S can be. The deﬁnition that we give slightly generalizes the deﬁnition of elasticity of a domain given in [46] and studied in (among others) [12, 28, 29, 30]. Here is a theorem that has appeared in a paper by Valenza [46] and was improved by Narkiewicz [36]. Theorem 9.8. Let R be a ring of algebraic integers. Then ρ(R) = D 2 where D is the Davenport constant of the class group of R if R is not a UFD (and if R is a UFD, then ρ(R) = 1). We remind the reader that the Davenport constant is the invariant of a ﬁnite abelian group G that is the smallest positive integer, n, such that any sequence of n (not necessarily distinct) elements of G has a subsequence that sums to 0. The Davenport constant has been studied rather extensively in the realm of pure group theory and has become a topic of much interest in commutative algebra because of its natural connection with factorization. There are many remaining open questions relating to the Davenport constant (e.g., “Is there a closed form representation of

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the Davenport constant?”). The answers to many of these questions would have far-reaching ramiﬁcations in factorization theory. For a more in-depth look at the theory of the Davenport constant and some of its applications, see [12, 13, 26]. We now produce a theorem that is useful in determining factorization behavior in rings of integers. This theorem is a hybrid of a couple of theorems that can be found in [16] and [20]. Theorem 9.9. Let F/Q be a Galois extension, R the ring of algebraic integers of F , and S the multiplicative set of integral norms from R to Z. Then R is an HFD if and only if ρ(S) = 1, and additionally, R is a UFD if and only if S has unique factorization into irreducibles. It should be noted that, in general, for Galois extensions we have that ρ(R) ≥ ρ(S). Perhaps this is not too surprising since one would expect that the set of norms would lose some of the factorization information that is contained in R. It is a subtle point that although ρ(R) ≥ ρ(S) in general, one cannot lose “too much” information in the sense that ρ(R) = 1 if and only if ρ(S) = 1. We would also like to highlight that the “Galois” assumption is very necessary here. In the non-Galois case, an example of a non-Galois extension with normal closure having Galois group S5 can be produced to show the failure of this theorem in general. Mainly because of its more complicated unit structure, the real case cannot be disposed of as easily as the imaginary case. We supply the following conjecture. Conjecture 9.10. There are inﬁnitely many real quadratic HFDs (even inside √ Z[ 2]). This result makes use of the “boundary map” which we deﬁne below. Definition 9.11. Let R be an HFD with quotient ﬁeld K. We deﬁne the boundary map ∂R : K \ {0} −→ Z via ∂R (α) = n − k ···πn with each πi , ξj ∈ Irr(R). In the where α ∈ K \ {0} is of the form α = πξ11πξ22···ξ k case that R = K we declare that ∂R ≡ 0.

The boundary map is a homomorphism from the multiplicative group K \ {0} to the additive group Z. The deﬁnition of the boundary map assumes that the domain, R, is an HFD and this is a necessary (and suﬃcient) condition for the boundary map to be well-deﬁned. In Zaks’ paper [48] a function called the length function was deﬁned. A length function is a function from the nonzero nonunits of R to the natural numbers such that f (st) = f (s) + f (t) for all s, t nonzero, nonunit elements of R. Of course, HFDs are precisely the domains which admit a well-deﬁned length function such that f (x) = 1 for all irreducible elements x ∈ R. The boundary map may be thought of as a natural generalization of this length function to a homomorphism from the set of nonzero elements of the quotient ﬁeld of R to the integers. In [18], the boundary map proved to be a useful tool in showing that, in the case of orders in rings of algebraic integers, the integral closure of an HFD is again an HFD. Since the time of the paper [18], more work has been done on exploring

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properties of the boundary map and attempts have been made, with some success, to glean properties of overring behavior of HFDs. One recent avenue of interest is the so-called “boundary valuation domains” introduced by J. Maney. Definition 9.12. Let R be an HFD with quotient ﬁeld K and boundary map ∂R . R is called a boundary valuation domain if for all α ∈ K such that ∂R (α) = 0, either α or α−1 is an element of R. This deﬁnition was motivated by an attempt to understand the overring behavior of HFDs in the spirit of “ordinary” valuation domains. A good example of a boundary valuation domain is any Noetherian valuation domain, and a more general example is given by the construction F + xK[[x]], where F K are ﬁelds. In [34], many interesting properties of the boundary valuation domains are explored, and insights into the overring behavior of HFDs are obtained. Additionally, boundary valuation domains are completely characterized using groups of divisibility. Recently, Maney has also shown that the class of boundary valuation domains coincides with the class of atomic pseudo-valuation domains. We close this section by pointing to some interesting recent work along these lines. In a fairly recent sequence of papers [38, 39, 40, 41] M. Picavet-L’Hermitte has done some striking work (often utilizing t-closedness, seminormality, and the concept of weak factoriality) that has advanced the understanding of the factorization behavior in orders of rings of integers. Finally, we leave the reader with a list of questions that the present author would like to see answered. This list of questions is by no means exhaustive (and is motivated by the present author’s current tastes). There are myriad interesting open questions with respect to the structure and behavior of HFDs (and certainly many, many more in factorization theory in general). It seems clear that despite recent vigorous progress, factorization theory has very far to go. Questions: 1. If R[x] is an HFD, does it follow that R[x, y] is an HFD? (Note: it is known that the answer to this is “yes” if R is Noetherian.) 2. If R[[x]] is an HFD, then is R[[x, y]] an HFD? (Note: this one might be quite diﬃcult; the analogous question for UFDs has been open for nearly 40 years [43].) 3. If the integral closure of an HFD is atomic, then is it an HFD? 4. Find a general ideal-theoretic characterization of HFDs (if it can be done). 5. If R is an HFD, ﬁnd general conditions for which R[[x]] is an HFD. 6. Classify all (non-Noetherian) polynomial HFDs. 7. Under what conditions is an aﬃne ring an HFD? √ 8. Is there an inﬁnite number of (real) quadratic HFDs (inside Z[ 2])? 9. Find a closed-form representation of the Davenport constant (if one exists).

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10. Characterize the relationship between the elasticity of an order in a ring of integers and the elasticity of its set of norms. (Note: this is only partially known even in the case that the order is a full ring of integers and its quotient ﬁeld is Galois over Q.)

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[13] S.T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems, Lecture Notes in Pure and Applied Mathematics, Marcel-Dekker, 189 (1997), 73–112. [14] S.T. Chapman and W.W. Smith, Factorization in Dedekind domains with ﬁnite class group, Israel J. Math 71 (1990), 65–95. [15] S.T. Chapman and W.W. Smith, On the HFD, CHFD and k-HFD properties in Dedekind domains, Comm. Algebra 20 (1992), 1955–1987. [16] J. Coykendall, Normsets and determination of unique factorization in rings of algebraic integers, Proc. Amer. Math. Soc. 124 (1996), no. 6, 1727–1732. [17] J. Coykendall, A characterization of polynomial rings with the half-factorial property, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 189 (1997), 291–294. [18] J. Coykendall, The half-factorial property in integral extensions, Comm. Algebra 27 (1999), 3153–3159. [19] J. Coykendall, Half-factorial domains in quadratic ﬁelds, J. Algebra 235 (2001), 417–430. [20] J. Coykendall, Elasticity properties preserved in the normset, J. Number Theory 94 (2002), no. 2, 213–218. [21] J. Coykendall, On the integral closure of a half-factorial domain, J. Pure Appl. Algebra 180 (2003), 25–34. [22] J. Coykendall, The half-factorial property in power series rings, preprint. [23] J. Coykendall, T. Dumitrescu and M. Zafrullah, The half-factorial property and domains of the form A + xB[x], submitted. [24] J. Coykendall and J. Dover, Sets with few intersection numbers from Singer subgroup orbits, European J. Combin. 22 (2001), no. 4, 455–464. [25] J. Coykendall and M. Zafrullah, AP-domains and unique factorization, J. Pure Appl. Algebra 189 (2004), 27–35. [26] A. Geroldinger and R. Schneider, On Davenport’s Constant, J. Combin. Theory Ser. A 61 (1992), 147–152. [27] R. Gilmer, W. Heinzer and W.W. Smith, On the distribution of prime ideals within the ideal class group, Houston J. Math. 22 (1996), 51–59. [28] N. Gonzalez, Elasticity and ramiﬁcation, Comm. Algebra 27 (1999), 1729–1736. [29] N. Gonzalez, Elasticity of A + XB[X] domains, J. Pure Appl. Algebra 138 (1999), 119–137. [30] N. Gonzalez, S. Pellerin and R. Robert, Elasticity of A+ XI[X] domains where A is a UFD, J. Pure Appl. Algebra 160 (2001), no. 2-3, 183–194.

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[31] F. Halter-Koch, Factorization of Algebraic Integers, Ber. Math. Stat. Sektion im Forschungszentrum 191 (1983). [32] I. Kaplansky. Commutative Rings. University of Chicago Press, 1974. [33] H. Kim, Examples of half-factorial domains, Canad. Math. Bull. 43 (2000), no. 3, 362–367. [34] J. Maney, Boundary valuation domains, J. Algebra 273 (2004), 373–383. [35] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, SpringerVerlag, 1990. [36] W. Narkiewicz, A note on elasticity of factorizations, J. Number Theory 51 (1995), no. 1, 46–47. [37] S. Pellerin and R. Robert, Elasticity of A + XI[X] domains where A is a Dedekind domain, J. Pure Appl. Algebra 176 (2002), no. 2-3, 195–212. [38] M. Picavet-L’Hermitte, Some remarks on half-factorial orders, Rend. Circ. Mat. Palermo 52 (2003), no. 2, 297–307. [39] M. Picavet-L’Hermitte, Weakly factorial quadratic orders, Arab. J. Sci. Eng. Sect. C Theme Issues 26 (2001), no. 1, 171–186. [40] M. Picavet-L’Hermitte, Factorization in some orders with a PID as integral closure, Algebraic number theory and Diophantine analysis (Graz, 1998), 365– 390, de Gruyter, Berlin, 2000. [41] M. Picavet-L’Hermitte, Seminormality and t-closedness of algebraic orders, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 205 (1999), 521–540. [42] M. Roitman, Polynomial extensions of atomic domains, J. Pure Appl. Algebra 87 (1993), 187–199. [43] P. Samuel, On unique factorization domains, Illinois J. Math. 5 (1961), 1–17. [44] H.M. Stark, A complete determination of the complex quadratic ﬁelds of classnumber one, Michigan Math. J. 14 (1967), 1–27. [45] H.M. Stark, On complex quadratic ﬁelds with class-number two, Math. Comp. 29 (1975), 289–302. [46] R.J. Valenza, Elasticity of factorization in number ﬁelds, J. Number Theory 36 (1990), no. 2, 212–218. [47] A. Zaks, Half-factorial domains, Bull. Amer. Math. Soc. 82 (1976), 721–723. [48] A. Zaks, Half-factorial domains, Israel J. of Math. 37(1980), 281-302.

Chapter 4

C-Monoids and Congruence Monoids in Krull Domains by

Franz Halter-Koch1 Abstract Only recently, several types of auxiliary monoids were used for the investigation of the arithmetic of congruence monoids in Dedekind domains and of noetherian domains satisfying some natural ﬁniteness conditions. In this chapter, we introduce C-monoids as a common generalization of all these types of auxiliary monoids and investigate their algebraic and arithmetical properties. We discuss congruence monoids in Krull domains as examples of arithmetical interest.

1

Introduction

In a recent paper [9], devoted to the arithmetic of congruence monoids in Dedekind domains, we introduced abstract congruence monoids (AC-monoids) and C0 monoids as appropriate tools for the arithmetical investigations of the more complicated arithmetically interesting objects. A similar concept (that of a Z-monoid) was used by W. Hassler [17] for the investigation of the multiplicative arithmetic of a large class of ﬁnitely generated integral domains. In this chapter, we introduce C-monoids as a common generalization of C0 -, Z- and (all interesting) AC-monoids. C-monoids have a nice algebraic structure: They are v-noetherian (Theorem 4.8) and their complete integral closures are Krull monoids with ﬁnite class group and have a nonempty conductor (Theorem 4.6). Every Krull monoid with ﬁnite class group is a C-monoid (Theorem 4.7). The arithmetically most important examples of C-monoids are congruence monoids in Krull domains. Among them are the congruence monoids in Dedekind domains investigated in [9] and the noetherian integral domains considered in [17]. 1 This

work was supported by the Austrian Science Fund FWF (Project-Nr. P16770-N12)

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By means of a suitable transfer principle, the arithmetical investigation of a C-monoid can essentially be reduced to that of a C0 -monoid (see [8, Theorem 7.2]). Apart from the arithmetical results already proved in [9], we present here a necessary and suﬃcient criterion for a C-monoid to have ﬁnite elasticity (Theorem 5.1), which is even new for congruence monoids of natural numbers (see [8, Theorem 7.8]). For ﬁnitely generated integral domains in general, a stronger criterion for ﬁnite elasticity was proved by F. Kainrath [18, Theorem 6.1]. This chapter is organized as follows. In Section 2 we ﬁx our notations concerning monoids and factorization theory. In Section 3 we recall and reﬁne the concept of class semigroups. Section 4 is devoted to the algebraic theory of C-monoids and contains the main results of this chapter. In Section 5 we present a criterion for the ﬁniteness of the elasticity of a C-monoid, and ﬁnally in Section 6 we discuss congruence monoids in Krull domains as a main application of this new concept.

2

Preliminaries and Notations

Let N denote the set of positive integers and N0 = N ∪ {0}. For m, n ∈ Z with m ≤ n, we set [m, n] = {x ∈ Z | m ≤ x ≤ n}. For a subset X ⊂ N0 , we denote by gcd(X) ∈ N0 its greatest common divisor. In particular, gcd(X) = 0 if and only if X ⊂ {0}. For any set X, we denote by |X| ∈ N0 ∪ {∞} its cardinality. For convenience and to ﬁx notations, we recall some notions from the theory of monoids and nonunique factorizations which we use in this chapter. Our main reference for the theory of divisibility and the theory of v-ideals is [16, Ch. 10 and 11]. For the theory of Krull monoids we refer to [16, Ch. 20, 22 and 23] and to the survey article [3]. For the basic concepts of nonunique factorizations, we refer to the survey articles in [1], in particular to [7] and [15]. Basic notions. By a semigroup we mean a nonempty set with a commutative and associative law of composition possessing a unit element. By a monoid we mean a cancellative semigroup. Unless stated otherwise, we use multiplicative notation and denote the unit element by 1. For subsets A, B of a monoid H, we set AB = {ab | a ∈ A , b ∈ B}, and for n ∈ N, we set An = A · . . . · A. Semigroup and monoid homomorphisms are assumed to respect the unit element. Subsemigroups and submonoids are assumed to contain the unit element.

For each monoid H, we ﬁx a quotient group q(H) of H, and we denote by H the complete integral closure of H in q(H), consisting of all elements x ∈ q(H) for which there exists some c ∈ H such that cxn ∈ H for all n ∈ N. For subsets X, Y ⊂ q(H), we deﬁne (X : Y ) = {z ∈ q(H) | zY ⊂ X}. If H ⊂ D is a submonoid, we tacitly assume that q(H) ⊂ q(D). We denote by H × the group of invertible elements of H and by Hred = {aH × | a ∈ H} the associated reduced monoid of H. We call H reduced , if H × = {1} (and in this case we will identify H with Hred ). For a monoid homomorphism ϕ : H → D we denote by ϕred : Hred → Dred the induced homomorphism of reduced monoids, given by ϕred (aH × ) = ϕ(a)D× . For an integral domain R, we denote by R• = R \ {0} its multiplicative monoid. Let H be a monoid and a, b ∈ H. We call a a divisor of b and we write a | b , if b ∈ aH. We call a and b associated and write a b , if aH × = bH × . An element u ∈ H is called an atom if u ∈ / H × , and for all a, b ∈ H, u = ab

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implies a ∈ H × or b ∈ H × . We denote by A(H) the set of all atoms of H, and we call the monoid H atomic , if every a ∈ H \ H × is a product of atoms. An element p ∈ H is called a prime , if H \ pH is a submonoid of H, and H is called factorial , if every a ∈ H \ H × is a product of primes. Every prime is an atom, and a monoid is factorial if and only if it is atomic and every atom is a prime. If H is atomic and p ∈ H is a prime, then every a ∈ q(H) has a representation a = pn bc−1 , where b, c ∈ H, p bc and n ∈ Z. The exponent n is uniquely determined by the classes aH × and pH × in Hred , and we call n = vp (a) the p-adic value of a. The map vp : q(H) → Z is a surjective group homomorphism, called the p-adic valuation. If P is a set of primes of an atomic monoid H and x ∈ q(H), then suppP (x) = {p ∈ P | vp (x) = 0} is called the support of x (in P ). Free monoids. For a set P , we denote by F (P ) the free (abelian) monoid with basis P . F (P ) is a reduced factorial monoid and P is the set of primes of F (P ). Every nonempty subset X of F (P ) possesses a unique greatest common divisor, denoted by gcd(X) ∈ F(P ). Every a ∈ F(P ) has a unique representation in the form pνp , where νp ∈ N0 and νp = 0 for all but ﬁnitely many p ∈ P , a= p∈P

and then νp = vp (a) for all p ∈ P . Sets of lengths and elasticity. Let H be an atomic monoid. For a ∈ H \ H × , we denote by L(a) = {r ∈ N | a = u1 · . . . · ur with u1 , . . . , ur ∈ A(H)} the set of lengths of a . The monoid H is called a BF-monoid , if L(a) is ﬁnite for all a ∈ H \ H × . If H is a BF-monoid and a ∈ H, then ρ(a) =

max L(a) ∈ Q>0 min L(a)

is called the elasticity of a and ρ(H) = sup{ρ(a) | a ∈ H \ H × } the elasticity of H. The elasticity is one of the best investigated invariants of nonunique factorizations (see [2] for a survey). Submonoids and class groups. Let D be a monoid and H ⊂ D a submonoid. For x ∈ q(D), we set [x]D/H = xq(H) ∈ q(D)/q(H); we call D/H = {[a]D/H | a ∈ D} the class monoid and q(D)/q(H) = q(D/H) the class group of D modulo H. If H ⊂ D× is a subgroup, then [a]D/H = aH ∈ q(D)/H for all a ∈ D, there are natural epimorphisms D → D/H → D/D× = Dred ⊂ q(D)/D× = q(Dred ), and Hred = H/H × is a submonoid of D/H × . In this special case, we write D/H multiplicatively. In the general situation however, we use additive notation for D/H, so that [ab]D/H = [a]D/H + [b]D/H for all a, b ∈ D, and [1]D/H is the zero element of D/H. A submonoid H ⊂ D is called • coﬁnal , if aD ∩ H = ∅ for all a ∈ D (equivalently, D/H = q(D/H)). • saturated , if q(H) ∩ D = H (equivalently, H = {a ∈ D | [a]D/H = [1]D/H }).

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Halter-Koch • divisor-closed , if a ∈ H, b ∈ D and b | a implies b ∈ H (equivalently, yD ∩ H = yH for all y ∈ D).

If H ⊂ D is divisor-closed, then H ⊂ D is saturated, A(H) = A(D)∩H, and factorizations of an element of H into atoms of H amounts to the same as factorizations into atoms of D. A subset E ⊂ H is a divisor-closed submonoid if and only if H \ E is a prime s-ideal of H [16, Deﬁnition 6.1]. For any subset E ⊂ H, we denote by [[E]] the smallest divisor-closed submonoid of H containing E. In particular, [[∅]] = H × . Divisor homomorphisms, divisor theories and Krull monoids. A monoid homomorphism ϕ : H → D is called a divisor homomorphism , if ϕ(a) | ϕ(b) implies a | b for all a, b ∈ H. If H ⊂ D is a submonoid, then the injection H → D is a divisor homomorphism if and only if H is a saturated submonoid of D. An arbitrary monoid homomorphism ϕ : H → D is a divisor homomorphism if and only if ϕred : Hred → Dred is a divisor homomorphism. A divisor theory is a divisor homomorphism ϕ : H → F (P ) into a free monoid F (P ) such that for every p ∈ P there exists a ﬁnite set E ⊂ H such that p = gcd(ϕ(E)). A monoid homomorphism ϕ : H → F (P ) is a divisor theory if and only ∼ if it induces an isomorphism Hred → ϕ(H) and the injection ϕ(H) → F (P ) is a divisor theory. If ϕ : H → F (P ) and ϕ : H → F (P ) are divisor theories, then there ∼ exists a unique isomorphism Φ : F (P ) → F (P ) such that Φ ◦ ϕ = ϕ . A monoid H is a Krull monoid , if it possesses a divisor theory. Every saturated submonoid of a Krull monoid is a Krull monoid. If ∂ : H → F is a divisor theory, then ∂ is coﬁnal, and we call C(H) = F/∂(H) the (divisor) class group of H. It is (up to canonical isomorphisms) uniquely determined by H. A monoid H is factorial if and only if it is a Krull monoid with trivial divisor class group. A monoid H is a Krull monoid if and only if Hred is a Krull monoid, and then C(H) = C(Hred ). An integral domain R is a Krull domain if and only if R• is a Krull monoid, and then C(R• ) = C(R) is the divisor class group of R as deﬁned in [5]. Products. For two monoids H1 , H2 , we denote by H1×H2 their direct product, and we view H1 and H2 as (divisor-closed) submonoids of H = H1×H2 . Then every a ∈ H has a unique factorization in the form a = a1 a2 with a1 ∈ H1 and a2 ∈ H2 . We have (H1×H2 )× = H1××H2× , q(H1×H2 ) = q(H1 )×q(H2 ) and H 1 ×H2 = H1×H2 . If H1 ⊂ D1 and H2 ⊂ D2 are submonoids, then H1×H2 ⊂ D1×D2 , and there is an ∼ isomorphism D1 /H1×D2 /H2 → D1×D2 /H1×H2 , given by a1 q(H1 ), a2 q(H2 ) → a1 a2 q(H1 ×H2 ). Also, H1 ×H2 is saturated [ coﬁnal ] in D1 ×D2 if and only if both H1 ⊂ D1 and H2 ⊂ D2 are saturated [ coﬁnal ]. A monoid H is factorial and P ⊂ H is a maximal set of representatives of pairwise nonassociated primes of H if and only if H = H × × F(P ). Every Krull monoid H splits in the form H = H × ×H0 , where H0 is a reduced Krull monoid, and there is a divisor theory H0 → F (P ). If H1 and H2 are Krull monoids with divisor theories ϕ1 : H1 → F1 and ϕ2 : H2 → F2 , then H1 × H2 is a Krull monoid, and ϕ1 × ϕ2 : H1 × H2 → F1 × F2 is a divisor theory.

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3

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Class Semigroups

Class semigroups were introduced in [9, Deﬁnition 4.1] as a appropriate reﬁnement of the usual concept of ordinary class groups in algebraic number theory. Later on, in Propositions 3.6 and 3.7 we shall prove that, for saturated and coﬁnal submonoids, the two concepts coincide. We recall the deﬁnition given in [9] in a reﬁned version. Definition 3.1. Let D be a monoid, S ⊂ D a submonoid and A ⊂ D a subset. Two elements y, y ∈ D are called (A, S)-equivalent, if y −1 A ∩ S = y −1 A ∩ S (equivalently: if x ∈ S, then xy ∈ A if and only if xy ∈ A). By deﬁnition, (A, S)-equivalence is an equivalence relation on D. For each y ∈ D, we denote by [y]SA the (A, S)-equivalence class of y. For a subset T ⊂ D, we deﬁne CT (A, S) =

[y]SA | y ∈ T , and in particular C(A, S) = CS (A, S)

∗

and C (A, S) = C(S\S × )∪{1} (A, S). We shall prove in Lemma 3.2 that (A, S)equivalence is a congruence relation on S. The quotient law on C(A, S) will be written additively, that is, [y1 ]SA + [y2 ]SA = [y1 y2 ]SA for all y1 , y2 ∈ S. Endowed with the quotient law, C(A, S) is an additive abelian semigroup with zero element [1]SA , and the canonical map S → C(A, S) , deﬁned by y → [y]SA , is an epimorphism. Consequently, if T ⊂ S is a submonoid, then CT (A, S) is a subsemigroup of C(A, S). In particular, C ∗ (A, S) is a subsemigroup of C(A, S). We call C(A, S) the class semigroup and C ∗ (A, S) the reduced class semigroup of A in S. Lemma 3.2. Let D be a monoid, S ⊂ D a submonoid and A ⊂ D a subset. 1. (A, S)-equivalence is a congruence relation on S. 2. For every subset T ⊂ D, there is a bijective map CT (A, S) → {y −1 A ∩ S | y ∈ T } ,

given by

[y]SA → y −1 A ∩ S.

3. If y ∈ D and [y]SA ∩ A = ∅, then [y]SA ⊂ A. Proof. 1. We must prove that, for all x, y, y ∈ S, [y]SA = [y ]SA implies [xy]SA = [xy ]SA . Thus let x, y, y ∈ S be such that y −1 A ∩ S = y −1 A ∩ S. By symmetry, it suﬃces to prove that (xy)−1 A ∩ S ⊂ (xy )−1 A ∩ S. If z ∈ (xy)−1 A ∩ S, then zx ∈ y −1 A ∩ xS = (y −1 A ∩ S) ∩ xS = (y −1 A ∩ S) ∩ xS = (y −1 A ∩ S) ∩ xS, and therefore z ∈ (xy )−1 A ∩ S. 2. By the very deﬁnition of (A, S)-equivalence, for all y, y ∈ T we have [y]SA = S [y ]A if and only if y −1 A ∩ S = y −1 A ∩ S. 3. If y ∈ [y]SA ∩ A and z ∈ [y]SA , then [z]SA = [y ]SA . Hence 1 ∈ y −1 A ∩ S = −1 z A ∩ S, and thus z ∈ A. Lemma 3.3. Let H ⊂ D be a submonoid. D × ∗ 1. [u]D H = [1]H for all u ∈ H , C (H, D) = C(D\D× )∪H × (H, D), and there is a × × monomorphism Ψ : D /H → C(H, D), given by Ψ(uH × ) = [u]D H for all u ∈ D× .

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Halter-Koch 2. C(H, D) is ﬁnite if and only if both C ∗ (H, D) and D× /H × are ﬁnite.

3. There is an isomorphism Φ : C(H, D) → C(Hred , D/H × ), given by Φ [y]D H =

D/H × [yH × ]Hred for all y ∈ D. It satisﬁes Φ C ∗ (H, D) = C ∗ (Hred , D/H × ). 4. If U ⊂ D× is a subgroup, then there is an epimorphism

D C(HU, D), given by Ψ [y]H = [y]D HU for all y ∈ D.

Ψ : C(H, D) →

5. If D ⊂ D is a submonoid and H = H ∩ D , then there is an epimorphism D CD (H, D) → C(H , D ), given by [y]D H → [y]H for all y ∈ D . In particular, ∗ × × ∗ if C (H, D) is ﬁnite and D = D ∩ D , then C (H , D ) is also ﬁnite. Proof. 1. The map Ψ0 : D× → C(H, D), deﬁned by Ψ0 (u) = [u]D H , is a group homomorphism, and we shall prove that Ker(Ψ0 ) = H × . Then it follows in particular D × that [u]D and C ∗ (H, D) = C(D\D× )∪H × (H, D). H = [1]H for all u ∈ H D If u ∈ D× , then u−1 H ∩ D = u−1 H. Therefore [u]D H = [1]H holds if and only if u−1 H = H, which is equivalent to u ∈ H × . 2. By 1., since C(H, D) = C ∗ (H, D) ∪ Ψ(D× /H × ). 3. If ρ : D → D/H × denotes the canonical epimorphism, then ρ(D× ) = (D/H × )× and ρ(H) = Hred . For all y, y ∈ D, we have y −1 H ∩ D = y −1 H ∩ D if and only if ρ(y)−1 ρ(H) ∩ ρ(D) = ρ(y )−1 ρ(H) ∩ ρ(D). Hence Φ is an isomorphism as asserted. 4. It is suﬃcient to prove that (H, D)-equivalence on D implies (HU, D)equivalence. We must prove that, for all y, y ∈ D, y −1 H ∩ D = y −1 H ∩ D entails y −1 HU ∩ D ⊂ y −1 HU ∩ D. Thus suppose that y −1 H ∩ D = y −1 H ∩ D and z ∈ y −1 HU ∩ D. Then zy = cu for some u ∈ U and c ∈ H, and therefore u−1 z ∈ y −1 H ∩ D = y −1 H ∩ D, which implies z ∈ y −1 HU ∩ D.

D D D 5. We shall prove that, for all y, y ∈ D , [y]D H = [y ]H implies [y]H = [y ]H . −1 −1 Thus suppose that y, y ∈ D and y H ∩ D = y H ∩ D. By symmetry, it suﬃces to prove that y −1 H ∩ D ⊂ y −1 H ∩ D . If z ∈ y −1 H ∩ D , then z ∈ y −1 H ∩ D = y −1 H ∩D, hence y z ∈ H ∩D = H and z ∈ y −1 H ∩D . Therefore the assignment D ∗ [y]D H → [y]H deﬁnes an epimorphism CD (H, D) → C(H , D ), and C (H , D ) is the ∗ × × image of CD \D (H, D), which is a subset of C (H, D), if D \ D ⊂ D \ D× .

The following Lemma 3.4 justiﬁes the subsequent restriction to coﬁnal submonoids. Lemma 3.4. Let H ⊂ D be a submonoid and D0 = {y ∈ D | yD ∩ H = ∅}. 1. D0 is the greatest monoid D such that H ⊂ D ⊂ D and H is coﬁnal in D. 2. D0× = D× and q(H) ∩ D = q(H) ∩ D0 . In particular, H is saturated in D if and only if H is saturated in D0 .

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∼ 0 = 3. There is an isomorphism Φ : C(H, D0 ) → CD0 (H, D) satisfying Φ [y]D H D [y]D H for all y ∈ D0 . If y0 ∈ D \ D0 , then C(H, D) = CD0 (H, D) [y0 ]H

and C ∗ (H, D) = Φ C ∗ (H, D0 ) [y0 ]D H . Proof. 1. Obvious by the deﬁnition. 2. If u ∈ D× , then uu−1 = 1 ∈ H implies u ∈ D0× . If a ∈ q(H) ∩ D, then there exists some y ∈ H such that ay ∈ H and thus a ∈ D0 . 3. It is easily checked that, for all y, y ∈ D0 , we have y −1 H ∩ D = y −1 H ∩ D if and only if y −1 H ∩D0 = y −1 H ∩D0 . Thus Φ is an isomorphism as asserted. Since y0−1 H ∩ D = ∅ for all y0 ∈ D \ D0 , Lemma 3.2.2 implies C(H, D) = CD0 (H, D) [y0 ]D H . Hence we obtain

C ∗ (H, D) = C(D0 \D× )∪{1} (H, D) ∪ CD\D0 (H, D) = Φ C ∗ (H, D0 ) [y0 ]D H . 0

Lemma 3.5. Let H1 ⊂ D1 and H2 ⊂ D2 be coﬁnal submonoids. ∼

1. There is an isomorphism Φ : C(H1 ×H2 , D1 ×D2 ) → C(H1 , D1 ) × C(H2 , D2 ) satisfying

D1 D2 1×D2 Φ [y1 y2 ]D = [y for all (y1 , y2 ) ∈ D1 ×D2 . ] , [y ] 1 H1 2 H2 H1×H2 2. C ∗ (H1 ×H2 , D1 ×D2 ) is ﬁnite if and only if the following two conditions are satisﬁed: (a) C ∗ (H1 , D1 ) and C ∗ (H1 , D1 ) are both ﬁnite. (b) Either D1 = D1× , or D2 = D2× , or the groups D1× /H1× and D2× /H2× are both ﬁnite. Proof. 1. If (y1 , y2 ) ∈ D1 × D2 , then (y1 y2 )−1 (H1 × H2 ) ∩ (D1 × D2 ) = (y1−1 H1 ∩ D1 ) × (y2−1 H2 ∩ D2 ) = ∅, since H1 ⊂ D1 and H2 ⊂ D2 are coﬁnal. Thus we obtain, for all (y1 , y2 ), (y1 , y2 ) ∈ D1 × D2 , (y1 y2 )−1 (H1 × H2 ) ∩ (D1 × D2 ) = (y1 y2 )−1 (H1 ×H2 ) ∩ (D1 ×D2 ) if and only if y1−1 H1 ∩ D1 = (y1 )−1 H1 ∩ D1 and y2−1 H2 ∩ D2 = (y2 )−1 H2 ∩ D2 . Therefore there is an isomorphism Φ as asserted.

2. Since (D1×D2 ) \ (D1××D2× ) = (D1 \ D1× )×(D2 \ D2× ) (D1 \ D1× )×D2×

D1× × (D2 \ D2× ) , the isomorphism Φ given in 1. maps C ∗ (H1 × H2 , D1 × D2 ) onto

C = C ∗ (H1 , D1 )×C ∗ (H2 , D2 ) ∪ CD1 \D× (H1 , D1 )×CD× (H2 , D2 ) ∪ 1 2

CD× (H1 , D1 )×CD2 \D× (H2 , D2 ) . 1

2

By Lemma 3.3.1 there are isomorphisms CD× (H1 , D1 ) ∼ = D1× /H1× and CD2× (H2 , D2 ) 1 ∼ = D2× /H2× . Hence C is ﬁnite if and only if (a) and (b) are satisﬁed.

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Proposition 3.6. Let H ⊂ D be a submonoid. 1. C(H, D) is a group if and only if C ∗ (H, D) is a group, and in this case we have either D = D× or C(H, D) = C ∗ (H, D). 2. If C(H, D) is a group, then H ⊂ D is coﬁnal. 3. If C(H, D) is a torsion group, then H ⊂ D is coﬁnal and saturated. × Proof. 1. By deﬁnition, C(H, D) = C ∗ (H, D) ∪ C , where C = {[u]D H | u ∈ D } ∗ is a subgroup of C(H, D). Hence C(H, D) is a group if and only if C (H, D) is a D group. Thus suppose that D = D× , a ∈ D \ D× and u ∈ D× . Then [a]D H , [ua]H ∈ ∗ D D D ∗ C (H, D), and therefore also [u]H = [ua]H − [a]H ∈ C (H, D).

2. Let C(H, D) be a group. For every c ∈ D, there exists some c ∈ D such that D D D D D −[c]D H = [c ]H , and therefore [cc ]H = [c]H + [c ]H = [1]H , which implies cc ∈ H. 3. Let C(H, D) be a torsion group, a, b ∈ H and c ∈ D be such that a = bc. D n−1 D D Let n ∈ N be such that n[b]D ]H = [a]D H = [1]H . Then [ab H + (n − 1)[b]H = D D D n−1 ∈ H implies c ∈ H. Hence H ⊂ D is saturated. [c]H + n[b]H = [c]H , and ab Proposition 3.7. Let D be a monoid and H ⊂ D a coﬁnal submonoid. 1. There are epimorphisms θ : C(H, D) → D/H and θ∗ : C ∗ (H, D) → D/D× H ,

∗ D given by θ [y]D H = [y]D/H for all y ∈ D, and θ ([y]H = [y]D/D × H for all y ∈ (D \ D× ) ∪ {1}. 2. [1]D H ⊂ H ⊂ [1]D/H . 3. The following conditions are equivalent: (a) H is saturated in D. (b) [y]D H = [y]D/H ∩ D for all y ∈ D. (c) [1]D H = [1]D/H ∩ D. (d) The epimorphism θ : C(H, D) → D/H deﬁned in 1. is an isomorphism. D Proof. 1. We assert that, for all y, y ∈ D, [y]D H = [y ]H implies yq(H) = y q(H). D D Indeed, suppose that y, y ∈ D and [y]H = [y ]H . Since H is coﬁnal in D, there exists some z ∈ D such that yz ∈ H. Hence y z ∈ H, and thus y y −1 = (y z)(yz)−1 ∈ q(H). Now the assignment [y]D H → [y]D/H = yq(H) deﬁnes a map θ : C(H, D) → D/H, and obviously θ is an epimorphism. Hence the map θ1 : C(H, D) → D/D× H, de× × D ﬁned by θ1 ([y]D H ) = yD q(H), is also an epimorphism. If y ∈ D , then θ1 ([y]H ) = D ∗ ∗ ∗ × θ1 ([1]H ), and therefore θ = θ1 | C (H, D) : C (H, D) → D/D H is also an epimorphism.

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2. By Lemma 3.2.3 we obtain [1]D H ⊂ H, and by deﬁnition we have [1]D/H = q(H) ⊃ H. D D 3. (a) ⇒ (b) If y ∈ D and y ∈ [y]D H , then [y ]H = [y]H , hence [y ]D/H = [y]D/H by 1., and therefore y ∈ [y]D/H ∩ D. To prove the reverse inclusion, suppose that y ∈ [y]D/H ∩ D = yq(H) ∩ D. Then there exist u, u ∈ H such that u y = uy. By symmetry, it is suﬃcient to prove that y −1 H ∩ D ⊂ y −1 H ∩ D. If x ∈ y −1 H ∩ D, then xy u = xyu ∈ H. Hence xy ∈ q(H) ∩ D = H, and thus x ∈ y −1 H ∩ D. (b) ⇒ (c) and (b) ⇒ (d) are obvious. (c) ⇒ (a) If [1]D H = [1]D/H ∩ D, then 2. implies H = [1]D/H ∩ D = q(H) ∩ D, and therefore H ⊂ D is saturated. (d) ⇒ (c) If (c) fails, then [1]D H [1]D/H ∩ D by 2., and thus there exists some D D . Since [u] u ∈ [1]D/H ∩ D \ [1]D H H = [1]H and [u]D/H = [1]D/H , it follows that θ is not injective. The following example is due to A. Geroldinger. Example 3.8. To illustrate the diﬀerence between D/H and C(H, D), we consider the additive monoid H = (N × N≥6 ) ∪ {(2n , 3) , (2n , 4) , (2n + 2, 5) | n ∈ N} ∪ {(0, 0} ⊂ D = N20 . Since q(H) = q(D) = Z2 , we have D/H = 0, but nevertheless C(H, D) is inﬁnite, since any two elements of the set {(2n , 0) | n ∈ N} are not (H, D)-equivalent. Indeed, n D n n suppose that m, n ∈ N, m ≤ n and [(2m , 0)]D H = [(2 , 0)]H . Then (2 , 3) + (2 , 0) = n+1 n m m n−m n−m , 3) ∈ H implies (2 , 3)+(2 , 0) = (2 (2 +1), 3) ∈ H, hence 2 +1 = 2l (2 for some l ∈ N and thus m = n. Note that H is a ﬁnitely primary monoid of rank 2 and exponent 6 satisfying

= D, and (6, 6) + D ⊂ H (for the theory of ﬁnitely primary monoids see [6]). H Theorem 3.9. Let D be a monoid and S ⊂ H ⊂ D submonoids such that S is saturated in H and H/S is ﬁnite. If T ⊂ D is a subset such that CT (H, D) is ﬁnite, then CT (S, D) is also ﬁnite. Proof. For y, y ∈ T , we set y ∼ y if either y −1 H ∩ D = y −1 H ∩ D = ∅ or if there exists some z ∈ y −1 H ∩ D = y −1 H ∩ D such that [zy]H/S = [zy ]H/S . Since H/S is ﬁnite and {y −1 H ∩ D | y ∈ T } is ﬁnite by Lemma 3.2.2, ∼ is an equivalence relation with only ﬁnitely many equivalence classes on T . We shall prove that, for all y, y ∈ T , y ∼ y implies y −1 S ∩ D = y −1 S ∩ D. Then {[y]D S | y ∈ T } is also ﬁnite. If y, y ∈ T and y −1 H ∩ D = y −1 H ∩ D = ∅, then y −1 S ∩ D = y −1 S ∩ D = ∅. Thus we may assume that there exists some z ∈ y −1 H ∩ D = y −1 H ∩ D such that [zy]H/S = [zy ]H/S . Then y −1 y = (zy)−1 (zy ) ∈ q(S), and since S is saturated in H, we obtain y −1 S ∩ D = y −1 (H ∩ q(S)) ∩ D = y −1 H ∩ D ∩ y −1 q(S) = y −1 H ∩ D ∩ y −1 q(S) = y −1 (H ∩ q(S)) ∩ D = y −1 S ∩ D.

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The arithmetical importance of C ∗ (H, D) is (inter alia) revealed by the following Proposition 3.10 (see also [9, Proposition 4.5]). Proposition 3.10. Let H ⊂ D be a submonoid such that C ∗ (H, D) is ﬁnite, and D × . V = u ∈ D× [ua]D H = [a]H for all a ∈ D \ D 1. V ⊂ D× is a subgroup of ﬁnite index, H × ⊂ V , and V (H \ H × ) ⊂ H. 2. There exists some α ∈ N such that (D× : V ) | α and q 2α D ∩ H = q α (q α D ∩ H) for all q ∈ D \ D× . × ∗ Proof. 1. D× operates on the ﬁnite set C = {[a]D H | a ∈ D \ D } ⊂ C (H, D) by D D × × means of u [a]H = [ua]H for all u ∈ D and a ∈ D \ D . This operation deﬁnes a homomorphism ϕ : D× → Perm(C) into the (ﬁnite) permutation group of C, V = Ker(ϕ) is a subgroup of ﬁnite index, and H × ⊂ V by Lemma 3.3.1. If u ∈ V D and a ∈ H \ H × , then [ua]D H = [a]H ⊂ H and thus ua ∈ H by Lemma 3.2.3.

2. For every g ∈ C ∗ (H, D), there exist γg , λg ∈ N such that (λg + γg )g = λg g, and then (λg + γg )g = λg g for all γg , λg ∈ N such that γg | γg and λg ≥ λg . Let α ∈ N be such that (D× : V ) | α, γg | α and α ≥ λg for all g ∈ C ∗ (H, D). Then 2αg = αg for all g ∈ C ∗ (H, D). If q ∈ D \ D× and a ∈ D, then D D D D α D [q 2α a]D H = 2α[q]H + [a]H = α[q]H + [a]H = [q a]H ,

and thus q 2α a ∈ H if and only if q α a ∈ H.

4

Algebraic Theory of C-Monoids

Definition 4.1. Let H be a monoid. 1. H is called a C-monoid , if it is a submonoid of a factorial monoid F such that H ∩ F × = H × and C ∗ (H, F ) is ﬁnite. Then it follows by Proposition 3.10 that there exists some α ∈ N and a subgroup V ⊂ F × such that H × ⊂ V , (F × : V ) | α, V (H \ H × ) ⊂ H and q 2α F ∩ H = q α (q α F ∩ H) for all q ∈ F \ F × . We refer to these properties by saying that H is deﬁned in F with exponent α and subgroup V ⊂ F × . 2. Let F = F × ×F(P ) be a factorial monoid and H ⊂ F a submonoid. (a) H is called dense in F , if vp (H) ⊂ N0 is a numerical monoid for all p ∈ P. (b) A subset E ⊂ P is called H-essential a ∈ H \ F ×.

if E = suppP (a) for some

(c) H is called simple in F , if every minimal H-essential subset of P is a singleton.

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Note that the conditions (a) and (c) do not depend on the particular choice of P . 3. H is called a C0 -monoid , if it is a C-monoid deﬁned in a factorial monoid F possessing only ﬁnitely many pairwise nonassociated prime elements. By deﬁnition, every factorial monoid is a C-monoid. If H is a C-monoid deﬁned in a factorial monoid F , then F is far from being uniquely determined by H. However, in Theorem 4.6 we shall prove that there is a canonical choice for F . Without giving details, we mention that the Z-monoids used in [17] and the ACmonoids deﬁned in factorial monoids which were introduced in [9] are C-monoids, and that the C0 -monoids as deﬁned above coincide with the C0 -monoids introduced in [9]. For Z-monoids, this follows immediately from the deﬁnitions, and for AC-monoids deﬁned in a factorial monoid this is true by [9, Theorem 5.4]. The conformity of the concepts of C0 -monoids will be explained after the subsequent Theorem 4.3. The following technical Lemma 4.2 is basic for the further theory of C-monoids. Lemma 4.2. Let F = F × × F(P ) be a factorial monoid, H ⊂ F a submonoid, α ∈ N, and let ρα : F → F be deﬁned by np ρα u p pnp −αlp =u p∈P

p∈P

for all u ∈ F × and (np )p∈P ∈ N0 , where (lp )p∈P ∈ N0 if np < 2α, and α ≤ np − αlp < 2α, if np ≥ 2α. (P )

(P )

is chosen so that lp = 0,

1. The following assertions are equivalent: (a) For all p ∈ P and a ∈ pα F , we have a ∈ H if and only if pα a ∈ H. (b) If u ∈ F and (np )p∈P , (np )p∈P ∈ N0 are such that, for each p ∈ P , either np = np or [ np ≡ np mod α and min{np , np } ≥ α ], then (P )

u

pnp ∈ H

implies

u

p∈P

pnp ∈ H .

p∈P

(c) For all x, y ∈ F we have xy ∈ H if and only if ρα (x)y ∈ H. 2. Let H be a C-monoid deﬁned in F with subgroup V ⊂ F × and exponent α ∈ N. (a) The equivalent conditions stated in 1. hold. (b) If Q ⊂ P is H-essential, u ∈ V and (lp )p∈Q ∈ NQ , then u pαlp ∈ H . p∈Q

82

Halter-Koch (c) If H is dense in F , then every ﬁnite subset of P is contained in an H-essential subset of P . (d ) If x, y ∈ H, then suppP (x) ⊂ suppP (y) if and only if x ∈ [[y]].

Proof. 1. (a) ⇒ (b) The assertion follows by induction on n= |np − np | . p∈P

(b) ⇒ (c) The elements xy and ρα (x)y satisfy the conditions stated in (b). (c) ⇒ (a) If p ∈ P and a ∈ pα F , then ρα (pα a) = ρα (a). 2. (a) Note that 1.(a) is equivalent to p2α F ∩ H = pα (pα F ∩ H) for all p ∈ P , and this holds by Proposition 3.10.2. (b) Suppose that a ∈ H \ F × , Q = suppP (a), u ∈ V and (lp )p∈Q ∈ NQ . If a=ε pnp , where ε ∈ F × and (np )p∈Q ∈ NQ , p∈Q

then uε−α ∈ V and a ∈ H \ H × implies uε−α aα = u

pαnp ∈ H .

p∈Q

Since αnp ≡ αlp mod α and min{αnp , αlp } ≥ α for all p ∈ Q, the assertion follows by 1.(b). (c) If H is dense in F and {p1 , . . . , pd } ⊂ P , then there exist elements a1 , . . . , ad ∈ H such that vpi (ai ) > 0 for all i ∈ [1, d], and if a = a1 · . . . · ad , then {p1 , . . . , pd } ⊂ suppP (a). (d) If x ∈ [[y]], then x | y n for some n ∈ N, and therefore suppP (x) ⊂ suppP (y). Thus suppose that x, y ∈ H and E = suppP (y) ⊃ suppP (x). Then x=ε pnp and y = η p mp , p∈E

p∈E

where ε, η ∈ F × , np ∈ N0 and mp ∈ N for all p ∈ E. Let k ∈ N be such that kmp > np for all p ∈ E. Then (b) implies pα(kmp −np ) ∈ H , x−1 y kα = xα−1 (ε−1 η)α p∈E

and therefore x ∈ [[y]]. The following Theorem 4.3 characterizes C0 -monoids without referring to the class semigroup, and in Proposition 4.4 we shall see that locally C-monoids behave like C0 -monoids. Theorem 4.3 (Characterization of C0 -monoids). Let F = F × ×[p1 , . . . , ps ] be a factorial monoid with pairwise nonassociated prime elements p1 , . . . , ps , and let H ⊂ F be a submonoid such that H ∩ F × = H × . Then the following assertions are equivalent:

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(a) H is a C0 -monoid deﬁned in F . (b) There exists some α ∈ N and a subgroup V ⊂ F × such that (F × : V ) | α, V (H \ H × ) ⊂ H, and for all j ∈ [1, s] and a ∈ pα j F we have a ∈ H if and only if pα j a ∈ H. Proof. (a) ⇒ (b) By Proposition 3.10. (b) ⇒ (a) Let {u1 , . . . , ut } ⊂ F × be a set of representatives for F ×/V , and consider the ﬁnite set B = uτ pn1 1 · . . . · pns s τ ∈ [1, t], (n1 , . . . , ns ) ∈ [0, 2α − 1]s . Let ρα : F → F be the reduction map deﬁned in Lemma 4.2, and deﬁne ϕ : F → B by ϕ(uc) = uτ ρα (c), whenever u ∈ uτ V and c ∈ [p1 , . . . , ps ]. We assert that F × ∗ [x]F H = [ϕ(x)]H for all x ∈ F \ F . Whence in particular C (H, F ) is ﬁnite. To prove this assertion, suppose that x = uc ∈ F \ F × , where u ∈ F × and c ∈ [p1 , . . . , ps ] \ {1}. Let τ ∈ [1, t] be such that u ∈ uτ V . Then ϕ(x) = uτ ρα (c), and we must prove that for all y ∈ F we have yx ∈ H if and only if yϕ(x) ∈ H. If y = vb ∈ F , where v ∈ F × and b ∈ [p1 , . . . , ps ], then yx = uvbc, yϕ(x) = uτ vbρα (c), and by Lemma 4.2.1 we have yϕ(x) ∈ H if and only if uτ vbc ∈ H. Since bc = 1 and (uτ v)−1 (uv) ∈ V it follows that yx ∈ H if and only if yϕ(x) ∈ H. Let F = F × ×[p1 , . . . , ps ] be as in Theorem 4.3 and H ⊂ F a submonoid such that H ∩ F × = H × . Let V ⊂ F × be a subgroup and ≡ the congruence relation modulo V on F × (observe that every congruence relation on F × is of this form). Then F ×/V = F ×/ ≡, and the following two assertions are equivalent: • V (H \ H × ) ⊂ H. • For all u, v ∈ F × , u ≡ v implies u−1 H ∩ [p1 , . . . , ps ] \ {1} = v −1 H ∩ [p1 , . . . , ps ] \ {1}. Therefore our concept of a C0 -monoid coincides with that of [9, Deﬁnition 5.1]. Proposition 4.4. Let H be a C-monoid deﬁned in a factorial monoid F with exponent α and subgroup V ⊂ F × . Let P1 be a set of pairwise nonassociated prime elements of F , U ⊂ F × a subgroup and F1 = U ×F(P1 ) ⊂ F . Then H1 = H ∩ F1 is a C-monoid deﬁned in F1 with exponent α and subgroup U ∩ V . In particular, (a) If P1 is ﬁnite, then H1 is a C0 -monoid. (b) If U = F × , then H1 ⊂ H is divisor-closed. Proof. F1 is factorial, F1× = U , and clearly H1× ⊂ U ∩ H1 . If ε ∈ U ∩ H1 , then ε ∈ F × ∩ H = H × and ε−1 ∈ U ∩ H ⊂ H1 , hence ε ∈ H1× . Whence H1× = F1× ∩ H1 . Since F1× = F1 ∩ F × , the reduced class semigroup C ∗ (H1 , F1 ) is ﬁnite by Lemma 3.3.5, and therefore H1 is a C-monoid deﬁned in F1 . If q ∈ F1 \ F1× ⊂ F \ F × and a ∈ F1 , then q α a ∈ H if and only if q 2α a ∈ H, and therefore q α a ∈ H1 if and only if q 2α a ∈ H1 . Since H × ∩ H1 = F × ∩ H ∩ H1 = F1× ∩ H1 = H1× , we obtain (V ∩ U )(H1 \ H1× ) ⊂ V (H \ H × ) ⊂ H and therefore (V ∩ U )(H1 \ H1× ) ⊂ H ∩ F1 = H1 . Since ﬁnally (U : V ∩ U ) | (F × : V ) | α, it follows that H1 is a C-monoid deﬁned in F1 with exponent α and subgroup U ∩ V . The more precise assertions (a) and (b) are obvious.

84

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Proposition 4.5. Let H be a C-monoid deﬁned in a factorial monoid F with exponent α and subgroup V ⊂ F × . 1. Hred is a C-monoid deﬁned in F/H × with exponent α and subgroup V /H × . In particular, if H is a C0 -monoid, then so is Hred . 2. Let S ⊂ H be a saturated submonoid such that H/S is ﬁnite. Then S is a C-monoid, and if H is a C0 -monoid, then so is S. Proof. 1. F/H × is factorial, (F/H × )× = F × /H × , and (F/H × )× ∩ Hred = (F × ∩ H)/H × = {1}. By Lemma 3.3.3 we have C ∗ (H, F ) ∼ = C ∗ (Hred , F/H × ). Hence Hred × is a C-monoid in F/H . The remaining assertions are easily checked. 2. Since S ⊂ H is saturated, we have F × ∩ S = F × ∩ H ∩ S = H × ∩ S = S × and, by Theorem 3.9, C ∗ (S, F ) is ﬁnite. Theorem 4.6. Let H be a C-monoid deﬁned in F = F × ×F(P ) with subgroup V and exponent α. 1. For p ∈ P , set dp = gcd(vp (H)) ∈ N0 . Deﬁne P0 = {pdp | p ∈ P, dp = 0} ⊂ F (P ), and F0 = {a ∈ F | vp (a) ∈ dp N0 for all p ∈ P } = F × ×F(P0 ) ⊂ F . Then H is a C-monoid deﬁned in F0 with subgroup V and exponent α, and H is dense in F0 .

is a Krull monoid, (H : H)

= ∅, and C(H)

is a ﬁnite group whose exponent 2. H divides α. 3. If H is dense in F , then the following assertions hold. (a) H is coﬁnal in F .

= q(H)∩F ⊂ q(H), and if {p1 , . . . , pd } ⊂ P is an H-essential subset (b) H F F α

satisfying {[p1 ]F H , . . . , [pd ]H } = {[p]H | p ∈ P }, then (p1 · . . . · pd ) H ⊂ H.

(c) For every q ∈ F \ F × , we have q α ∈ H.

→ F (P ), deﬁned by (d ) The map ∂ : H ∂(a) = pvp (a) , p∈P

is a divisor theory, and there is an epimorphism C ∗ (H, F ) → C(H). (e) If H is deﬁned in another factorial monoid F such that H is dense in F , then there exists an isomorphism Φ : Fred → Fred such that Φ(aF × ) = × aF for all a ∈ H.

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In particular, if H is dense in F , then (up to isomorphisms) Fred is uniquely determined by H.

=H

× ×H0 with a reduced Krull monoid H0 , and let H0 → 4. Suppose that H

××F(P ), F (P ) be a divisor theory. Then H is a C-monoid deﬁned in F1 = H H is dense in F1 , and if H is a C0 -monoid, then P is ﬁnite. Proof. 1. By deﬁnition, we have H ⊂ F0 , F0× ∩ H = F × ∩ H = H × , and C ∗ (H, F0 ) is ﬁnite by Lemma 3.3.5. Hence H is a C-monoid deﬁned in F0 . It is now obvious that H is deﬁned in F0 with subgroup V and exponent α, and that H is dense in F0 . 2. and 3. Replacing F by F0 as deﬁned in 1., we may assume that H is dense

is in F . Then it suﬃces to prove the assertions 3.(a) to 3.(d). Indeed, by 3.(d) H

a Krull monoid and C(H) is ﬁnite, by 3.(b) (H : H) = ∅, and by 3.(c) the exponent

divides α. To prove 3.(e), suppose that H is a C-monoid which is deﬁned of C(H)

be the complete integral closure of H inside and dense both in F and F ; let H

→H

such that

those inside F . Then there is an isomorphism Φ0 : H F and H

→ F → Fred and H

→ F → F Φ0 | H = idH . Since the maps H red are divisor theories, the assertions 3.(e) follows from the uniqueness of divisor theories. 3.(a) If a = εp1 · . . . · pn ∈ F , where n ∈ N0 , ε ∈ F × and p1 , . . . , pn ∈ P , then there exist elements a1 , . . . , an ∈ H such that vpi (ai ) > 0 for all i ∈ [1, n], and thus a | a1 · . . . · an . Hence H is coﬁnal in F . 3.(b) We deﬁne a homomorphism ×

π : q(F ) → F /V ×(Z/αZ)

(P )

np by π ε p = εV, (np + αZ)p∈P , p∈P

where ε ∈ F × and (np )p∈P ∈ Z(P ) . We set R = π −1 (π(H)) ⊂ q(F ), and we shall

= R ∩ F . π(H) is a subsemigroup of an abelian torsion prove that R = q(H) and H group, and therefore it is itself a group. Hence R is a group, and H ⊂ R implies q(H) ⊂ R.

⊂ q(F ), then there exists some c ∈ H such that cz k ∈ H for all k ∈ N. If z ∈ H Hence vp (cz k ) = vp (c)+kvp (z) ≥ 0 for all p ∈ P and k ∈ N, which implies vp (z) ≥ 0 for all p ∈ P and thus z ∈ F . Therefore it remains to show that R ⊂ q(H) and

R ∩ F ⊂ H. Suppose that z ∈ R, and let x ∈ H be such that π(z) = π(x). Since H is dense in F , there exists some x0 ∈ H such that Q = suppP (x0 ) ⊃ suppP (x) ∪ suppP (z). We set pnp and z = v p mp , x=u p∈Q

p∈Q

where u, v ∈ F × , v −1 u ∈ V , np ∈ N0 , mp ∈ Z, and np ≡ mp mod α for all p ∈ Q. By Lemma 4.2 we have y0 =

p∈Q

pα ∈ H

and v −1 uy0l ∈ H

for all l ∈ N .

86

Halter-Koch

Let l ∈ N be such that mp + lα ≥ α for all p ∈ Q. Then xy0 = u pnp +α ∈ H , v −1 uy0l z = u pmp +lα ∈ F , p∈Q

p∈Q

hence v −1 uy0l z ∈ H, and z = (v −1 uy0l )−1 (v −1 uy0l z) ∈ q(H). Assume now that z ∈ R ∩ F . Then mp ≥ 0 for all p ∈ P . Let {p1 , . . . , pd } ⊂ P F F be an H-essential subset satisfying {[p1 ]F H , . . . , [pd ]H } = {[p]H | p ∈ P }. Then α w0 = (p1 · . . . · pd ) ∈ H by Lemma 4.2, and we assert that w0 z k ∈ H for all k ∈ N. F Let Q = Q1 . . . Qd be a partition such that Qi ⊂ {p ∈ Q | [p]F H = [pi ]H } for all i ∈ [1, d]. For all k ∈ N, we have w0 z k = v k

d i=1

pα i

(u−1 v)k w0 xk = v k

pkmp ∈ F,

d i=1

p∈Qi

pα i

pknp ∈ H ,

p∈Qi

and we consider the elements zk = v k

d

pikmi +α ∈ F

and xk = v k

i=1

d

i +α pkn ∈F i

i=1

where mi =

p∈Qi

mp and ni =

np .

p∈Qi

−1 k v) w0 xk ]F We have [xk ]F H = [(u H , and thus xk ∈ H, hence zk ∈ H by Lemma 4.2, F k F k and since [zk ]H = [w0 z ]H , it follows that w0 z ∈ H. 3.(c) If q ∈ F \ F × and a ∈ q α F ∩ H, then q α a ∈ H and therefore q α = −1 α

a (q a) ∈ q(H) ∩ F = H. 3.(d) If ϕ : F → q(F )/q(H) denotes the natural homomorphism deﬁned by

Hence H

⊂ F is a saturated ϕ(a) = aq(H), then ϕ−1 (1) = F ∩ q(H) = H.

is a Krull monoid, and the composite map ∂ : H

→ F = submonoid. Therefore H F × ×F(P ) → F (P ) is a divisor homomorphism. It is given by

, pvp (a) for all a ∈ H ∂(a) = p∈Q

and and we assert that it is a divisor theory. Indeed, if p ∈ P , then pα ∈ H, since vp (H) is a numerical monoid, there exists some z ∈ H such that vp (z) =

vp (p−nα z) = 1, and nα + 1 for some n ∈ N. Then p−nα z ∈ F ∩ q(H) = H, p = gcd ∂(pα ), ∂(p−nα z) .

We It remains to prove that there exists an epimorphism C ∗ (H, F ) → C(H).

∼

= have C(H) C(∂) = F (P )/∂ H = q(F (P ))/q(∂ H), and we assert that q(∂ H) = ×

F q(H) ∩ q(F (P )) ⊂ q(F (P )). Indeed, suppose that z ∈ F(P ). If z ∈ q(∂ H), −1 ×

then z = a b, where a, b ∈ F(P ) and aε, bη ∈ H for some ε, η ∈ F . Then z =

Conversely, if z ∈ F × q(H),

then z = ε0 (εa)−1 (ηb), (εη −1 )(aε)−1 (bη) ∈ F × q(H).

Then z ∈ F(P ) implies where ε0 , ε, η ∈ F × , a, b ∈ F(P ) and εa, ηb ∈ H. −1 −1

ε0 ε η = 1 and z = a b ∈ q(∂ H), since ∂(εa) = a and ∂(ηb) = b.

C-Monoids and Congruence Monoids in Krull Domains

87

By Proposition 3.7.1, there exists an epimorphism θ : C ∗ (H, F ) → F/F × H. We

and

= q(F )/F × q(H) combine it with the natural epimorphism F/F × H → F/F × H consider the isomorphisms

= F × q(F (P ))/F × q(H)

q(F )/F × q(H) ∼

∩ q(F (P )) = q(F (P ))/q(∂ H)

∼

= C(H) = q(F (P ))/F × q(H) to obtain the desired epimorphism. 4. Replacing F by F0 as deﬁned in 1., we may assume that H is a C-monoid deﬁned in a factorial monoid F = F × ×F(P ) such that H is dense in F , and if H

⊂ F , and the composite is a C0 -monoid, then P is ﬁnite. By 3. we have H ⊂ H

splits in the form map ∂ : H → F → F (P ) is a divisor theory. The Krull monoid H ×

=H

×H0 , where H0 is a reduced Krull monoid, ∂ | H0 is injective, and thus we H

× ×F(P ) ⊂ F , may assume that H0 → F (P ) is a divisor theory. Then H ⊂ F1 = H and by Proposition 4.4 H is a C-monoid deﬁned in F1 . By construction, H is dense in F1 .

Theorem 4.7. 1. Let H be a Krull monoid with ﬁnite class group. Then H is a C-monoid, and H is a C0 -monoid if and only if Hred is ﬁnitely generated. 2. Let H be a C-monoid deﬁned in a factorial monoid F such that C(H, F ) is a group. Then H is a Krull monoid. Proof. 1. Let ∂ : H → F (P ) be a divisor theory, and H = H × ×H0 , where H0 is a reduced Krull monoid. Then ∂ | H0 is injective and thus we may assume that H0 → F (P ) is a divisor theory. Then H ⊂ F = H × ×F(P ), F × = H × , and by Proposition 3.7 we have C(H, F ) ∼ = F/H ∼ = F0 /H0 = C(H). Hence C(H, F ) is ﬁnite, and H is a C-monoid. By [12, Satz 4], Hred is ﬁnitely generated if and only if P is ﬁnite. If P is ﬁnite, then H is a C0 -monoid by deﬁnition, and if H is a C0 -monoid, then P is ﬁnite by

since H is a Krull monoid). Theorem 4.6 (note that H = H, 2. By Proposition 3.6.3, H ⊂ F is saturated, and thus it is a Krull monoid.

Theorem 4.8. Every C-monoid is v-noetherian. The proof of Theorem 4.8 depends on the following Lemma 4.9, which is a reﬁnement of Dickson’s Finiteness Theorem. Lemma 4.9. Let s ∈ N, ∅ = M ⊂ Ns0 , and let Min(M ) be the set of all minimal points of M . Then there exists a ﬁnite set M such that Min(M ) ⊂ M ⊂ M with the following property: If n = (n1 , . . . , ns ) ∈ M , n∗ = (n∗1 , . . . , n∗s ) ∈ Min(M ) and n∗ ≤ n, then there exists some n = (n1 , . . . , ns ) ∈ M such that n∗ ≤ n ≤ n and {i ∈ [1, s] | n∗i < ni } = {i ∈ [1, s] | n∗i < ni }.

88

Halter-Koch

Proof of Lemma 4.9. For n∗ = (n∗1 , . . . , n∗s ) ∈ Min(M ) and I ⊂ [1, s], we denote by M (n∗ , I) the set of all n = (n1 , . . . , ns ) ∈ M such that n ≥ n∗ and I = {i ∈ [1, s] | ni > n∗i }. Let M0 (n∗ , I) be the set of all minimal points of M (n∗ , I). It is ﬁnite by Dickson’s Theorem [16, Theorem 1.1], and therefore the set M = M0 (n∗ , I) n∗ ∈Min(M) I⊂[1,s]

is also ﬁnite and, by construction, it has the desired property. Proof of Theorem 4.8. Let H be a C-monoid deﬁned in a factorial monoid F = F × × F(P ) with subgroup V and exponent α. By [16, Theorem 24.1] we must prove that for every subset X ⊂ H there exists a ﬁnite subset Y ⊂ X such that Y −1 ⊂ X −1 . We may assume that X = ∅, we ﬁx an element x0 ∈ X and distinct primes p1 , . . . , ps ∈ P such that vp (x0 ) = 0 for all p ∈ P = P \ {p1 , . . . , ps }. We deﬁne homomorphisms v : F → Ns0 and π : F → F ×/V ×(Z/αZ)s ×C ∗ (H, F ) as follows: If x = εpn1 1 · . . . · pns s a, where ε ∈ F × , n1 , . . . , ns ∈ N0 and a ∈ F(P ), then v(x) = (n1 , . . . , ns ) ∈ Ns0 and

× s ∗ π(x) = εV, n1 + αZ, . . . , ns + αZ, [a]F H ∈ F /V ×(Z/αZ) ×C (H, F ) .

Let X = X1 ∪ . . . ∪ Xt be a partition of X such that |π(Xτ )| = 1 for all τ ∈ [1, t]. For τ ∈ [1, t], we set Mτ = v(Xτ ) ⊂ Ns0 , and we choose a ﬁnite subset Mτ ⊂ Mτ with the property of Lemma 4.9. Let Yτ ⊂ Xτ be a ﬁnite subset satisfying the following two conditions: 1. For every n ∈ Mτ there exists some x ∈ Yτ such that v(x) = n. 2. If there exist elements x1 , x2 ∈ Xτ such that v(x1 ) = v(x2 ) ∈ Mτ and x−1 1 x2 ∈ F × F (P ) \ H, then Yτ also contains two elements with these properties. Let Y ⊂ X be a ﬁnite set such that Y1 ∪ . . . ∪ Yt ∪ {x0 } ⊂ Y . Then we have X −1 ⊂ Y −1 and vp (x) ≥ 0 for all x ∈ Y −1 and p ∈ P . We shall prove that Y −1 ⊂ X −1 . Suppose that z ∈ Y −1 , τ ∈ [1, t] and x ∈ Xτ , say x=ε

s i=1

pni i a

and z = η

s

plii z0 ,

i=1

where ε, η ∈ F × , n = (n1 , . . . , ns ) = v(x), l1 , . . . , ls ∈ Z and a, z0 ∈ F(P ). Let n∗ = (n∗1 , . . . , n∗s ) ∈ Min(Mτ ) and n = (n1 , . . . ns ) ∈ Mτ be such that ∗ n ≤ n ≤ n and {i ∈ [1, s] | n∗i < ni } = {i ∈ [1, s] | n∗i < ni }. For i ∈ [1, s], we set ni = n∗i + ki α and ni = n∗i + ki α, where ki , ki ∈ N0 and either ki = ki = 0 or ki ≥ ki ≥ 1. Let x∗ , x ∈ Yτ be elements such that v(x∗ ) = n∗ and v(x ) = n . Suppose that s s n∗ n x∗ = ε∗ pi i a∗ and x = ε p i i a , i=1

i=1

C-Monoids and Congruence Monoids in Krull Domains

89

where ε∗ , ε ∈ F × are such that ε∗−1 ε, ε−1 ε ∈ V , and a∗ , a ∈ F(P ) are such ∗ F F that [a]F H = [a ]H = [a ]H . Therefore we obtain zx = εη

s

n∗ +li +ki α

pi i

z0 a , zx = ε η

i=1 ∗

s

n∗ +li +ki α

pi i

z0 a , zx∗ = ε∗ η

i=1

s

n∗ +li

pi i

z 0 a∗ ,

i=1

zx , zx ∈ H, and we must prove that zx ∈ H. For all i ∈ [1, s], we have n∗i + li + ki α ≥ n∗i + li + ki α ≥ n∗i + li ≥ 0, and either n∗i + li + ki α = n∗i + li + ki α or n∗i + li + ki α ≥ n∗i + li + ki α ≥ α. Hence Lemma 4.2 implies y = ε η

s

n∗ +li +ki α

pi i

z 0 a ∈ H ,

i=1 F F −1 × × and since [a]F εy]F H = [a ]H , we obtain [zx]H = [ε H . If y ∈ F , then y ∈ H \ H , and ε−1 εy ∈ V (H \ H × ) ⊂ H implies zx ∈ H. If y ∈ F × , then z0 = a = 1 and n∗i + li + ki α = 0, hence n∗i + li + ki α = n∗i + li = 0 for all i ∈ [1, s]. Thus we obtain y = ε η = zx ∈ F × ∩ H = H × and zx = εηa ∈ F × F (P ). Assume that zx ∈ H. Then x−1 x = (zx )−1 zx = ε−1 εa ∈ F × F (P ) \ H, and since v(x) = v(x ) ∈ Mτ , there exists some x1 ∈ Yτ such that × −1 x−1 x1 ∈ F × F (P ) \ H or x−1 implies 1 x ∈ F F (P ) \ H. In any case, z ∈ Y zx1 ∈ H. If x−1 x1 ∈ F × F (P ) \ H, then zx1 = (zx )(x−1 x1 ) ∈ F × F (P ) \ H, a contradiction. −1 × × × = If x−1 1 x ∈ F F (P )\H, then zx = (zx1 )(x1 x ) ∈ H implies zx1 ∈ H ∩F −1 × × H and therefore x1 x ∈ H , again a contradiction.

Proposition 4.10. Let H be a C-monoid deﬁned in F = F × ×F(P ) with exponent α. For an H-essential subset E ⊂ P , we deﬁne uE = pα ∈ H , p∈E

and for any subset Q ⊂ P , we set HQ = {x ∈ H | suppP (x) ⊂ Q} = F (Q) ∩ H.

F××

1. {HQ | Q ⊂ P } is the set of all divisor-closed submonoids of H. In particular, every divisor-closed submonoid of a C-monoid is a C-monoid. 2. For every subset Q ⊂ P , we have HQ = [[{uE | E ⊂ Q is H-essential}]]. In particular, if Q ⊂ P is H-essential, then HQ = [[uQ ]]. 3. The following assertions are equivalent: (a) H is a C0 -monoid. (b) H has only ﬁnitely many divisor-closed submonoids. (c) H has only ﬁnitely many prime s-ideals.

90

Halter-Koch

Proof. Note that uE ∈ H by Lemma 4.2. 1. If Q ⊂ P , then HQ = (F ××F(Q))∩H is a C-monoid, and it is a divisor-closed submonoid of H by Proposition 4.4. Thus let S ⊂ H be a divisor-closed submonoid and set suppP (x) . Q= x∈S

Then S ⊂ HQ by deﬁnition. If x ∈ HQ , then suppP (x) ⊂ Q, and thus there exist x1 , . . . , xm ∈ S such that suppP (x) ⊂ suppP (x1 ) ∪ . . . ∪ suppP (xm ) = suppP (x1 · . . . · xm ). Hence x ∈ [[x1 · . . . · xm ]] ⊂ S by Lemma 4.2.2(d). 2. If Q ⊂ P and E ⊂ Q is H-essential, then uE ∈ HQ . Hence [[{uE | E ⊂ Q is H-essential}]] ⊂ HQ . Conversely, suppose that Q ⊂ P , x ∈ HQ and E = suppP (x). Since E = suppP (uE ), Lemma 4.2.2(d) implies x ∈ [[uE ]] ⊂ [[{uE | E ⊂ Q is H-essential}]]. 3. The equivalence of (b) and (c) follows from the deﬁnitions. (a) ⇒ (b) If H is a C0 -monoid, then we may assume that P is ﬁnite. Then H has only ﬁnitely many divisor-closed submonoids by 1. (b) ⇒ (a) By Theorem 4.6 we may assume that H is dense in F . If Q and Q are H-essential subsets of P and Q = Q , then HQ = HQ . Hence P has only ﬁnitely many H-essential subsets. However, by Lemma 4.2.2(c) every ﬁnite subset of P is contained in an H-essential subset of P . Therefore P is ﬁnite and H is a C0 -monoid. Theorem 4.11. Let H1 and H2 be monoids. Then H1 ×H2 is a C-monoid if and only if the following two conditions are satisﬁed: (a) H1 and H2 are both C-monoids. 1 × /H × and H 2 × /H × (b) Either H1 = H1× , or H2 = H2× , or the groups H 1 2 are both ﬁnite. Proof. If H1 × H2 is a C-monoid, then H1 and H2 are C-monoids by Proposition 4.10.1, since they are divisor-closed submonoids. Thus suppose that H1 and H2 are C-monoids. Then Theorem 4.6 implies that for i ∈ {1, 2} the monoid Hi is a C-monoid deﬁned in a factorial monoid Fi = i × × F(Pi ) such that there is a divisor theory H i → F (Pi ). Then H1 × H2 ⊂ H × × × H1 × H2 × F(P1 ) × F(P2 ) = (H1 ×H2 ) × F(P1 ∪ P2 ) = F1 × F2 , and there is a divisor theory H 1 ×H2 = H1 × H2 → F (P1 ) × F(P2 ) = F (P1 ∪ P2 ). Again by Theorem 4.6 and by the uniqueness of divisor theories, H1 ×H2 is a C-monoid if × and only if it is a C-monoid deﬁned in F1 ×F2 , and since (H 1 ×H2 ) ∩ (H1 ×H2 ) = × × × × × (H1 ∩ H1 )× (H2 ∩ H2 ) = H1 × H2 = (H1 × H2 ) , this is the case if and only if C ∗ (H1 × H2 , F1 × F2 ) is ﬁnite. The criterion given in Lemma 3.5 completes the proof. We close this section with some remarks concerning ﬁnitary monoids and Gmonoids. We recall the relevant deﬁnitions from [10]. A subset U of a monoid H is called an almost generating set , if U ⊂ H \ H × , and there exists some n ∈ N such that (H \ H × )n ⊂ U H. A monoid H is called

C-Monoids and Congruence Monoids in Krull Domains

91

• ﬁnitary , if it is a BF-monoid and possesses a ﬁnite almost generating set. • a G-monoid , if there exists an element a ∈ H lying in all nonempty prime s-ideals of H. A v-noetherian monoid is a G-monoid if and only if it has only ﬁnitely many prime s-ideals, and then it is also ﬁnitary [10, Proposition 4.5 and Theorem 4.7]. In [9, Theorem 5.9] it is proved that every C0 -monoid is ﬁnitary. We can do a little bit more. Proposition 4.12. Let H be a C-monoid. Then H is a G-monoid if and only if it is a C0 -monoid. Proof. By Theorem 4.8, H is v-noetherian, and therefore H is a G-monoid if and only if it has only ﬁnitely many prime s-ideals. Thus the assertion follows by Proposition 4.10.3. Example 4.13. The Hilbert monoid H = 2N ∪ {1} is a C-monoid deﬁned in N [8, Theorem 7.4]. Hence it is v-noetherian by Theorem 4.8. Since (H \ {1})2 ⊂ 2H, it follows that {2} is an almost generating set, and therefore H is ﬁnitary. For every prime number p, the set 2pN is a prime s-ideal of H. Hence H has inﬁnitely many prime s-ideals and thus it is not a G-monoid (see also [10, Remark 4.11.2]).

5

The Elasticity of C-Monoids

In [8, Theorem 7.2] we proved that a C-monoid H fulﬁlls all relevant ﬁniteness conditions of nonunique factorizations: It is locally tame, has ﬁnite catenary degree, and the Structure Theorem for Sets of Lengths holds for H. In particular, it is a BF-monoid. Monotone chains of factorizations in C-monoids are studied in [4]. The purpose of this section is to present a criterion for the elasticity of a C-monoid to be ﬁnite. Theorem 5.1. Let H be a C-monoid deﬁned in the factorial monoid F = F ××F(P ) with exponent α. 1. Assume that a ∈ H, and set supp∗P (a) = p ∈ suppP (a) {p} is H-essential . (a) If a ∈ A(H), then

vp (a) ≤ (2α − 1) |C ∗ (H, F )| .

p∈supp∗ P (a)

(b) We have the estimates 1 (2α − 1)|C ∗ (H, F )|

vp (a) ≤ min L(a)

p∈supp∗ P (a)

≤

vp (a)

p∈supp∗ P (a)

+ (3α − 1) | suppP (a)| .

92

Halter-Koch (c) If every minimal H-essential subset of suppP (a) is a singleton, then

max L(a) ≤

vp (a)

and

ρ(a) ≤ (2α − 1) |C ∗ (H, F )| .

p∈supp∗ P (a)

2. If H is simple in F , then ρ(H) ≤ (2α − 1) |C ∗ (H, F )|. Otherwise ρ(H) = ∞. Proof. 1.(a) Suppose that

vp (a) > (2α − 1) |C ∗ (H, F )| .

p∈supp∗ P (a)

Then there exist (not necessarily distinct) primes p1 , . . . , p2α ∈ supp∗P (a) such that F [p1 ]F H = . . . = [p2α ]H and a = p1 · . . . · p2α b for some b ∈ F . We set q = p1 · . . . · pα , α F F a = pα+1 · . . . · p2α b, a1 = p2α 1 b and a1 = p1 b. Then [a1 ]H = [a]H , hence α α F a1 ∈ H, and by Lemma 4.2 we obtain p1 ∈ H and a1 ∈ H. Since [p1 ]H = [q]F H and F [a1 ]F = [a ] , it follows that q ∈ H and a ∈ H, and therefore a = qa is not an H H atom of H. (b) If r = min L(a) and a = u1 ·. . .·ur with u1 , . . . , ur ∈ A(H), then (a) implies

vp (a) =

p∈supp∗ P (a)

r

vp (ui ) ≤ r(2α − 1) |C ∗ (H, F )| .

i=1 p∈supp∗ P (a)

Hence the ﬁrst inequality holds. To prove the second one, we consider F0 = F × ×F(P0 ), where P0 = suppP (a), and H0 = F0 ∩ H. By Proposition 4.4, H0 is a C0 -monoid deﬁned in F0 , and H0 is a divisor-closed submonoid of H. Hence we may replace H ⊂ F by H0 ⊂ F0 , and then the inequality follows by [9, Proposition 5.6]. (c) If every minimal H-essential subset of suppP (a) is a singleton, then every atom u ∈ A(H) dividing a satisﬁes vp (u) ≥ 1 for at least one p ∈ P ∗ . Hence max L(a) ≤

p∈P ∗

vp (a)

and

ρ(a) =

max L(a) ≤ (2α − 1) |C ∗ (H, F )| . min L(a)

2. If H is simple in F and a ∈ H, then every minimal H-essential subset of suppP (a) is a singleton, and thus ρ(a) ≤ (2α − 1) |C ∗ (H, F )| by 1.(c). Hence ρ(H) ≤ (2α − 1) |C ∗ (H, F )|. Assume now that H is not simple in F . Then there exists some a ∈ H such that suppP (a) is a minimal H-essential subset of P and |suppP (a)| ≥ 2. Then supp∗P (a) = ∅, and 1.(b) implies min L(an ) ≤ (3α − 1) | suppP (a)| for all n ∈ N , and thus ρ(H) = ∞.

hence

lim ρ(an ) = ∞ ,

n→∞

C-Monoids and Congruence Monoids in Krull Domains

6

93

Congruence Monoids in Krull Domains

We recall the deﬁnition and the elementary properties of congruence monoids from [9]. Let R be an integral domain. A map σ = (σ1 , . . . , σm ) : R• → {±1}m is called a sign vector of R, if there exist distinct ring monomorphisms w1 , . . . , wm : R → R such that σj (a) = sign{wj (a)} for all a ∈ R• and all j ∈ [1, m]. For m = 0, the empty sequence will also be considered as a sign vector. Let {0} = f R be an ideal and σ = (σ1 , . . . , σm ) a sign vector of R. Two elements a, b ∈ R• are called congruent modulo fσ, a ≡ b mod fσ, if a ≡ b mod f and σ(a) = σ(b). Congruence modulo fσ is a congruence relation on R• , and the semigroup of congruence classes is denoted by R/fσ. For a ∈ R• , we denote by [a]fσ ∈ R/fσ the congruence class containing a. Note that (R/fσ)× = {[a]fσ | a + f ∈ (R/f)× } is the group of invertible elements of R/fσ. For m = 0, the congruence modulo fσ is just the ordinary congruence modulo f, and we write f instead of fσ. If ∅ = Γ ⊂ R/fσ is a multiplicatively closed subset (not necessarily containing the unit element [1]fσ ), then the (multiplicative) monoid HΓ = a ∈ R• | [a]fσ ∈ Γ ∪ {1} ⊂ R• is called the congruence monoid deﬁned in R modulo fσ by Γ. It is called • regular modulo f , if aR + f = R for all a ∈ H. • singular modulo f , if aR + f = R for all a ∈ H \ {1}. Note that H is regular modulo f if and only if Γ ⊂ (R/fσ)× , and H is singular modulo f if and only if Γ ∩ (R/fσ)× = ∅. A submonoid H ⊂ R• is called a congruence monoid in R, if there exists an ideal f = {0} of R, a sign vector σ of R and a multiplicatively closed subset ∅ = Γ ⊂ R/fσ such that H = HΓ . Every such ideal f is called an ideal of deﬁnition for H. If f is an ideal of deﬁnition for H and R/f is noetherian, then H has a largest ideal of deﬁnition (see [9, Proposition 3.4]). The most interesting examples of congruence monoids are Hilbert monoids of natural numbers and orders in Krull domains. Regular Hilbert monoids were discussed in [13] and regular congruence monoids in Dedekind domains in [14]. For the analytical number theory of congruence monoids we refer to [11] and for the deeper arithmetical theory of congruence monoids in Dedekind domains we refer to [9]. The arithmetical theory of Hilbert monoids is also discussed in [8]. In this chapter we investigate congruence monoids in Krull domains and give a common generalization of the theory developed in [9] and the investigations of W. Hassler [17] on certain noetherian domains. Lemma 6.1. Let R be an integral domain, f = {0} an ideal of R, σ a sign vector of R, ∅ = Γ ⊂ R/fσ a multiplicatively closed subset, and H = HΓ ⊂ R• . 1. Suppose that either H is singular modulo f or that Γ ∩ (R/fσ)× is a subgroup of (R/fσ)× . Then R× ∩ H = H × .

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Halter-Koch 2. If (R/fσ)× is a torsion group, then either H is singular modulo f or Γ ∩ (R/fσ)× is a subgroup of (R/fσ)× .

Proof. 1. Obviously, H × ⊂ R× ∩ H, and if H is singular modulo f, then R× ∩ H = {1} = H × . If Γ ∩ (R/fσ)× is a subgroup of (R/fσ)× and a ∈ R× ∩ H, then × −1 [a−1 ]fσ = [a]−1 ∈ H and therefore a ∈ H × . fσ ∈ Γ ∩ (R/fσ) . Hence a 2. Observe that every nonempty multiplicatively closed subset of a torsion group is a subgroup. If H is a congruence monoid in a Dedekind domain, f is its largest ideal of deﬁnition, and H = HΓ for some Γ ⊂ R/fσ such that Γ ∩ (R/fσ)× is a group, then H is regular modulo f if and only if H is a Krull monoid (see [9, Theorem 3.7]). In a Krull domain, we have the following weaker result. Proposition 6.2. Let R be a Krull domain, {0} = f R an ideal and σ a sign vector of R, ∅ = Γ ⊂ R/fσ a multiplicatively closed subset, H = HΓ ⊂ R• and H ∗ = {a ∈ H | a + f ∈ (R/f)× }. Suppose that either H is singular modulo f, or that Γ ∩ (R/fσ)× is a subgroup of (R/fσ)× . 1. H ∗ is a saturated submonoid of R• . In particular, H ∗ is a Krull monoid. 2. If H is regular modulo f, then H is a Krull monoid. Proof. 1. If H is singular modulo f, then H ∗ = {1}. Thus let Γ ∩ (R/fσ)× be a subgroup of (R/fσ)× . If a, b ∈ H ∗ and b = ac for some c ∈ R• , then [a]fσ ∈ × ∗ Γ ∩ (R/fσ)× , and [c]fσ = [a]−1 fσ [b]fσ ∈ Γ ∩ (R/fσ) implies c ∈ H . 2. If H is regular modulo f, then H = H ∗ is a Krull monoid by 1.

Theorem 6.3. Let R be a Krull domain, H ⊂ R• a congruence monoid in R and f an ideal of deﬁnition for H. Let C(R) and R/f be ﬁnite, and suppose that either R is noetherian or f is divisorial. Then H is a C-monoid. Proof. Being a Krull monoid, R• splits in the form R• = R× × D, where D is a reduced Krull monoid, and there is a divisor theory D → F0 = F (P0 ). Then the composite map ∂ : R• → D → F0 is also a divisor theory. For every a ∈ F0 , ∂ −1 (aF0 ) ∪ {0} is a divisorial ideal of R (see [16, Theorem 20.5]). Let σ be a sign vector of R and ∅ = Γ ⊂ R/fσ a multiplicatively closed subset such that H = HΓ . By deﬁnition, H ⊂ R× ×D ⊂ R× ×F0 = F , F is factorial, and F × ∩ H = R× ∩ H. Since R/f is ﬁnite, Lemma 6.1 implies R× ∩ H = H × . Thus it remains to prove that C ∗ (H, F ) is ﬁnite. We consider the group G = q(F )/q(R• ). It is ﬁnite, since it is isomorphic to C(R), and since R• ⊂ F is coﬁnal, we have G = {zq(R• ) | z ∈ F }. Therefore it is × suﬃcient to prove that for each g ∈ G the set {[x]F H | x ∈ F ∩ g \ F } is ﬁnite. × Let g ∈ G be given. If z = εz0 ∈ F ∩ (−g), where ε ∈ R and z0 ∈ F0 , then z(F ∩ g) = zF ∩ q(R• ) = R× ×(z0 F0 ∩ q(D)) = R× ×(z0 F0 ∩ F0 ∩ q(D)) = R× ×(z0 F0 ∩ D) = ∂ −1 (z0 F0 ) .

C-Monoids and Congruence Monoids in Krull Domains

95

We ﬁx an element w ∈ F ∩ −g, and we set E = {x ∈ F ∩ g | x(F ∩ −g) ⊂ f} ,

a = w(F ∩ g) ∪ {0} and b = (wE) ∪ {0} .

Then a is a divisorial ideal of R, and b ⊂ a. We shall prove the following assertions: 1) b is an ideal of R; 2) af ⊂ b; 3) a/b is ﬁnite; 4) If x, y ∈ F ∩ g \ F × and F wx ≡ wy mod bσ, then [x]F H = [y]H . × Once this is done, the ﬁniteness of the set {[x]F H | x ∈ F ∩ g \ F } follows. Indeed, by 4) there is a surjective map × , given by [wx]bσ → [x]F Z = [wx]bσ x ∈ F ∩g\F × → [x]F H x ∈ F ∩g\F H

and clearly Z ⊂ {[a]bσ | a ∈ a}. Since a/b is ﬁnite and every residue class modulo b splits into ﬁnitely many residue classes modulo bσ (by [9, Lemma 3.2]), it follows that Z is ﬁnite. 1) We prove that b, b ∈ b and c ∈ R implies b + b ∈ b and cb ∈ b. We may assume that b, b , b + b and c are all diﬀerent from zero. Then b = wx, b = wx and b + b = wy, where x, x , y ∈ F ∩ g, x(F ∩ −g) ⊂ f and x (F ∩ −g) ⊂ f. If z ∈ F ∩ −g, then yz = yw(w−1 z) = (xw + x w)w−1 z = xz + x z ∈ f. Hence b + b ∈ b, and cx(F ∩ −g) ⊂ f implies cb = wcx ∈ b. 2) If 0 = a ∈ a and c ∈ f, then a = wx, where x ∈ F ∩ g, ca = w(cx), and cx(F ∩ −g) ⊂ cR ⊂ f. 3) Since fa ⊂ b, it follows that a/b is an R/f-module, and since R/f is ﬁnite, it is suﬃcient to prove that a/b is a ﬁnitely generated R/f-module. If R is noetherian, then a is ﬁnitely generated, and consequently a/b is also ﬁnitely generated. If f is divisorial, then all prime ideals containing f are minimal over f. Hence they are divisorial, and there are only ﬁnitely many of them, say p1 , . . . , pk (see [16, Proposition 6.6 and Theorem 24.2]). If S = R \ (p1 ∪ . . . ∪ pk ), then S −1 R is a semilocal Krull domain and {S −1 p1 , . . . , S −1 pk } = max(S −1 R). Hence S −1 R is one-dimensional, and therefore it is a principal ideal domain, and a/b = S −1 (a/b) = S −1 a/S −1 b is a ﬁnitely generated module over S −1 (R/f) = R/f (the latter argument is due to F. Kainrath). 4) Suppose that x, y ∈ F ∩ g \ F × = F ∩ g \ R× and wx ≡ wy mod bσ. Then wx − wy ∈ b = wE ∪ {0}, say wx − wy = wu for some u ∈ E ∪ {0}. We must prove that, for all z ∈ F , zx ∈ H implies zy ∈ H. If z ∈ F and zx ∈ H, then z ∈ F ∩ −g, and therefore zu ∈ f. Since zu = (zw−1 )(wu) = zx − zy and σ(zx) = σ(zw−1 )σ(wx) = σ(zw−1 )σ(wy) = σ(zy), it follows that zx ≡ zy mod fσ. Since x ∈ / R× , we have zx = 1 and hence zy ∈ H. Corollary 6.4. Let A be a noetherian domain, R its integral closure and f = AnnA (R/A) = {0}. 1. R is a Krull domain, and A• is a congruence monoid deﬁned in R modulo f by A/f. 2. If C(R) and R/f are ﬁnite, then A• is a C-monoid.

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Proof. By [5, Theorem 4.3], R is a Krull domain, and obviously A• is a congruence monoid deﬁned in R modulo f by A/f (see also [11] and [9, Example 3.3.1]). Thus the assertions follow by Theorem 6.3. An even stronger version of Corollary 6.4 was proved by W. Hassler [17, Theorem 3.2], who also showed the ﬁniteness of the unit factor group R× /A× .

Bibliography [1] D. D. Anderson (ed.), Factorization in integral domains, Marcel Dekker, 1997. [2] D. F. Anderson, Elasticity of factorizations in integral domains: a survey, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 1–29. [3] S.T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 73–112. [4] A. Foroutan and A. Geroldinger, Monotone chains of factorizations in Cmonoids, these Proceedings. [5] R.M. Fossum, The divisor class group of a Krull domain, Springer, 1973. [6] A. Geroldinger, On the structure and arithmetic of ﬁnitely primary monoids, Czech. Math. J. 46 (1996), 677–695. [7] A. Geroldinger, The catenary degree and tameness of factorizations in weakly Krull domains, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 113–153. [8] A. Geroldinger and F. Halter-Koch, Transfer principles in the theory of nonunique factorizations, these Proceedings. [9] A. Geroldinger and F. Halter-Koch, Congruence monoids, Acta Arith. 112 (2004), 263–296. [10] A. Geroldinger, F. Halter-Koch, W. Hassler, and F. Kainrath, Finitary Monoids, Semigroup Forum 67 (2003), 1–21. [11] A. Geroldinger, F. Halter-Koch, and J. Kaczorowski, Non-unique factorizations in orders of global ﬁelds, J. Reine Angew. Math. 459 (1995), 89–118. [12] F. Halter-Koch, Halbgruppen mit Divisorentheorie, Expo. Math. 8 (1990), 27– 66. [13] F. Halter-Koch, Arithmetical Semigroups Deﬁned by Congruences, Semigroup Forum 42 (1991), 59–62. [14] F. Halter-Koch, Ein Approximationssatz f¨ ur Halbgruppen mit Divisorentheorie, Result. Math. 19 (1991), 74–82. 97

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[15] F. Halter-Koch, Finitely generated monoids, ﬁnitely primary monoids and factorization properties of integral domains, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 73–112. [16] F. Halter-Koch, Ideal systems. An introduction to multiplicative ideal theory, Marcel Dekker, 1998. [17] W. Hassler, Factorization in ﬁnitely generated domains, J. Pure Appl. Algebra 186 (2004), 151–168. [18] F. Kainrath, Elasticity of ﬁnitely generated domains, Houston J. Math. 30 (2004).

Chapter 5

Monotone Chains of Factorizations in C-Monoids by

1

Andreas Foroutan and Alfred Geroldinger1

Introduction and Main Result

C-monoids were recently introduced by F. Halter-Koch as a common generalization of various types of auxiliary monoids studied in factorization theory. C-monoids are suitably deﬁned submonoids of factorial monoids (cf. Deﬁnition 2.1), and they include Krull monoids with ﬁnite class groups and congruence monoids in Krull domains satisfying some natural ﬁniteness conditions. In particular, let A be a noetherian domain, R its integral closure and f = AnnA (R/A) = {0}. Then the multiplicative monoid A• of A is a congruence monoid in R, and if the class group C(R) and R/f are ﬁnite, then A• is a C-monoid ([12, Theorem 3.2] and [9, Corollary 6.4]). C-monoids have nice algebraic and arithmetical properties. Every C-monoid H

is a Krull monoid with ﬁnite class is v-noetherian, its complete integral closure H

group and the conductor (H : H) is nonempty ([9, Theorem 4.6]). Moreover, H is locally tame, has ﬁnite catenary degree, satisﬁes the Structure Theorem for Sets of Lengths, and there is an explicit criterion for the ﬁniteness of its elasticity ([7, Theorem 5.9], [5, Theorem 7.2] and [9, Theorem 5.9]). This chapter continues the investigations of the arithmetic of C-monoids. We discuss the main result which is formulated below (Theorem 1.1). Let H be a monoid (i.e., a commutative, cancellative semigroup with unit element) and H × its group of invertible elements. Then H is atomic, if every nonunit a ∈ H \ H × has a factorization (product decomposition) into atoms (irreducible elements) of H, and H is factorial, if it is atomic and every nonunit has a unique factorization 1 This

work was supported by the Austrian Science Fund FWF (Project-Nr. P16770-N12)

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into atoms. Suppose that H is atomic but not factorial. Then there exists some a ∈ H \ H × having two distinct factorizations z and z , say z = u1 · . . . · ul · v1 · . . . · vm

and z = u1 · . . . · ul · w1 · . . . · wn

where ui , vj , wk are atoms and vj H × = wk H × for all i ∈ [1, l], j ∈ [1, m] and k ∈ [1, n]. Then |z| = l + m (resp. |z | = l + n) is called the length of z (resp. the length of z ), and d(z, z ) = max{m, n} is called the distance between z and z . A simple observation shows that for every N ∈ N there exist some aN ∈ H \ H × and factorizations zN , zN of aN such that d(zN , zN ) ≥ N . The monoid H has ﬁnite catenary degree, if there exists some constant M ∈ N such that for every a ∈ H \H × each two factorizations z, z of a can be concatenated by an M -chain of factorizations (i.e., there exists a ﬁnite sequence of factorizations (z = z0 , z1 , . . . , zk+1 = z ) of a such that d(zi−1 , zi ) ≤ M for every i ∈ [1, k + 1]). It is well-known that C-monoids have ﬁnite catenary degree, but up to now there was no information about length or structure of concatenating chains. Theorem 1.1. Let H be a C-monoid. Then there exists some constant M ∈ N having the following property: for every a ∈ H and for each two factorizations z, z of a there exist factorizations z = z0 , z1 , . . . , zk+1 = z of a such that, for every i ∈ [1, k + 1], d(zi−1 , zi ) ≤ M

and (either |z1 | ≤ . . . ≤ |zk | or |z1 | ≥ . . . ≥ |zk | ).

Thus there exists some M ∈ N such that for every a ∈ H and each two factorizations of a there is a concatenating M -chain for which the associated sequence of lengths (|z0 |, |z1 |, . . . , |zk |, |zk+1 |) is monotone, with possible exceptions at the ﬁrst and the last step. These exceptions do not occur in Krull monoids with ﬁnite class groups (see [4, Theorem 5.1]). Let A be a one-dimensional local noetherian domain A with integral closure R such that f = AnnA (R/A) = {0} and R/f is ﬁnite, whence A• is a C-monoid. W. Hassler proved an arithmetical property for A• implying Theorem 1.1, but on the other hand he could show that these exceptions actually occur. Thus there is a domain A as above for which there is no M ∈ N such that for every a ∈ A• each two factorizations of a can be concatenated by a monotone M -chain of factorizations (see [11, Theorem 4.1 and Example 6.3]). The ﬁrst step in the proof of Theorem 1.1 is to reduce the problem to those C-monoids which are submonoids of ﬁnitely generated factorial monoids (see [5, Theorem 7.2]). This allows to apply Dickson’s Finiteness Theorem on minimal points of subsets S ⊂ Ns0 . Furthermore, these simpler C-monoids are ﬁnitary in the sense of [8], and we can use the machinery of tamely generated ideals as developed in [6].

2

Preliminaries

Our terminology and our notations are consistent with [7], [9] and [5]. For convenience we recall some key notions. We denote by N the set of positive integers and

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set N0 = N ∪ {0}. For integers m, n ∈ Z let [m, n] = {x ∈ Z | m ≤ x ≤ n}, and for a set X we denote by |X| ∈ N0 ∪ {∞} its cardinality. Basic Notions on Monoids. Throughout this chapter, a semigroup is a nonempty set with a commutative and associative law of composition possessing a unit element. We use multiplicative notation and denote the unit element by 1. By a monoid, we mean a cancellative semigroup. Let H be a monoid. Then H × denotes its group of invertible elements and Hred = {aH × | a ∈ H} the associated reduced monoid of H. We say that H is reduced, if H × = {1}. An element u ∈ H is called irreducible (or an atom of H), if u ∈ / H × and, for all a, b ∈ H, u = ab × × implies that a ∈ H or b ∈ H . We denote by A(H) the set of atoms of H, and we say that H is atomic if every a ∈ H \ H × is a product of atoms. An element p ∈ H is called a prime of H, if p ∈ / H × , and if a, b ∈ H, then p | ab implies that p | a or p | b. Every prime is an atom, and a monoid is factorial if and only if it is atomic and every atom is a prime. If H is atomic and p ∈ H is a prime element of H, then every a ∈ H may be written in the form a = pn b where b ∈ H and n ∈ N0 . The exponent n is uniquely determined by the classes aH × and pH × in Hred , and we call n = vp (a) the p-adic value of a. For a set P , we denote by F (P ) the free abelian monoid with basis P . Every a ∈ F(P ) has a unique representation in the form pvp (a) where vp (a) ∈ N0 and vp (a) = 0 for all but ﬁnitely many p ∈ P . a= p∈P

For a, b ∈ F(P ), we set |a| =

p∈P

vp (a)

and d(a, b) = max

b a , . gcd(a, b) gcd(a, b)

Note that | · | : F (P ) → N0 is a homomorphism, and d : F (P ) × F(P ) → N0 is a metric. For every element a ∈ H we set [[a]] = {b ∈ H | b | an for some n ∈ N} ⊂ H (then [[a]] is the smallest divisor-closed submonoid containing a). A subset a ⊂ H is called an s-ideal of H, if aH = a. Finitary Monoids. Let H be a monoid with H = H × . A subset U ⊂ H \ H × is called an almost generating set of H, if there exists some n ∈ N such that (H \ H × )n ⊂ U H. We denote by M(U ) the smallest possible n ∈ N for which the above inclusion holds. If U is an almost generating set, then U [θ] = {uθ | u ∈ U } is an almost generating set for every θ ∈ N (cf. [8, Lemma 3.4.1]). The monoid H is called ﬁnitary, if it satisﬁes the ascending chain condition on principal ideals, and if it has some ﬁnite almost generating set U . Finitary monoids were introduced in [8] as a powerful tool in factorization theory. Main examples of ﬁnitary monoids are ﬁnitely generated monoids, strongly primary monoids, v-noetherian G-monoids and abstract congruence monoids. Let H be a ﬁnitary monoid, a ⊂ H an s-ideal, U ⊂ H \ H × a ﬁnite almost generating set and u ∈ U . We denote by Hu the set of all a ∈ H without a divisor in [[u]] \ H × , by a(U, u) the set of all a ∈ a ∩ u2 H such that [[u]] is maximal in the set {[[v]] | v ∈ U, a ∈ v 2 H}, and we set a[U, u] = Hu ∩ (u[[u]])−1 a(U, u) .

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A simple calculation gives that a⊂ u[[u]] a[U, u] ∪ a \ U [2] H u∈U

(cf. [6, Proposition 5.5]), and we say that a is U -generated, if equality holds. Clearly, the s-ideal H is U -generated. A ﬁnite almost generating set W ⊂ H is called full, if there exists some m ∈ N such that H[W, w] ⊂ H \ W m H for all w ∈ W , and every ﬁnitary monoid has such an almost generating set. Class semigroups. Class semigroups were recently introduced in [7, Section 4] as a reﬁnement of ordinary class groups in algebraic number theory. We repeat the deﬁnition for a situation which is suﬃcient for our requirements. Let D be a monoid and H ⊂ D a submonoid. Two elements y, y ∈ D are called H-equivalent, −1 if y −1 H ∩ D = y H ∩ D. H-equivalence is a congruence relation on D. For y ∈ D, D let [y]H denote the congruence class of y, and let ∗ D × C(H, D) = {[y]D H | y ∈ D} and C (H, D) = {[y]H | y ∈ (D \ D ) ∪ {1}}.

Then C(H, D) is a semigroup with unit element [1]D H (called the class semigroup of H in D) and C ∗ (H, D) ⊂ C(H, D) is a subsemigroup (called the reduced class semigroup of H in D). If C(H, D) is a torsion group, then H ⊂ D is saturated and coﬁnal, and if H ⊂ D is saturated and coﬁnal, then C(H, D) is isomorphic to the factor group of the quotient groups of D and H (cf. [9, Section 3]). C-monoids and C0 -monoids. C-monoids generalize abstract congruence monoids deﬁned in factorial monoids and Z-monoids (cf. [7] and [12]). For their relevance and their properties we refer to the paper of F. Halter-Koch in this volume (see [9]). Here we only give the deﬁnition and a characterization of C0 -monoids which is important for the present investigations. Definition 2.1. A monoid H is called a C-monoid, if it is a submonoid of a factorial monoid F such that H ∩ F × = H × and C ∗ (H, F ) is ﬁnite. A C0 -monoid is a C-monoid H deﬁned in a factorial monoid F having only ﬁnitely many pairwise nonassociated prime elements. Theorem 2.2. Let F = F × × [p1 , . . . , ps ] be a factorial monoid with pairwise nonassociated prime elements p1 , . . . , ps , and let H ⊂ F be a submonoid such that H ∩ F × = H × . Then the following assertions are equivalent: 1. H is a C0 -monoid deﬁned in F . 2. There exists some α ∈ N and a subgroup V ⊂ F × such that (F × : V ) | α, V (H \ H × ) ⊂ H, and for all j ∈ [1, s] and a ∈ pα j F we have a ∈ H if and a ∈ H. only if pα j In particular, every C0 -monoid in the sense of Deﬁnition 2.1 is a C0 -monoid in the sense of [7, Deﬁnition 5.1], and conversely. Let H ⊂ F be a C0 -monoid with all notations as in Theorem 2.2. Then a simple inductive argument shows that H satisﬁes the following congruence condition (C), which will turn out to be crucial for our investigations:

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(C) If u ∈ F and (n1 , . . . , ns ), (n1 , . . . , ns ) ∈ Ns0 are such that,

either ni = ni or ni ≡ ni mod α and min{ni , ni } ≥ α for each i ∈ [1, s], then u

s

pni ∈ H

implies

i=1

u

s

pni ∈ H .

i=1

Factorization Theory. Let H be a monoid. The monoid Z(H) = F (A(Hred )) is called the factorization monoid of H. Let π : Z(H) → Hred denote the unique homomorphism satisfying π(u) = u for all u ∈ A(Hred ). Then H is atomic if and only if π is surjective. For every a ∈ H the elements in Z(a) = π −1 (aH × ) are called factorizations of a. It was a crucial new idea, introduced in [4], to study arithmetical invariants (such as sets of lengths, catenary and tame degrees) with respect to a given subset of factorizations X ⊂ Z(H). We shall make substantial use of this idea and give the necessary deﬁnitions. Suppose that H is an atomic monoid, a ∈ H and X ⊂ Z(H) a subset. Note that in case X = Z(H), the forthcoming deﬁnitions coincide with the usual ones. Let ZX (a) = Z(a) ∩ X denote the set of factorizations of a which are in X and X for k ∈ N0 let ZX k (a) = {z ∈ Z (a) | |z| = k}. The set X ⊂ Z(H) is called M -dense for some nonnegative integer M ∈ N0 , if for every b ∈ H and every z ∈ Z(b) there exists some z ∈ ZX (b) such that d(z, z ) ≤ M . We say that X is a dense subset, if it is M -dense for some M ∈ N0 . We denote by LX (a) = {|z| | z ∈ ZX (a)} ⊂ N0 the set of lengths of a with respect to X. H is called a BF-monoid, if all sets of lengths LZ(H) (a) are ﬁnite. Finitary monoids are BF-monoids. Sets of lengths and all invariants derived from them, such as the elasticity of a monoid, are central objects of interest in factorization theory (cf. [2], [1], [3], [13]). Distances: Let L ⊂ Z. If m ∈ L and d ∈ N are such that [m, m + d] ∩ L = {m, m + d}, then m and m + d are called successive lengths of L, and d is called a distance of L. We denote by ∆(L) ⊂ N the set of all distances of L, and call ∆(LX (a)) ⊂ N ∆X (H) = a∈H

the set of distances of H with respect to X. For subsets Z, Z ⊂ Z(H) we set d(Z, Z ) = min{d(z, z ) | z ∈ Z, z ∈ Z } ∈ N0 . Let a ∈ H. If z, z ∈ ZX (a), then z = z if and only if d(z, z ) = 0; if z = z , then 2 ≤ d(z, z ) ≤ sup LX (a). We set X δ X (a) = sup{d({z}, ZX k (a)) | z ∈ Z (a),

|z| and k are successive lengths in LX (a)} ∈ N0 ∪ {∞} and δ X (H) = sup{δ X (a) | a ∈ H} ∈ N0 ∪ {∞}.

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Chains of Factorizations: If a ∈ H, z, z ∈ ZX (a) and N ∈ N0 , then an N -chain of X-factorizations from z to z is a ﬁnite sequence (z0 , z1 , . . . , zk ) in ZX (a) such that z = z0 , zk = z and d(zi−1 , zi ) ≤ N for all i ∈ [1, k]. We denote by cX (a) the smallest N ∈ N0 ∪ {∞} such that for each two factorizations z, z ∈ ZX (a) there is an N -chain of X-factorizations from z to z . We set cX (H) = sup{cX (a) | a ∈ H} ∈ N0 ∪ {∞}. If cX (H) < ∞, then ∆X (H) is ﬁnite. We say that H has ﬁnite catenary degree, if cZ(H) (H) < ∞. A chain of X-factorizations (z0 , . . . , zk ) is called monotone, if either |z0 | ≤ . . . ≤ |zk | or |z0 | ≥ . . . ≥ |zk |. We denote by cX m (a) the smallest N ∈ N0 ∪ {∞} such that for each two factorizations z, z ∈ ZX (a) there is a monotone N -chain of Xfactorizations from z to z . We call X cX m (H) = sup{cm (a) | a ∈ H} ∈ N0 ∪ {∞}

the monotone catenary degree of H with respect to X. Tameness: For a ∈ H and u ∈ Z(H), let tX (a, u) denote the smallest integer N ∈ N0 with the following property: if z ∈ ZX (a) and ZX (a) ∩ uZ(H) = ∅, then there exists some z ∈ ZX (a) ∩ uZ(H) such that d(z, z ) ≤ N . For subsets H ⊂ H and U ⊂ Z(H) we deﬁne tX (H , U ) = sup{tX (a, u) | a ∈ H , u ∈ U } ∈ N0 ∪ {∞}. H is called locally tame, if tZ(H) (H, u) < ∞ for all u ∈ A(Hred ). Let a ⊂ H be an s-ideal. A subset E ⊂ a is called a tame generating set of a (with bound M ∈ N), if for every a ∈ a there exists some e ∈ E satisfying e | a, sup L(e) ≤ M and tZ(H) (a, Z(e)) ≤ M (cf. [6, Deﬁnition 3.3 and Proposition 3.8]). For a ∈ H and u ∈ Z(H), let tX m (a, u) denote the smallest integer N ∈ N0 with the following property: if z ∈ ZX (a) and z ∈ ZX (a) ∩ uZ(H), then there exists some z ∈ ZX (a) ∩ uZ(H) such that d(z, z ) ≤ N and (either |z| ≤ |z | ≤ |z| or |z| ≤ |z | ≤ |z|). For subsets H ⊂ H and U ⊂ Z(H) we deﬁne X tX m (H , U ) = sup{tm (a, u) | a ∈ H , u ∈ U } ∈ N0 ∪ {∞}.

Whenever an arithmetical invariant appears without an upper index specifying the set of factorizations X ⊂ Z(H), then it is supposed that X = Z(H), so L(a) = LZ(H) (a), c(H) = cZ(H) (H) and so on. All arithmetical invariants introduced above are in fact invariants of the associated reduced monoid Hred . Hence we may suppose that H is reduced whenever this is convenient.

3

Reduction of the Theorem to the Main Proposition

In this section we formulate a Main Proposition and show how it implies Theorem 1.1. The proof of the Main Proposition will be given in Sections 4 and 5.

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105

Proposition 3.1 (Main Proposition). Let H be a reduced C0 -monoid deﬁned in a factorial monoid F having a ﬁnite group of units F × . Then there exist a ﬁnite set A ⊂ A(H), a ﬁnite almost generating set W of H, sets of factorizations Xw ⊂ Z(H) for every w ∈ W, a set of factorizations Y ⊂ Z(H) and a constant M ∈ N such that the following conditions are satisﬁed: 1. W is full and H=

w[[w]]H[W, w] ∪ (H \ W[2] H).

w∈W w 2. A is an almost generating set, AXw ⊂ Xw , A−1 Xw ∩Z(H) ⊂ Xw and ZX k (a)∩ Xw AZ(H) = ∅ for every a ∈ H, w ∈ W and every k ∈ L (a) with k ≥ M.

3. Y is M-dense. 4. Let w ∈ W, X = Xw and a ∈ w[[w]]H[W, w]. H[W, w], b ∈ Z(b) and a ∈ [[w]] such that a = b. Furthermore, for every z ∈ ZY (a) there are factorizations z = z0 , . . . , zk = z ∈ Z(a) such d(zi−1 , zi ) = 2 for every i ∈ [1, k].

Then there exist some b ∈ wa b and ZY (a) ⊂ Z(wa ) · some z ∈ ZX (wa ) · b and that |z0 | = . . . = |zk | and

5. tXw (H, v) < ∞ and ∆Xw (H) is ﬁnite for every v ∈ A and every w ∈ W. 6. δ Xw (H) < ∞ for every w ∈ W. Lemma 3.2. Let H be a reduced BF-monoid, X ⊂ Z(H) and A ⊂ A(H) such that AX ⊂ X and A−1 X ∩ Z(H) ⊂ X. 1. If there is some M ∈ N such that ZX (a) ∩ AZ(H) = ∅ for every a ∈ H with max LX (a) > M , then cX (H) ≤ sup{M, tX (H, A)}. 2. If there is some M ∈ N such that ZX k (a) ∩ AZ(H) = ∅ for every a ∈ H and every k ∈ LX (a) with k ≥ M , then X cX m (H) ≤ sup{M, tm (H, A)}.

Proof. 1. If tX (H, A) = ∞, then there is nothing to show. Suppose that this is not the case, and let a ∈ H. We prove that cX (a) ≤ max{M, tX (H, A)} = M ∗ . We proceed by induction on max LX (a). If max LX (a) ≤ M , then cX (a) ≤ max LX (a) ≤ M . Suppose that max LX (a) > M , and let z, z ∈ ZX (a). By assumption there exists some u ∈ A such that ZX (a) ∩ uZ(H) = ∅, and we set b = u−1 a. Then there are y = ux, y = ux ∈ ZX (a), with x, x ∈ Z(b), such that d(z, ux) ≤ tX (H, u) and d(z, ux ) ≤ tX (H, u). Since

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A−1 X ∩ Z(H) ⊂ X, it follows that x, x ∈ ZX (b). Since max LX (b) < max LX (a), there exists an M ∗ -chain (x, x1 , . . . , xk , x ) of X-factorizations from x to x . Since AX ⊂ X, (z, ux, ux1 , . . . , uxk , ux , z ) is an M ∗ -chain of X-factorizations from z to z. 2. [4, Lemma 3.5]. We brieﬂy analyze the assumption occurring in part 1. of Lemma 3.2. If A = A(H), then it holds with M = 1. Suppose that X = Z(H). Then there exists some ﬁnite set A ⊂ A(H) such that the assumption holds if and only if H is ﬁnitary. Lemma 3.3. Let H be a reduced atomic monoid and X ⊂ Z(H). 1. If u ∈ A(H) such that uX ⊂ X and u−1 X ∩ Z(H) ⊂ X, then X X X X tX m (H, u) ≤ δ (H) t (H, u) + sup ∆ (H) + t (H, u). 2. If there exists some a ∈ H with |ZX (a)| ≥ 2, then 2 + sup ∆X (H) ≤ cX (H). In particular, if cX (H) < ∞, then ∆X (H) is ﬁnite. Proof. 1. follows from [4, Lemma 3.7], and 2. from [4, Lemma 3.3]. Proof of Theorem 1.1. A suitable transfer homomorphism (as derived in [5, Lemma 3.5 and Theorem 7.2]) shows that it suﬃces to prove Theorem 1.1 for C0 -monoids deﬁned in factorial monoids having ﬁnite groups of units. By [9, Proposition 4.5], we may restrict to reduced monoids. Let H be a reduced C0 -monoid deﬁned in a factorial monoid F having a ﬁnite group of units F × . Thus H is as in the Main Proposition, and let A, W, Xw , Y and M be as there. First we show that, for every X ∈ {Xw | w ∈ W}, we have cX m (H) < ∞. By Proposition 3.1(5. and 6.) and Lemma 3.3.1, it follows that tX m (H, u) < ∞ for all u ∈ A. Therefore, Lemma 3.2.2 implies that cX (H) < ∞. m Now we show that the assertion of Theorem 1.1 holds with the constant w M = max{M(W[2] ), M, 2, cX m (H) | w ∈ W} ∈ N.

Let a ∈ H and z, z ∈ Z(a). If a ∈ H \ W[2] H, then d(z, z ) ≤ max L(a) ≤ M(W[2] ). Suppose that a ∈ W[2] H whence a ∈ w[[w]]H[W, w] for some w ∈ W and set X = Xw . Since Y is M-dense, there are z, z ∈ ZY (a) such that d(z, z) ≤ M and d(z , z ) ≤ M. By Proposition 3.1.4, there are y, y ∈ ZX (H), some b ∈ Z(H) and 2-chains of factorizations, all factorizations having the same lengths, from z to yb ∈ Z(a) and from z to y b. Since cX m (H) < ∞, there exists a monotone M -chain of factorizations ( y b, y1 b, . . . , yk b, y b) from yb to y b. Thus we obtain a monotone M -chain of factorizations from z to z , and the assertion follows.

Monotone Chains of Factorizations in C-Monoids

4

107

Construction of A, W, Xw , Y and M

We ﬁx our notations which remain valid throughout the rest of this chapter. Let H be as in the Main Proposition, whence H ⊂ F = F × × [p1 , . . . , ps ] is a reduced C0 -monoid in the factorial monoid F , where F × is ﬁnite and p1 , . . . , ps ∈ F are pairwise nonassociated primes of F . Let α ∈ N be as in Theorem 2.2. In particular, we have H ∩ F × = H × = {1}, and H satisﬁes the congruence condition (C). By [7, Theorem 5.9], H is ﬁnitary, locally tame and has ﬁnite catenary degree. For x ∈ F we call supp(x) = {i ∈ [1, s] | vpi (x) > 0} the support of x. A subset E ⊂ [1, s] is called H-essential, if there exists some a ∈ H such that E = supp(a), and we denote by EH the set of all nonempty H-essential subsets of [1, s]. Construction 4.1. We deﬁne A = {u ∈ A(H) | vpi (u) < 4α for all i ∈ [1, s]}. Since F × is ﬁnite, A is ﬁnite, and obviously every uI = i∈I pα i , where I ∈ EH , has some irreducible divisor lying in A. Since {uI | I ∈ EH } is an almost generating set of H by [7, Proposition 5.6], A is an almost generating set by [8, Lemma 3.4.2]. Lemma 4.2. There exists a map Θ : H → H satisfying the following two properties: 1. For every a ∈ H and every i ∈ [1, s] we have: if vpi (a) < 4α, then vpi (a) = vpi (Θ(a)). If vpi (a) ≥ 4α, then vpi (a) ≡ vpi (Θ(a)) mod α and vpi (Θ(a)) ∈ [2α, 3α − 1]. 2. Θ(A(H)) ⊂ A. Proof. Let a ∈ H and i ∈ [1, s]. If vpi (a) < 4α, we set ki = 0. If vpi (a) ≥ 4α, let ki ∈ N such that vpi (a) − ki α ∈ [2α, 3α − 1]. We deﬁne Θ(a) = a ·

s

iα p−k . i

i=1

Then (C) implies that Θ(a) ∈ H. Suppose that a ∈ A(H) and assume to the contrary that Θ(a) = a1 a2 with a1 , a2 ∈ H \ H × . We construct a1 , a2 ∈ H \ H × such that a = a1 a2 , a contradiction. We set a1 = a1 ·

s i=1

l

pi1,i

α

and a2 = a2 ·

s

l

pi2,i

α

i=1

where l1,1 , ..., l1,s , l2,1 , ..., l2,s ∈ N0 are deﬁned as follows. Let i ∈ [1, s]. If ki = 0, then l1,i = l2,i = 0 whence vpi (aj ) = vpi (aj ) for j ∈ [1, 2]. Suppose that ki ∈ N. If vpi (a1 ) ≥ α, then l1,i = ki and l2,i = 0. If vpi (a1 ) < α, then vpi (a2 ) = vpi (Θ(a)) − vpi (a1 ) ≥ α and we set l1,i = 0 and l2,i = ki . Then (C) implies that a1 , a2 ∈ H \ F × = H \ {1}, and by construction, we have a = a1 a2 .

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Lemma 4.3. Let u ∈ A(H), i ∈ [1, s] and n ∈ N0 . ∈ A(H). 1. If vpi (u) − nα ≥ 2α, then up−nα i 2. If vpi (u) ≥ 4α, then upnα ∈ A(H). i . Since pnα Proof. 1. Suppose that vpi (u) − nα ≥ 2α and set a = up−nα i a = u ∈ H, i it follows from (C) that a ∈ H. Assume to the contrary that a = a1 a2 with a1 , a2 ∈ H \ {1} and vpi (a1 ) ≥ vpi (a2 ). Then vpi (a1 ) ≥ α whence a1 pnα ∈ H \ {1} i and u = (a1 pnα ) · a , a contradiction. 2 i 2. Suppose that vpi (u) ≥ 4α. Again (C) implies that upnα ∈ H. Assume to i the contrary that upnα = a1 a2 with a1 , a2 ∈ H \ {1}. Without restriction we may i suppose that vpi (a1 ) ≥ 2α. We set vpi (a2 )−vpi (Θ(a2 ))−nα

a1 = a1 pi

vpi (Θ(a2 ))−vpi (a2 )

and a2 = a2 pi

.

Obviously, we have u = a1 a2 . We assert that a1 , a2 ∈ H \ {1} which implies a contradiction. Since vpi (a1 ) ≡ vpi (a1 ) mod α, and

vpi (a1 ) ≥ 2α

vpi (a1 ) = vpi (a1 ) + vpi (a2 ) − vpi (Θ(a2 )) − nα = vpi (u) + nα − vpi (Θ(a2 )) − nα ≥ 4α − 3α = α,

it follows that a1 ∈ H \ {1}. If vpi (a2 ) < 4α, then vpi (Θ(a2 )) = vpi (a2 ) whence a2 = a2 ∈ H \ {1}. If vpi (a2 ) ≥ 4α, then vpi (a2 ) = vpi (Θ(a2 )) ∈ [2α, 3α − 1] and since vpi (a2 ) ≡ vpi (a2 ) mod α, it follows that a2 ∈ H \ {1}. Let I ∈ EH be an H-essential subset. In the next step we deﬁne the subset XI ⊂ Z(H) (later, the elements of W will be indexed by EH and we shall write XI = XwI , see 4.6). Let I ⊂ I denote the set of all j ∈ I for which there exists some atom u ∈ A(H) with supp(u) ⊂ I

and vpj (u) ≥ 4α,

(∗)

and we set ΛI = {(I, j) | j ∈ I}. Let λ = (I, j) ∈ ΛI and nλ ∈ N0 . We ﬁx some atom uλ ∈ A(H) having property (∗) and set (n ) pλ = pj and uλ λ = uλ pλnλ α . (nλ )

Then Lemma 4.3 implies that uλ

∈ A(H).

Construction 4.4. We deﬁne ΛI ΦI : NA 0 × N0

(m, n)

−→ Z(H) (n ) u mu uλ λ −→ u∈A

and set

λ∈ΛI

ΛI . XI = Φ NA 0 × N0

Monotone Chains of Factorizations in C-Monoids

109

For X = XI , we obviously have AX ⊂ X and A−1 X∩Z(H) ⊂ X. Furthermore, for every a ∈ H and every k ∈ LX (a) with k ≥ |ΛI |+1, it follows that ZX k (a)∩AZ(H) = ∅. Thus 2. in the Main Proposition holds, provided that M ≥ |ΛI | + 1 (cf. 4.6). We continue with a construction of a set Y which is dense in the set of all factorizations. Clearly, every tame generating set of a monoid gives rise to a dense subset of factorizations. We perform this construction for the canonical tame generating sets of complete ideals. Lemma 4.5. For I ∈ EH , we deﬁne wI =

α(s+1−|I|)

pi

i∈I

and we set W = {wI | I ∈ EH }. Then there exists some θ ∈ N with the following properties: 1. W [θ] is a full almost generating set of H and w[[w]]H[W [θ] , w] ∪ (H \ W [2θ] H). H= w∈W [θ]

2. For every I ∈ EH ,

λ∈ΛI

uλ divides wIθ .

3. There exists some M ∈ N with the following property: for every w ∈ W [θ] and every a ∈ w[[w]]H[W [θ] , w] there exist some a ∈ [[w]] and some b ∈ H[W [θ] , w] such that a = wa b and t(a, Z(wb)) ≤ M . Proof. We show that 1., 2., and 3. hold for all suﬃciently large θ ∈ N. 1. The ﬁrst assertion follows from [7, Proposition 5.7.1], and the second assertion from [6, Proposition 5.5]. 2. Since supp( λ∈ΛI uλ ) ⊂ I = supp(wI ), the assertion follows from [9, Lemma 4.2.2.(d)]. 3. We are going to apply [6, Theorem 5.10]. To do so, we must prove the following assertion: Assertion: There exists some M ∈ N such that for every w ∈ W and every a ∈ H(W [θ] , wθ ) there exist a ∈ [[w]] and some b ∈ H[W [θ] , wθ ] such that a = wθ a b and t(a, Z(b)) ≤ M . Once this is done, then the set

E= wθ H[W [θ] , wθ ] ∪ H \ W [2θ] H w∈W

is a tame generating set of H. In fact, the proof of [6, Theorem 5.10] shows the more precise statement of the Lemma. It remains to prove the Assertion. Let I ∈ EH , w = wI and a ∈ H(W [θ] , wθ ). By [7, Lemma 5.8.4], there exist some a ∈ [[w]] and some b ∈ H[W [θ] , wθ ] such that a = wθ a b and vpi (b) ≤ 2α − 1 for all i ∈ I. Since H[W [θ] , wθ ] ⊂ [[w]]−1 [[w]]H(W [θ] , wθ ) ∩ H,

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we may apply [7, Lemma 5.8.5 and Proposition 5.6.2] and [6, Lemma 5.9.1] to obtain that t(a, Z(b)) ≤ M for some bound M ∈ N depending only on H ⊂ F . Construction 4.6. Let W , θ and M be as in Lemma 4.5. We deﬁne W = W [θ]

and XwI = XI for every I ∈ EH .

For a ∈ H \ W[2] H, we set Z (a) = Z(a). For w ∈ W, we ﬁx a factorization w ∈ Z(w) ∩ λ∈ΛI uλ Z(H). If a = wa b ∈ w[[w]]H[W, w] where a and b are as in Lemma 4.5.3, then we ﬁx some b ∈ Z(b) and set Z (a) = wZ(a )b. We deﬁne Y= Z (a) ⊂ Z(H). a∈H

Then Y is M -dense by construction, and we deﬁne M = max{M, |ΛI | + 1 | I ∈ EH }. Thus 1., 2. and 3. of the Main Proposition hold true.

5

Proof of the Main Proposition

It remains to prove 4., 5. and 6. Let I ∈ EH , w = wIθ ∈ W, Λ = ΛI , X = XI and Φ = ΦI . 4. Let a ∈ w[[w]]H[W, w]. The ﬁrst assertion is clear by Construction 4.6. Let z = wyb ∈ ZY (a), where a = wa b, b ∈ H[W, w], b ∈ Z(b), y ∈ Z(a ) and w ∈ Z(w) are as in 4.6. We have to ﬁnd a 2-chain of factorizations, all having the same lengths, from wy to some factorization in ZX (wa ). We consider factorizations z of the form (n ) z = uλ λ · y0 · v1 · . . . · vl (∗) λ∈Λ

where all nλ ∈ N0 , y0 ∈ F(A) and v1 , . . . , vl ∈ A([[w]]) \ A. Then for every i ∈ [1, l] z ) denote the number of all there is some j ∈ I such that vpj (vi ) ≥ 4α. Let l( z ) ≥ l, and if l( z ) = 0, then l = 0 (i, j) ∈ [1, l] × I such that vpj (vi ) ≥ 4α. Then l( and z ∈ ZX (H). Clearly, wy has form (∗), and we show that for any given z of form (∗) there exists some z of form (∗) with d( z , z ) = 2 and l( z ) < l( z ). Then the assertion follows. Let z be as in (∗), i ∈ [1, l], j ∈ I such that vpj (vi ) ≥ 4α and λ0 = (I, j) ∈ Λ. By Lemma 4.3 we obtain that (n

)

vpj (vi )−vpj (Θ(vi ))

uλ0 = uλ0λ0 · pj and

vpj (Θ(vi ))−vpj (vi )

vi = vi · pj

∈ A(H)

∈ A(H).

Monotone Chains of Factorizations in C-Monoids (n

111

)

Then vi uλ0λ0 = vi uλ0 ,

z = uλ0

(nλ )

uλ

· y0 · vi · vi−1 · v1 · . . . · vl ∈ Z(a),

λ∈Λ\{λ0 }

z ) < l( z ). d( z , z ) = 2 and l( 5. If tX (H, v) < ∞ for every v ∈ A, then Lemma 3.2.1 and Lemma 3.3.2 imply that ∆X (H) is ﬁnite. Thus it suﬃces to prove the ﬁrst assertion. Recall that π : Z(H) → H denotes the factorization homomorphism and, by Construction 4.4, we have Λ Φ : NA 0 × N0

−→ Z(H) (n ) −→ u mu uλ λ .

(m, n)

u∈A

λ∈Λ

Λ If (m, n) ∈ NA 0 ×N0 , then z = Φ(m, n) ∈ X ⊂ Z(H) has length |z| =

u∈A

mu +|Λ|.

For v ∈ A we set Λ 2 Rv = {(m, n; m , n ) ∈ (NA 0 × N0 ) | π(Φ(m, n)) = π(Φ(m , n )) and mv > 0}.

By Dickson’s Theorem (cf. [10, Theorem 1.1]), Rv has only ﬁnitely many minimal points. We set mu + |Λ|, mu + |Λ| (m, n; m , n ) is a minimal point in Rv , t(v) = max u∈A

u∈A

and assert that

tX (H, v) ≤ t(v).

Let a ∈ H, z ∈ ZX (a) and ZX (a) ∩ vZ(H) = ∅. There exist (m, n), (m , n ) ∈ Λ X NA 0 × N0 such that z = Φ(m, n) and Φ(m , n ) ∈ Z (a) ∩ vZ(H). Then mv > 0, ; m ,n ) ∈ Rv such m, n (m, n; m , n ) ∈ Rv and there exists some minimal point ( that ) ≤ (m, n; m , n ). ; m , n ( m, n Since a=

u mu

u∈A

=

λ∈Λ

u

mu −g mu

u∈A

=

=

u∈A

u

g u um

u∈A mu −g mu

u∈A

(nλ )

uλ

λ∈Λ

u

m g u

u

m g u

u∈A

u

mu −g mu

λ∈Λ

u∈A

(n f λ)

uλ

(nλ −n f λ )α

pλ

λ∈Λ (n f) uλ λ

(nλ −n f λ )α

pλ

λ∈Λ (n f +n −n f) uλ λ λ λ

∈ F,

λ∈Λ

v + m v > 0, it follows that and mv − m (nf +n −nf ) mu +g mu umu −g uλ λ λ λ ∈ ZX (a) ∩ vZ(H), z = u∈A

λ∈Λ

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Foroutan and Geroldinger

and clearly d(z, z ) ≤ max

m u + |Λ|,

u∈A

m u + |Λ| ≤ t(v).

u∈A

6. For every d ∈ ∆X (H) we set Λ 2 Rd+ = {(m, n; m , n ) ∈ (NA 0 × N0 ) | π(Φ(m, n)) = π(Φ(m , n )) and ( mu + |Λ|) = ( mu + |Λ|) + d} u∈A

u∈A

and Λ 2 Rd− = {(m, n; m , n ) ∈ (NA 0 × N0 ) | π(Φ(m, n)) = π(Φ(m , n )) and ( mu + |Λ|) = ( mu + |Λ|) − d}. u∈A

u∈A

Rd−

Rd+

and have only ﬁnitely many minimal points, Again by Dickson’s Theorem, and we set δ(d) = max mu + |Λ|, mu + |Λ| (m, n; m , n ) u∈A

u∈A

is a minimal point in Rd+ or Rd− .

By 5., ∆X (H) is ﬁnite, and we assert that δ X (H) ≤ max{δ(d) | d ∈ ∆X (H)} ∈ N. Let a ∈ H, z ∈ ZX (a) and |z|, k ∈ LX (a) successive lengths. We have to verify that X d({z}, ZX k (a)) ≤ max{δ(d) | d ∈ ∆ (H)}.

Without restriction we suppose that |z| < k, and we work with Rd+ . Let (m, n), Λ (m , n ) ∈ NA 0 × N0 such that z = Φ(m, n), z = Φ(m , n ), |z | = k and d = X ; m ,n ) ∈ Rd+ such that m, n k − |z| ∈ ∆ (H). There exists some minimal point (

) ≤ (m, n; m , n ). ; m ,n ( m, n The same calculation as in 5. shows that (nf +n −nf ) mu +g mu z = umu −g uλ λ λ λ ∈ ZX (a). u∈A

Since

|z | − |z| =

λ∈Λ

(mu − m u + m u ) + |Λ| − mu + |Λ| = (m u − m u ) = d,

u∈A

we obtain that z ∈

u∈A

u∈A

ZX k (a)

and d(z, z ) ≤ max m u + |Λ|, m u + |Λ| ≤ δ(d). u∈A

u∈A

Acknowledgement: We would like to thank F. Halter-Koch and W. Hassler for pointing out various inaccuracies in previous versions of this manuscript.

Bibliography [1] D. D. Anderson (ed.), Factorization in integral domains, Marcel Dekker, 1997. [2] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorizations in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19. [3] S. T. Chapman and S. Glaz, Non-noetherian Commutative Ring theory, Kluwer Academic Publisher, 2000. [4] A. Foroutan, Monotone chains of factorizations, Int. J. of Commutative Rings 3 (2003). [5] A. Geroldinger and F. Halter-Koch, Transfer principles in the theory of nonunique factorizations, these Proceedings. [6] [7]

, Tamely generated ideals in ﬁnitary monoids, JP J. Algebra Number Theory Appl. 2 (2002), 205–239. , Congruence monoids, Acta Arith. 112 (2004), 263–296.

[8] A. Geroldinger, F. Halter-Koch, W. Hassler, and F. Kainrath, Finitary Monoids, Semigroup Forum 67 (2003), 1–21. [9] F. Halter-Koch, C-monoids and congruence monoids in Krull domains, these Proceedings. [10]

, Ideal systems. An introduction to multiplicative ideal theory, Marcel Dekker, 1998.

[11] W. Hassler, Properties of factorizations with successive lengths in onedimensional local domains, Int. J. of Commutative Rings. [12]

, Factorization in ﬁnitely generated domains, J. Pure Appl. Algebra 186 (2004), 151–168.

[13] F. Kainrath, Elasticity of ﬁnitely generated domains, Houston J. Math. 30 (2004).

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Chapter 6

Transfer Principles in the Theory of Non-unique Factorizations by

Alfred Geroldinger and Franz Halter-Koch1 Abstract A successful method to investigate phenomena of nonunique factorizations occurring in an arithmetically interesting object H (an integral domain or a monoid) is to construct a simple auxiliary monoid B and a homomorphism θ : H → B which allows to transfer arithmetical properties from B to H. In this chapter we present a very general transfer principle and apply it to obtain arithmetical results for (weakly) Krull monoids, K +M -domains and congruence monoids.

1

Introduction

It is well-known that in general the ring OK of integers of an algebraic number ﬁeld is not factorial, and classical philosophy in algebraic number theory states that the class group CK of OK measures the deviation from unique factorization. For a long time the only result justifying this opinion was the classical criterion that OK is factorial if and only if CK = 0. Only in 1960, L. Carlitz [8] proved that |CK | ≤ 2 if and only if OK is half-factorial, that is, any two factorizations of an element have the same length. A systematical investigation of phenomena of nonunique factorizations was initiated by W. Narkiewicz in a series of papers in the sixties and seventies (see [30, Ch. 9] for an overview). One of the most important insights of these investigations was the discovery of the connection between factorization problems in OK and combinatorial problems on zero-sum sequences in CK (see [29] and [31]). In the language we use today, this was the ﬁrst application of a transfer principle: Factorization properties of OK were investigated by means of 1 This

work was supported by the Austrian Science Fund FWF (Project-Nr. P16770-N12)

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115

factorization properties of the much simpler monoid B(CK ) of all zero-sum sequences of CK . In the sequel, problems of nonunique factorizations, not only in rings of integers of algebraic number ﬁelds, but more generally in arbitrary integral domains, received a great deal of attention. In particular, we mention the papers [4], [5] and [2], the survey articles published in [1] and the survey articles [9] and [10]. These papers also contain an extensive bibliography. The problems and phenomena of nonunique factorizations in an integral domain R are of a purely multiplicative nature and can be formulated in an arbitrary multiplicative monoid H. Apart from its generality, this point of view turned out to be an utmost powerful tool in the theory of nonunique factorizations in integral domains. Concepts and results can be derived in a suitable auxiliary monoid of a concise structure and, by means of suitable transfer principles, the consequences for integral domains can be obtained. As mentioned above, this point of view was implicitly already used by W. Narkiewicz. It was systematically enhanced by the authors. The survey articles [11], [16] and [25] review the state of knowledge at that time. Only recently, the investigations of W. Hassler [27], A. Foroutan [12] and the authors [21] showed the need for more general and powerful transfer principles. In [21], we proved strong arithmetical ﬁniteness results for C0 -monoids and applied them to study the arithmetic of congruence monoids in Dedekind domains. At the same time, W. Hassler noticed that C0 -monoids could also serve to investigate the multiplicative arithmetic of a large class of noetherian domains ([27]). For this purpose he introduced Z-monoids and derived a transfer principle to reduce their arithmetical structure to those of C0 -monoids. The transfer principle presented in this paper is a common generalization of all such principles published hitherto. It applies to C-monoids (and thus it contains Hassler’s transfer principle for Z-monoids), it generalizes the transfer principle for T-block monoids ([15], [16]), and a transfer principle used by F. Kainrath for certain noetherian domains ([28]). This chapter is organized as follows. In Section 2 we ﬁx the notations concerning the algebraic and arithmetical theory of monoids and gather the basic concepts of nonunique factorizations. In Section 3 we recall the concept of a transfer homomorphism and investigate the behavior of arithmetical invariants under transfer homomorphisms. Section 4 contains the main result of the chapter, a very general transfer principle (Theorem 4.1), which will be applied to (generalized) block monoids in Section 5, to K+M -domains in Section 6 and to C-monoids and congruence monoids in Section 7. In Section 7 we shall also present an explicit criterion for a congruence monoid of natural numbers to have ﬁnite elasticity (Theorem 7.8).

2

Notations and Preliminaries on Monoids

Let N denote the set of positive integers, N0 = N ∪ {0} and P the set of all prime numbers. For g ∈ N, we denote by Ng the set of all natural numbers coprime to g, and by Pg the set of all prime numbers not dividing g. For m, n ∈ Z with m ≤ n, we set [m, n] = {x ∈ Z | m ≤ x ≤ n}. For a set X, we denote by |X| ∈ N0 ∪ {∞} its

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cardinality. For convenience and to ﬁx notations, we recall some notions from the theory of monoids and the basic results from the theory of nonunique factorizations which we use in this chapter. Unless stated otherwise, proofs can be found in [26, Ch. 10] and in the survey articles [11], [16] and [25]. Basic notions. By a semigroup we mean a nonempty set with a commutative and associative law of composition possessing a unit element. By a monoid we mean a cancellative semigroup. Unless stated otherwise, we use multiplicative notation and denote the unit element by 1. For subsets A, B of a monoid H, we set AB = {ab | a ∈ A , b ∈ B}. Semigroup and monoid homomorphisms are assumed to respect the unit element. Subsemigroups and submonoids are assumed to contain the unit element. For each monoid H, we ﬁx a quotient group q(H) of H, and if H ⊂ D is a submonoid, we tacitly assume that q(H) ⊂ q(D). We denote by H × the group of invertible elements of H and by Hred = {aH × | a ∈ H} the associated reduced monoid of H. We call H reduced , if H × = {1} (and then Hred = H). For a monoid homomorphism ϕ : H → D we denote by ϕred : Hred → Dred the induced homomorphism of reduced monoids, given by ϕred (aH × ) = ϕ(a)D× . For an integral domain R, we denote by R• = R \ {0} its multiplicative monoid. Let H be a monoid and a, b ∈ H. We call a a divisor of b and we write a | b, if b ∈ aH. We call a and b associated and write a b, if aH × = bH × . An element u ∈ H is called an atom if u ∈ / H × , and for all a, b ∈ H, u = ab implies a ∈ H × or b ∈ H × . We denote by A(H) the set of all atoms of H, and we call the monoid H atomic , if every a ∈ H \ H × is a product of atoms. An element p ∈ H is called a prime , if H \ pH is a submonoid of H, and H is called factorial , if every a ∈ H \H × is a product of primes. Every prime is an atom, and a monoid is factorial if and only if it is atomic, and every atom is a prime. If H is atomic and p ∈ H is a prime, then every a ∈ q(H) has a representation a = pn bc−1 , where b, c ∈ H, p bc and n ∈ Z. The exponent n is uniquely determined by the classes aH × and pH × in Hred , and we call n = vp (a) the p-adic value of a. The map vp : q(H) → Z is a surjective group homomorphism, called the p-adic valuation. If P is a set of primes of an atomic monoid H and x ∈ q(H), then suppP (x) = {p ∈ P | vp (x) = 0} is called the support of x (in P ). Free monoids. For a set P , we denote by F (P ) the free (abelian) monoid with basis P , and in particular F (∅) = {1}. For every map f : P → S into a monoid S there exists a unique homomorphism f : F (P ) → S extending f . F (P ) is a reduced factorial monoid and P is the set of primes of F (P ). Every subset X of F (P ) possesses a unique greatest common divisor, denoted by gcd(X) ∈ F (P ). Every a ∈ F(P ) has a unique representation in the form a=

pνp ,

where

νp ∈ N0

and νp = 0 for all but ﬁnitely many p ∈ P ,

p∈P

and then νp = vp (a) for all p ∈ P . For a, b ∈ F(P ), we set |a| =

p∈P

vp (a)

and

a , d(a, b) = max gcd(a, b)

b gcd(a, b) .

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117

We call |a| the size of a and d(a, b) the distance between a and b. Note that | · | : F (P ) → N0 is a homomorphism, and d : F (P )×F(P ) → N0 is a metric. Products. For two monoids H1 , H2 , we denote by H1×H2 their direct product, and we view H1 and H2 as submonoids of H = H1 ×H2 . Then every a ∈ H has a unique factorization in the form a = a1 a2 with a1 ∈ H1 and a2 ∈ H2 . Note that a monoid H is factorial and P is a set of representatives of pairwise nonassociated primes of H if and only if H = H × ×F(P ). and sets of lengths. Let H be a monoid. The monoid Z(H) = Factorizations F A(Hred ) is called the factorization monoid of H. The unique homomorphism π : Z(H) → Hred satisfying π | A(Hred ) = id is called the factorization homomorphism of H. It is surjective if and only if H is atomic, and it is an isomorphism if −1 × and only if H is factorial. For a ∈ H, the elements in Z(a) = π (aH ) ⊂ Z(H) are called the factorizations of a , and L(a) = |z| z ∈ Z(a) ⊂ N0 is called the set of lengths of a. We call L(H) = {L(a) | a ∈ H} the system of sets of lengths of H. It is one of the best studied invariants in the theory of nonunique factorizations. H is called a BF-monoid (a monoid with bounded factorizations), if all sets L ∈ L(H) are ﬁnite and not empty. H is called half-factorial, if |L(a)| = 1 for all a ∈ H. Let now H be a BF-monoid. For a ∈ H \ H × , we call ρ(a) =

max L(a) ∈ Q>0 min L(a)

the elasticity of a. We call ρ(H) = sup{ρ(a) | a ∈ H \ H × } ∈ R≥0 ∪ {∞} the elasticity of H, and we say that H has accepted elasticity, if ρ(H) = ρ(a) for some a ∈ H. If H = H1 × H2 , then ρ(H) = max{ρ(H1 ), ρ(H2 )}, and H has accepted elasticity if and only if both H1 and H2 have accepted elasticity (see [24, Proposition 4]). Let H be a BF-monoid and λ : H → N0 a homomorphism satisfying λ−1 (0) = H . Then sup{λ(u) | u ∈ A(H), u not prime} ρ(H) ≤ . min{λ(u) | u ∈ A(H), u not prime} ×

For integral domains, this was proved in [3, Theorem 2.1], where λ is called a semi-length function. The proof given there remains literally valid for monoids. The Structure Theorem for Sets of Lengths. A ﬁnite set L ⊂ Z is called an almost arithmetical multiprogression with bound M ∈ N , if there exists some d ∈ [1, M ] and some subset D ⊂ [0, d] with {0, d} ⊂ D such that L = L ∪ L∗ ∪ L , where L ⊂ min L∗ + [−M, −1], L ⊂ max L∗ + [1, M ] and L∗ = [min L∗ , max L∗ ] ∩ (min L∗ + D + dZ) . We say that the Structure Theorem for Sets of Lengths holds for a monoid H, if H is a BF-monoid, and there exists some M ∈ N such that every L ∈ L(H) is an almost arithmetical multiprogression with bound M .

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The Structure Theorem for Sets of Lengths has an abstract ideal-theoretical and combinatorial basis (see [19] and [20]). In [21] we proved that it holds for AC-monoids, and as a consequence it holds for congruence monoids in Dedekind domains (see [21]) and for noetherian domains (see [27]), provided that some natural ﬁniteness conditions are satisﬁed. In this chapter, we shall prove that it also holds for C-monoids (Theorem 7.2), and consequently it continues to hold for congruence monoids in Krull domains (see [23]). Next we explain two more subtle invariants of nonunique factorizations, introduced by the ﬁrst author in order to investigate the system of sets of lengths (see [16]). Let H be an atomic monoid. Catenary degree: For a ∈ H, we denote by c(a) the smallest N ∈ N0 ∪ {∞} with the following property: For any two factorizations z, z ∈ Z(a), there exist factorizations z = z0 , z1 , . . . , zk = z in Z(a) such that d(zi−1 , zi ) ≤ N for all i ∈ [1, k] (we refer to this fact by saying that z and z can be concatenated by an N -chain). Then c(a) ≤ sup L(a), and we call c(H) = sup{c(a) | a ∈ H} ∈ N0 ∪ {∞} the catenary degree of H. Note that c(H) = 0 if and only if H is factorial, and c(H) ≥ 2 otherwise. If c(H) = 2, then H is half-factorial. If H = H1 × H2 , then c(H) = max{c(H1 ), c(H2 )} by [17, Lemma 3.3]. Tameness: For a ∈ H and x ∈ Z(H), we denote by t(a, x) the smallest N ∈ N0 ∪ {∞} with the following property: If Z(a) ∩ xZ(H) = ∅ and z ∈ Z(a), then there exists some z ∈ Z(a) ∩ xZ(H) such that d(z, z ) ≤ N (note that, if c ∈ H and x ∈ Z(c), then the condition Z(a) ∩ xZ(H) = ∅ is equivalent to c | a). For subsets H ⊂ H and X ⊂ Z(H), we deﬁne t(H , X) = sup{t(a, x) | a ∈ H , x ∈ X} ∈ N0 ∪ {∞} . For a ∈ H and x ∈ Z(H), we set t(a, X) = t({a}, X) and t(H , x) = t(H , {x}). H is called locally

tame , if t(H, u) < ∞ for all u ∈ A(Hred ), and H is called tame,

if t(H) = t H, A(Hred ) < ∞. We call t(H) the tame degree of H. Note that t(H) = 0 if and only if H is factorial, and t(H) ≥ 2 otherwise. If H = H1 × H2 , then t(H) = max{t(H1 ), t(H2 )}, and H is locally tame if and only if both H1 and H2 are locally tame. Submonoids and class groups. Let D be a monoid and H ⊂ D a submonoid. For x ∈ q(D), we set [x]D/H = xq(H) ∈ q(D)/q(H), we call D/H = {[a]D/H | a ∈ D} the class monoid and q(D)/q(H) = q(D/H) the class group of D modulo H. If H ⊂ D× is a subgroup, then [a]D/H = aH ∈ q(D)/H for all a ∈ D, there are natural epimorphisms D → D/H → D/D× = Dred ⊂ q(D)/D× = q(Dred ), and Hred = H/H × is a submonoid of D/H × . In this special case, we write D/H multiplicatively. In the general situation however, we use additive notation for D/H, so that [ab]D/H = [a]D/H + [b]D/H for all a, b ∈ D, and [1]D/H is the zero element of D/H. A submonoid H ⊂ D is called • coﬁnal , if aD ∩ H = ∅ for all a ∈ D (equivalently, D/H = q(D/H)).

Transfer Principles in the Theory of Non-Unique Factorizations

119

• saturated , if q(H) ∩ D = H (equivalently, H = {a ∈ D | [a]D/H = [1]D/H }). Block monoids and the Davenport constant. For an (additive) abelian group G and any subset G0 ⊂ G, let F (G0 ) be the free (multiplicative) monoid with basis G0 , and let σ : F (G0 ) → G be the unique homomorphism satisfying σ(g) = g for all g ∈ G0 . Then B(G0 ) = {S ∈ F(G0 ) | σ(S) = 0} is the monoid of all zero-sum sequences over G0 . It is a saturated submonoid of F (G0 ), called the block monoid over G0 , and D(G0 ) = |S| S ∈ A(B(G0 )) is called the Davenport constant. It is ﬁnite whenever G0 is ﬁnite [14, Proposition 2], and it plays a prominent role in the theory of nonunique factorization (see [11] or [30, Chapter 9.1]). For every invariant ∗(H) attached to a monoid H, we write ∗(G0 )

arithmetical instead of ∗ B(G0 ) .

If D is any atomic monoid, H ⊂ D is a saturated submonoid, G0 = {[u]D/H | u ∈ A(D)}, and if a1 , . . . , an ∈ D \ D× are such that a1 · . . . · an ∈ A(H), then n ≤ D(G0 ) (see [22, Section 3]). Class semigroups, the ω-invariant and the d-invariant. These notions were introduced and investigated in [21, Section 4], see also [23, Section 3]. We recall the deﬁnition and the basic properties. Let H ⊂ D be a submonoid. Two elements y, y ∈ D are called H-equivalent , if y −1 H ∩ D = y −1 H ∩ D ⊂ q(D). H-equivalence is a congruence relation on D, we denote by [y]D H the congruence class of an element y ∈ D, and for a subset T ⊂ D we set CT (H, D) = [y]D H y ∈ T}. In particular, C(H, D) = CD (H, D) and C ∗ (H, D) = C(D\D× )∪{1} (H, D) are semigroups under the induced law of composition which we write additively, [ab]D H = D + [b] for all a, b ∈ D. [a]D H H Let H be a monoid and a, b ∈ H. Then ωH (a, b) denotes the smallest N ∈ N0 ∪ {∞} with the following property: For any n ∈ N and a1 , . . . , an ∈ H, if a = a1 · . . . · an and b | a, then there exists a subset Ω ⊂ [1, n] such that |Ω| ≤ N and b aν . ν∈Ω

By deﬁnition, ωH (a, b) = 0 if either b a or b ∈ H × . If m ∈ N0 , p1 , . . . , pm are primes of H and b = p1 · . . . · pm , then ωH (a, b) ≤ m. For an additive abelian semigroup C, we denote by d(C) the smallest d ∈ N0 ∪ {∞} with the following property: For any m ∈ N and c1 , . . . , cm ∈ C there exists a subset J ⊂ [1, m] such that |J| ≤ d and m j=1

cj =

cj .

j∈J

If C is a ﬁnite additive abelian semigroup, then d(C) < ∞. If C is an abelian group, then D(C) = d(C) + 1 is the Davenport constant.

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× ×

If H ⊂ D is a submonoid satisfying H ∩ D = H , then ωH (a, b) ≤ ωD (a, b) + ∗ d C (H, D) for all a, b ∈ H. If H ⊂ D is a coﬁnal submonoid, then there is a natural epimorphism C(H, D) → D/H, which is an isomorphism if and only if H ⊂ D is saturated, see [23, Proposition 3.7].

Restriction to reduced monoids: All factorization properties explained hitherto hold for a monoid H if and only if they hold for the associated reduced monoid Hred . Hence we may without restriction assume that H is reduced, and we will do this whenever it will be convenient.

3

Transfer Homomorphisms

The notion of a transfer homomorphism was ﬁrst introduced in [25]. We recall the deﬁnition and investigate the behavior of the invariants of nonunique factorization under transfer homomorphisms. Definition 3.1. A monoid homomorphism θ : H → B is called a transfer homomorphism , if it has the following properties: (T 1)

B = θ(H)B × and θ−1 (B × ) = H × .

(T 2)

If u ∈ H, b, c ∈ B and θ(u) = bc, then there exist v, w ∈ H such that u = vw, θ(v) b and θ(w) c.

Note that θ : H → B is a transfer homomorphism if and only if θred : Hred → Bred is a transfer homomorphism. Proposition 3.2. Let θ : H → B be a transfer homomorphism and u ∈ H. 1. If n ∈ N, b1 , . . . , bn ∈ B and θ(u) b1 ·. . .·bn , then there exist u1 , . . . , un ∈ H such that u u1 · . . . · un and θ(uν ) bν for all ν ∈ [1, n]. 2. u is an atom of H if and only if θ(u) is an atom of B. 3. There is a unique homomorphism θ : Z(H) → Z(B) satisfying θ(uH × ) = θ(u)B × for all u ∈ A(H). It is surjective, induces the commutative diagram θ

Z(H) −−−−→ Z(B) ⏐ ⏐ ⏐ ⏐πB πH θ

red Hred −−− −→ Bred ,

and has the following properties:

(a) If z, z ∈ Z(H), then |θ(z)| = |z| and d θ(z), θ(z ) ≤ d(z, z ). (b) θ(ZH (u)) = ZB (θ(u)) and LH (u) = LB (θ(u)).

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(c) If z ∈ Z(u) and y ∈ Z(θ(u)), then there exists some y ∈ Z(u) such that θ(y) = y, θ(gcd(z, y)) = gcd(θ(z), y) and d(z, y) = d(θ(z), y). 4. H is atomic if and only if B is atomic. 5. If H is atomic, then L(H) = L(B), ρ(H) = ρ(B), H has accepted elasticity if and only if B has accepted elasticity, and H is a BF-monoid if and only if B is a BF-monoid. Proof. We may assume that H and B are both reduced. 1. By induction on n. 2. If u ∈ A(H) and θ(u) = bc for some b, c ∈ B, then there exist v, w ∈ H such that u = vw, θ(v) = b and θ(w) = c. Hence v = 1 or w = 1 and thus b = 1 or c = 1. If θ(u) ∈ A(B) and u = vw for some v, w ∈ H, then θ(u) = θ(v)θ(w) implies θ(v) = 1 or θ(w) = 1 and thus v = 1 or w = 1. 3. Observe that Z(H) = F (A(H)), Z(B) = F (A(B)), and that the map θ | A(H) : A(H) → A(B) is surjective. Hence there is a unique homomorphism θ : Z(H) → Z(B) such that θ | A(H) = θ | A(H). It is surjective, and πB ◦ θ | A(H) = θ ◦ πH | A(H) implies πB ◦ θ = θ ◦ πH . (a) is obvious by deﬁnition, and it remains to prove the properties (b) and (c). (b) If z ∈ Z(u), then πB (θ(z)) = θ(πH (z)) = θ(u), and therefore θ(z) ∈ Z(θ(u)). Conversely, if z = u1 · . . . · un ∈ Z(θ(u)), where u1 , . . . , un ∈ A(B), then θ(u) = u1 · . . . · un (observe that the ﬁrst product is performed in Z(B) and the second one in B). By 1. there exist u1 , . . . , un ∈ H such that u = u1 · . . . · un and θ(uν ) = uν for all ν ∈ [1, n]. By 2. we have u1 , . . . , un ∈ A(H), hence z = u1 · . . . · un ∈ Z(u) and z = θ(z). Now the assertion concerning the sets of lengths follows from (a). (c) If z = u1 ·. . .·un ∈ Z(u), then θ(z) = θ(u1 )·. . .·θ(un ) and, after renumbering if necessary, we may assume that y = θ(u1 ) · . . . · θ(ur )v 1 · . . . · v s , where r ∈ [0, n], s ∈ N0 , v 1 , . . . , v m ∈ A(B) and θ(u1 ) · . . . · θ(ur ) = gcd(θ(z), y). In B, we have θ(u) = θ(u1 )·. . .·θ(un ) = θ(u1 )·. . .·θ(ur )v 1 ·. . .·v s , hence θ(ur+1 ·. . .·un ) = v 1 ·. . .·vs , and therefore there exist v1 , . . . , vs ∈ H such that ur+1 · . . . · un = v1 · . . . · vs and θ(vi ) = v i for all i ∈ [1, s]. Hence v1 , . . . , vs ∈ A(H), and y = u1 · . . . · ur v1 · . . . · vs has the required properties. 4. Note that H is atomic if and only if Z(a) = ∅ for all a ∈ H. Since θ is surjective, it follows that B is atomic if and only if Z(θ(a)) = ∅ for all a ∈ H. Hence the assertion follows by 3.(b). 5. Obvious by 3.(b). Definition 3.3. Let θ : H → B be a transfer homomorphism of atomic monoids and θ : Z(H) → Z(B) the unique homomorphism satisfying θ(uH × ) = θ(u)B × for all u ∈ A(H) (see Proposition 3.2). We call θ the extension of θ to the factorization monoids. 1. For a ∈ H, we denote by c(a, θ) the smallest N ∈ N0 ∪{∞} with the following property: If z, z ∈ Z(a) and θ(z) = θ(z ), then there exist factorizations

122

Geroldinger and Halter-Koch z0 , . . . , zk ∈ Z(a) such that z0 = z, zk = z , θ(zi ) = θ(z) and d(zi−1 , zi ) ≤ N for all i ∈ [1, k] (that is, z and z can be concatenated by an N -chain in the −1 ﬁbre Z(a) ∩ θ (θ(z))). We set c(H, θ) = sup{c(a, θ) | a ∈ H} ∈ N0 ∪ {∞} .

2. For a ∈ H and x ∈ Z(H), we denote by t(a, x, θ) the smallest N ∈ N0 ∪ {∞} with the following property: If Z(a) ∩ xZ(H) = ∅, z ∈ Z(a) and θ(z) ∈ θ(x)Z(B), then there exists some factorization z ∈ Z(a) ∩ xZ(H) such that θ(z ) = θ(z) and d(z, z ) ≤ N . We set t(H, x, θ) = sup{t(a, x, θ) | a ∈ H} ∈ N0 ∪ {∞} . Theorem 3.4. Let θ : H → B be a transfer homomorphism of atomic monoids and θ : Z(H) → Z(B) its extension to the factorization monoids.

1. If a ∈ H and x ∈ Z(H), then t(a, x) = 0 or t θ(a), θ(x) ≤ t(a, x) ≤

t θ(a), θ(x) + t(a, x, θ).

In particular, if u ∈ A(Hred ), then t B, θ(u) ≤ t(H, u) ≤ t B, θ(u) + t(H, u, θ). 2. t(B) ≤ t(H), and if H is locally tame, then so is B. 3. If B is locally tame and t(H, u, θ) < ∞ for all u ∈ A(Hred ), then H is locally tame.

4. If a ∈ H, then c θ(a) ≤ c(a) ≤ max{c θ(a) , c(a, θ)}. 5. c(B) ≤ c(H) ≤ max{c(B), c(H, θ)} Proof. 1. We may suppose that H and B are both reduced. Let a ∈ H and x ∈ Z(H). If Z(a) ∩ xZ(H) = ∅, then t(a, x) = 0. Suppose that Z(a) ∩ xZ(H) = ∅ whence Z(θ(a)) ∩ θ(x)Z(B) ⊃ θ(Z(a) ∩ xZ(H)) = ∅. a) t(θ(a), θ(x)) ≤ t(a, x) : Let z ∈ Z(θ(a)), say z = θ(z) for some z ∈ Z(a). There exists some z ∈ Z(a) ∩ xZ(H) such that d(z, z ) ≤ t(a, x). Then θ(z ) ∈ Z(θ(a)) ∩ θ(x)Z(B) and d(θ(z), θ(z )) ≤ d(z, z ) ≤ t(a, x). b) t(a, x) ≤ t(θ(a), θ(x)) + t(a, x, θ) : Let z ∈ Z(a). There exists some y ∈ Z(θ(a)) ∩ θ(x)Z(B) such that d(θ(z), y) ≤ t(θ(a), θ(x)). By Proposition 3.2.3 (c), there is some y ∈ Z(a) such that θ(y) = y and d(z, y) = d(θ(z), y), and thus there is some z ∈ Z(a) ∩ xZ(H) such that θ(z ) = θ(y) and d(y, z ) ≤ t(a, x, θ). Therefore we obtain d(z, z ) ≤ d(z, y) + d(y, z ) ≤ t(θ(a), θ(x)) + t(a, x, θ) . 2. and 3. are now obvious, 4. follows from the subsequent more precise Lemma 3.5, and 5. is an immediate consequence of 4.

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Lemma 3.5. Let θ : H → B be a transfer homomorphism of atomic monoids, θ : Z(H) → Z(B) its extension to the factorization monoids and a ∈ H.

z, z ∈ Z θ(a)

z0 , z1 , . . . , zk ∈ Z(a) be such that θ(z0 ) = z

and θ(zk ) = z . Then θ(z0 ), . . . , θ(zk ) ∈ Z θ(a) and d θ(zi−1 ), θ(zi ) ≤ d(zi−1 , zi ) for all i ∈ [1, k]. In particular, c(θ(a)) ≤ c(a).

1. Let

and

2. Let z, z ∈ Z(a) and z 0 , z 1 , . . . , z k ∈ Z(θ(a)) be such that θ(z) = z 0 and θ(z ) = z k . Then there exists some m ≥ k and there exist factorizations z = z0 , z1 , . . . , zk , zk+1 , . . . , zm = z ∈ Z(a) such that θ(zi ) = z i ,

d(zi−1 , zi ) = d(z i−1 , z i )

for all

i ∈ [1, k]

and d(zj−1 , zj ) ≤ c(a, θ) for all In particular, c(a) ≤ max c(θ(a)), c(a, θ) . θ(zj ) = z k ,

j ∈ [k + 1, m] .

Proof. 1. By Proposition 3.2.3 (a). 2. By Theorem 3.2.3(c) and induction on k, we obtain z = z0 , z1 , . . . , zk ∈ Z(a) such that θ(zi ) = z i and d(zi−1 , zi ) = d(z i−1 , z i ) for all i ∈ [1, k]. The existence of zk+1 , . . . , zm = z ∈ Z(a) such that θ(zj ) = z k and d(zj−1 , zj ) ≤ c(a, θ) for all j ∈ [k + 1, m] follows by the very deﬁnition of c(a, θ). In particular, if N = max c(θ(a)), c(a, θ)

and z 0 , z 1 , . . . z k is an N -chain

concatenating θ(z) and θ(z ) in Z(θ(a)), then z0 , z1 , . . . , zm is an N -chain concatenating z and z in Z(a). Hence c(a) ≤ N . Proposition 3.6. Let θ : H → B and β : B → D be transfer homomorphisms of atomic monoids, and let θ : Z(H) → Z(B) and β : Z(B) → Z(D) be their extensions to the factorization monoids. Then β ◦ θ : H → D is a transfer homomorphism, and β ◦ θ : Z(H) → Z(D) is its extension to the factorization monoids. Moreover, if a ∈ H and x ∈ Z(H), then c(a, β◦θ) ≤ max{c(a, θ), c(θ(a), β)}

and

t(a, x, β◦θ) ≤ t(a, x, θ)+t(θ(a), θ(x), β) .

Proof. It is easily checked that β ◦ θ is a transfer homomorphism and that β ◦ θ is its extension to the factorization monoids. Let a ∈ H and z, z ∈ Z(a) be such that β ◦ θ(z) = β ◦ θ(z ). We must prove that z and z can be concatenated by a max{c(a, θ), c(θ(a), β)}-chain in the ﬁbre Z(a)∩(β◦θ)−1 (β◦θ(z)). Since θ(z), θ(z ) ∈ Z(θ(a)), there exist factorizations θ(z) = −1 z 0 , z 1 , . . . , z k = θ(z ) ∈ Z(θ(a)) ∩ β (β(θ(z)) such that d(z i−1 , z i ) ≤ c(θ(a), β) for all i ∈ [1, k]. By Lemma 3.5.2, there exists some m ≥ k and there exist factorizations z = z0 , z1 , . . . , zk , zk+1 , . . . , zm = z ∈ Z(a) such that θ(zi ) = z i for all i ∈ [1, k], θ(zj ) = z k for all j ∈ [k + 1, m], d(zi−1 , zi ) = d(z i−1 , z i ) for all i ∈ [1, k] and d(zj−1 , zj ) ≤ c(a, θ) for all j ∈ [k + 1, m].

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Suppose now that a ∈ H, x ∈ Z(H), Z(a) ∩ xZ(H) = ∅, z ∈ Z(a) and β ◦ θ(z) ∈ β ◦ θ(x)Z(D). We must prove that there exists some z ∈ Z(a) ∩ xZ(H) such that β ◦ θ(z ) = β ◦ θ(z) and d(z, z ) ≤ t(a, x, θ) + t(θ(a), θ(x), β). Since θ(x) ∈ Z(B), θ(z) ∈ Z(θ(a)) and Z(θ(a)) ∩ θ(x)Z(B) ⊃ θ Z(a) ∩ xZ(H) = ∅, there exists some y ∈ ZB (θ(a)) ∩ θ(x)Z(B) such that β(y) = β ◦ θ(z) and d(θ(z), y) ≤ t(θ(a), θ(x), β). By Proposition 3.2.3(c), there exists some y ∈ Z(a) such that θ(y) = y and d(z, y) = d(θ(z), y). Since θ(y) ∈ θ(x)Z(B), there exists some z ∈ Z(a) ∩ xZ(H) such that θ(z ) = θ(y) and d(y, z ) ≤ t(a, x, θ). Hence β ◦ θ(z ) = β(y) = β ◦ θ(z) and d(z, z ) ≤ d(z, y) + d(y, z ) ≤ t(θ(a), θ(x), β) + t(a, x, θ). Our next result presents very general estimates for the invariants c(a, θ) and t(a, x, θ). The estimate for t(a, x, θ) is essentially due to W. Hassler [27, Proposition 4.7]. Proposition 3.7. Let θ : H → B be a transfer homomorphism of reduced atomic monoids. For a ∈ H and u ∈ A(H), we denote by ω ∗ (a, u, θ) the smallest N ∈ N0 ∪ {∞} with the following property: If u | a and a = u1 ·. . .·un , where n ∈ N, u1 , . . . , un ∈ A(H) and θ(u) = θ(u1 ), then there exists a subset Ω ⊂ [1, n] such that 1 ∈ Ω, |Ω| ≤ N , and uν . u ν∈Ω

1. If u ∈ A(H) and a ∈ uH, then t(a, u, θ) ≤ ω ∗ (a, u, θ) ≤ ω(a, u) + 1. 2. If W = sup ω ∗ (c, u, θ) c ∈ H, u ∈ A(H) , then c(a, θ) ≤ W a ∈ H.

for all

Proof. 1. Suppose that u ∈ A(H) and a ∈ uH. a) ω ∗ (a, u, θ) ≤ ω(a, u) + 1 : If a = u1 · . . . · un , where n ∈ N, u1 , . . . , un ∈ A(H) and θ(u) = θ(u1 ), then there exists a subset Ω0 ⊂ [1, n] such that |Ω0 | ≤ ω(a, u) and u uν . ν∈Ω0

Then 1 ∈ Ω = Ω0 ∪{1}, and |Ω| ≤ ω(a, u)+1, which implies ω ∗ (a, u, θ) ≤ ω(a, u)+1. b) t(a, u, θ) ≤ ω ∗ (a, u, θ) : Suppose that z = u1 · . . . · un ∈ ZH (a), where u1 , . . . , un ∈ A(H) and θ(z) ∈ θ(u)Z(B). We may assume that θ(u) = θ(u1 ), and we must prove that there exists a factorization z ∈ Z(a) ∩ uZ(H) such that θ(z ) = θ(z) and d(z, z ) ≤ ω ∗ (a, u, θ). We may renumber u2 , . . . , un so that u | u1 · . . . · um for some m ∈ [1, n] such that m ≤ ω ∗ (a, u, θ). If u1 · . . . · um = ub, where b ∈ H, then θ(b) = θ(u2 ) · . . . · θ(um ), and since θ is a transfer homomorphism, there exist u2 , . . . , um ∈ H such that b = u2 · . . . · um and θ(uj ) = θ(uj ) for all j ∈ [2, m]. Now we obtain z = u·u2 ·. . .·um um+1 ·. . .·un ∈ Z(a)∩uZ(H), θ(z ) = θ(z) and d(z, z ) ≤ m ≤ ω ∗ (a, u, θ).

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2. Suppose that a ∈ H, z, z ∈ Z(a) and θ(z) = θ(z ). We may assume that z = u1 · . . . · un and z = u1 · . . . · un , where n ∈ N, uν , uν ∈ A(H) and θ(uν ) = θ(uν ) for all ν ∈ [1, n]. We must prove that z and z can be concatenated −1 by a W -chain in Z(a) ∩ θ (θ(z)), and we do this by induction on n. If n ≤ W , there is nothing to do. Thus suppose that n > W . After renumbering, we may assume that there exists some m ∈ [1, n] such that m ≤ W and u1 | u1 · . . . · um , say u1 · . . . · um = u1 b for some b ∈ H. Then θ(b) = θ(u2 ) · . . . · θ(um ), and thus there exist v2 , . . . , vm ∈ A(H) such that b = v2 · . . . · vm and θ(vj ) = θ(uj ) for −1 all j ∈ [2, m]. We obtain y = u1 v2 · . . . · vm um+1 · . . . · un ∈ Z(a) ∩ θ (θ(z)), −1 −1 −1 −1 −1 d(y, z ) ≤ m ≤ W , u−1 1 y ∈ Z(u1 a), u1 z ∈ Z(u1 a) and θ(u1 y) = θ(u1 z). By induction hypothesis there exists a W -chain of factorizations y0 , y1 , . . . , yk in −1 −1 −1 Z(u−1 (θ(u−1 1 a) ∩ θ 1 z)) concatenating u1 z and u1 y. Then u1 y0 , . . . , u1 yk , z is −1 a W -chain concatenating z and z in Z(a) ∩ θ (θ(z)).

4

A Transfer Theorem

Theorem 4.1. Let D be a monoid, P a set of pairwise nonassociated primes of D and T ⊂ D a submonoid such that D = F (P ) × T . Let H ⊂ D be a submonoid such that H∩D× = H × , and U ⊂ D× a subgroup such that H × ⊂ U and U (H\H × ) ⊂ H. Let ∼ be a congruence relation on D such that, for all u, v ∈ D, u∼v

and

u∈H

implies

v∈H.

Let P ∗ = {p ∈ P | p−1 H ∩ D = H \ H × }, P0 ⊂ P ∗ a subset and p = [p]∼ the congruence class of p for p ∈ P \ P0 . Let P = { p | p ∈ P \ P0 },

= F (P) × T /U D

: D → D β

and

the unique homomorphism satisfying β(p) = p for all p ∈ P \ P0 , β(p) = 1 for = tU ∈ T /U for all t ∈ T . Finally, we set all p ∈ P0 , and β(t) = β(H) H

and

|H : H → H . β=β

Then we have: −1 (H) = HU F (P0 ), β −1 (H × ) = H × U F (P0 ), β −1 (H × ) = H × and H ∩ 1. β × × D =H .

2. If D = F (P \ P0 )T \ D× ∪ {1} ⊂ D, then there is an isomorphism D) , β ∗ : CD (HU, D) → C ∗ (H,

given by

e D β∗ ([y]D e HU ) = [β(y)]H

D) is also for all y ∈ D . In particular, if C ∗ (H, D) is ﬁnite, then C ∗ (H, ﬁnite. is a transfer homomorphism. If H is atomic, then c(H, β) ≤ 2 3. β : H → H and t(a, uH × , β) ≤ ωH (a, u) + 1 for all a ∈ H and u ∈ A(H).

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The proof of Theorem 4.1 depends on the following technical lemma. Lemma 4.2. Let all assumptions and notations be as in Theorem 4.1. D 1. If u, v ∈ D and u ∼ v, then [u]D H = [v]H .

2. For p ∈ P , the following assertions are equivalent: (a) p ∈ P ∗ . D D × (b) pD× ∩ H = ∅ and [p]D H + [a]H = [a]H for all a ∈ D \ D .

3. If p ∈ P ∗ , q ∈ P and q ∼ p, then q ∈ P ∗ . 4. If c ∈ F(P ∗ ), then c−1 H ∩ D ⊂ H, c(H \ H × ) ⊂ H, and if c = 1, then cD× ∩ H = ∅. ), then b ∼ b , and β(b) 5. If b, b ∈ F(P \ P0 ) and β(b) = β(b = 1 if and only if b = 1. Proof of Lemma 4.2. 1. By deﬁnition, it suﬃces to prove that u ∼ v implies u−1 H ∩ D ⊂ v −1 H ∩ D. If u ∼ v and z ∈ u−1 H ∩ D, then zu ∈ H and zu ∼ zv, hence zv ∈ H and z ∈ v −1 H ∩ D. 2. (a) ⇒ (b) Suppose that p ∈ P ∗ . If e ∈ pD× ∩ H, then e = pu for some u ∈ D× ∩ p−1 H = (H \ H × ) ∩ D× = ∅, a contradiction. Hence pD× ∩ H = ∅. If D D a ∈ D \ D× , then [p]D H + [a]H = [pa]H , and therefore it is suﬃcient to prove that a−1 H∩D = (pa)−1 H∩D. If z ∈ a−1 H∩D, then az ∈ H\D× = H\H × = p−1 H∩D, hence paz ∈ H and z ∈ (pa)−1 H ∩D. Conversely, if z ∈ (pa)−1 H ∩D, then paz ∈ H, hence az ∈ p−1 H ∩ D ⊂ H and z ∈ a−1 H ∩ D. D (b) ⇒ (a) Suppose that p ∈ P , pD× ∩ H = ∅ and [pa]D H = [a]H for all × −1 × −1 a ∈ D \ D . We must prove that p H ∩ D = H \ H . If a ∈ p H ∩ D, then D pa ∈ H and pD× ∩ H = ∅ implies a ∈ / D× . Hence [a]D H = [pa]H ⊂ H and therefore × × × D a ∈ H \ H . Conversely, if a ∈ H \ H = H \ D , then [pa]H = [a]D H ⊂ H implies pa ∈ H and a ∈ p−1 H ∩ D. D 3. Suppose that p ∈ P ∗ , q ∈ P and q ∼ p. Then [q]D H = [p]H by 1., and by 2. it × suﬃces to prove that qD ∩ H = ∅. Assume to the contrary that qu ∈ H for some u ∈ D× . Then pu ∼ qu implies pu ∈ H, a contradiction.

4. Suppose that c = p1 · . . . · pn , where n ∈ N0 and p1 , . . . , pn ∈ P0 , and proceed by induction on n. If n = 0, there is nothing to do. Thus suppose that n ≥ 1, b = p1 · . . . · pn−1 , p = pn , c = bp, b−1 H ∩ D ⊂ H and b(H \ H × ) ⊂ H. If a ∈ c−1 H ∩ D, then ca = pba ∈ H implies ba ∈ p−1 H ∩ D ⊂ H, and thus a ∈ b−1 H ∩ D ⊂ H. / H × . Hence If a ∈ H \ H × , then ba ∈ H, and H × = H ∩ D× implies ba ∈ × −1 ba ∈ H \ H = p H ∩ D, and therefore ca = pba ∈ H. It remains to prove that cD× ∩ H = ∅. Assume the contrary. Then there exists some u ∈ D× such that a = cu = pbu ∈ cD× ∩ H. Then u ∈ c−1 H ∩ D× ⊂ H ∩ D× = H × , and pb = u−1 a ∈ H. Hence p ∈ b−1 H ∩ D ⊂ H, a contradiction.

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5. If b, b ∈ F(P \ P0 ) and β(b) = β(b ), then we may assume that b = p1 · . . . · pn and b = p1 ·. . .·pn , where n ∈ N0 , pi , pi ∈ P \ P0 and pi ∼ pi for all i ∈ [1, n]. Since ∼ is a congruence relation, we obtain b ∼ b . In particular, β(b) = p1 · . . . · pn = 1 implies n = 0 and b = 1. −1 (H) −1 (H ⊃ HU F (P0 ), β ×) ⊃ Proof of Theorem 4.1. 1. We obviously have β × −1 × × × × × H U F (P0 ), β (H ) ⊃ H and H ∩ D = H ∩ T /U ⊃ H . Thus it remains to verify the reverse inclusions. ). We set a = cbt and −1 (H) and a ∈ H be such that β(a) = β(a Let a ∈ β ), = β(b a = c b t , where c, c ∈ F(P0 ), b, b ∈ F(P \ P0 ) and t, t ∈ T . Then β(b) −1 −1 hence b ∼ b by Lemma 4.2.5, and t t ∈ U . Since b t ∈ c H ∩ D ⊂ H (by Lemma 4.2.4) and bt ∼ b t , we obtain bt ∈ H and a = cbt (t−1 t) ∈ HU F (P0 ). −1 (H × ), then β(a) × ⊂ D × = T × /U , and therefore = β(b)(tU )∈ H If a ∈ β × β(b) = 1 and t ∈ T . For the same reason, we have β(b ) = 1 and t ∈ T × , hence b = b = 1 and t ∈ T × ∩ H = H × . Whence a = ct (t−1 t) ∈ F(P0 )H × U . −1 (H ×) = β × ) ∩ H = F (P0 )H × U ∩ H, then a = cu, where If a ∈ β −1 (H × c ∈ F(P0 ) and u ∈ H U ⊂ D× . Hence c = 1 by Lemma 4.2.4, and a = u ∈ D× ∩ H = H × . ∩D × = H ∩ T × /U , then x = t U for some t ∈ T × . Let t ∈ T , If x ∈ H = x. This b ∈ F(P \ P0 ) and c ∈ F(P0 ) such that bct ∈ H and β(bct) = β(bt) −1 ∈ U ⊂ T × = D× , implies b = 1 and Lemma 4.2.4 shows that t = bt ∈ H. Then tt × × × × t ∈ D ∩ H = H and x = β(t) ∈ β(H ) ⊂ H .

F (P \ P0 )T \ D× = D \D × . Therefore it suﬃces to prove 2. Observe that β that, for all y, y ∈ F(P \ P0 )T , we have D [y]D HU = [y ]HU

e D De . if and only if [β(y)] e = [β(y )]H e H

/ D× . Suppose that y = bt and y = b t , where b, b ∈ F(P \P0 ), t, t ∈ T and y, y ∈ We must prove that y −1 HU ∩ D = y −1 HU ∩ D

implies

)−1 H −1 H ∩D ⊂ β(y ∩D β(y)

and −1 H )−1 H ∩D = β(y ∩D β(y)

implies

y −1 HU ∩ D ⊂ y −1 HU ∩ D .

−1 H ∩ D, Suppose ﬁrst that y −1 HU ∩ D = y −1 HU ∩ D and z = β(z) ∈ β(y) −1 (H) = HU F (P0 ) by where z = b1 t1 with b1 ∈ F(P \ P0 ) and t1 ∈ T . Then yz ∈ β 1., and therefore yz = bb1 tt1 = at0 c0 for some a ∈ H, t0 ∈ U and c0 ∈ F(P0 ). Since bb1 tt1 ∈ F(P \ P0 )T , we obtain c0 = 1 and yz ∈ HU . Hence z ∈ y −1 HU ∩ D = )β(z) )−1 H ∩ D. and β(z) ∈H ∈ β(y y −1 HU ∩D and y z ∈ HU , which implies β(y ∩D = β(y )−1 H ∩D and z ∈ y −1 HU ∩ D. Then −1 H Assume now that β(y) By 1., this implies ∈ H. yz ∈ HU , β(y)β(z) ∈ H and therefore β(y z) = β(y )β(z) y z = at0 c0 , where a ∈ H, t0 ∈ U and c0 ∈ F(P0 ). Since y = b t ∈ F(P \ P0 )T , −1 × we obtain c0 | z (in D) and y (c−1 / D× . By Lemma 0 z)t0 = a ∈ H \ D , since y ∈ −1 −1 4.2.4 we have c0 a = y zt0 ∈ H, and therefore z = y (c0 a)t0 ∈ y −1 HU ∩ D.

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Since CD (HU, D) ⊂ C ∗ (HU, D), it is suﬃcient to prove that C ∗ (HU, D) is ﬁnite. But if C ∗ (H, D) is ﬁnite, then C ∗ (HU, D) is ﬁnite by [23, Lemma 3.3]. is surjective, and β−1 (H × ) = H × by 1. In order 3. By deﬁnition, β : H → H to prove that β is a transfer homomorphism, we will show the following assertion: are such that β(u) = yz, then there exist T: If u ∈ H \ H × and y, z ∈ H v, w ∈ H such that u = vw, β(v) = y and β(w) = z. Suppose that u = cp1 ·. . .·pn t ∈ H \H × , where c ∈ F(P0 ), n ∈ N0 , p1 , . . . , pn ∈ P \P0 and t ∈ T . Since cD× ∩H ∈ {H × , ∅} (by Lemma 4.2.4), we obtain p1 ·. . .·pn t ∈ / D× . We may assume that y = p1 · . . . · pm (t1 U ) and z = pm+1 · . . . · pn (t2 U ), where / D× . We set m ∈ [0, n], t1 , t2 ∈ T , t = t1 t2 t0 for some t0 ∈ U , and p1 · . . . · pm t1 ∈ −1 v = p1 · . . . · pm t1 , w = pm+1 · . . . · pn t2 ∈ β (H), and since v , w ∈ F(P \ P0 )T , 1. implies that v = v0 t and w = wt for some v0 , w ∈ H and t , t ∈ U . By −1 / D× , hence v0 ∈ H \ H × , v0 t t t0 ∈ assumption, we have v0 = p1 · . . . · pm t1 t ∈ × × (H \ H )U ⊂ H \ H and v = c(v0 t t t0 ) ∈ H by Lemma 4.2.4. Now it is easily checked that u = vw, β(v) = y and β(w) = z. The estimate of t(a, uH × , β) follows by Proposition 3.7, and it remains to prove that c(H, β) ≤ 2. For this, we may assume that H is reduced. be the extension of β to the factorization monoids, a ∈ H Let β : Z(H) → Z(H) and z, z ∈ Z(a) such that β(z) = β(z ). We may assume that z = u1 · . . . · un and z = u1 · . . . · un , where n ∈ N, uν , uν ∈ A(H) and β(uν ) = β(uν ) for all −1 ν ∈ [1, n]. We must prove that there is a 2-chain of factorizations in Z(a)∩β (β(z)) concatenating z and z , and we may assume that n ≥ 3. We consider ﬁrst the case where p a for all p ∈ P0 . We set u1 = b1 t1 and 1 ) = β(b ) and t = t1 t for u1 = b1 t1 , where b1 , b1 ∈ F(P \ P0 ), t1 , t1 ∈ T , β(b 1 1 some t ∈ U . We proceed by induction on n + dF (P \P0 ) (b1 , b1 ) + δ, where 0, if t1 = t1 δ= 1, if t1 = t1 . If δ = 1, then u1 t = b1 t1 ∈ H, t−1 u2 ∈ H, and since β(u1 ) = β(u1 t) and β(u2 ) = β(t−1 u2 ), it follows that u1 t ∈ A(H) and t−1 u2 ∈ A(H). If z1 = (u1 t)(t−1 u2 )u3 · . . . · un , then z1 ∈ Z(a), β(z1 ) = β(z), d(z, z1 ) ≤ 2, and by the −1 induction hypothesis, there exists a 2-chain of factorizations in Z(a) ∩ β (β(z)) concatenating z1 and z . Thus we may now assume that δ = 0, hence t1 = t1 . If dF (P \P0 ) (b1 , b1 ) = 0, then u1 = u1 , and by the induction hypothesis there exists a 2-chain of factoriza−1 −1 −1 tions y0 , y1 , . . . yk in Z(u−1 (β(u−1 1 a) ∩ β 1 z)) concatenating u1 z and u1 z . Then −1 u1 y0 , u1 y1 , . . . , u1 yk is a 2-chain of factorizations in Z(a) ∩ β (β(z)) concatenating z and z . Now we consider the case where dF (P \P0 ) (b1 , b1 ) ≥ 1, and we set b = gcd(b1 , b1 ) ∈ 1 ) = β(b ), there exists some p ∈ P \ P0 such that p | b−1 b . F (P \ P0 ). Since β(b 1 1 −1 Then p b b1 , and since b−1 b1 u2 · . . . · un = b−1 b1 u2 · . . . · un , there exists some −1 b ) = β(b −1 b1 ), ν ∈ [2, n] such that p | uν . We assume that p | u2 . Since β(b 1 −1 there exists some p ∈ P \ P0 such that p ∼ p and p | b b1 | u1 . We set v1 = pp−1 b1 t1 and v2 = p p−1 u2 . Then v1 , v2 ∈ D, v1 ∼ u1 , v2 ∼ u2 and therefore

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v1 , v2 ∈ H. Since β(v1 ) = β(u1 ) and β(v2 ) = β(u2 ), we obtain v1 , v2 ∈ A(H) and z1 = v1 v2 u3 · . . . · un ∈ Z(H). We have z1 ∈ Z(a), β(z1 ) = β(z), d(z, z1 ) ≤ 2 and dF (P \P0 ) (pp−1 b1 , b1 ) = dF (P \P0 ) (b1 , b1 ) − 1. By the induction hypothesis, there −1

exists a 2-chain of factorizations in Z(a) ∩ β (β(z)) concatenating z1 and z . This completes the proof in the case where p a for all p ∈ P0 . In the general case, we proceed by induction on |a|F (P0 ) . Let p ∈ P0 be such that a ∈ pD, and apply the assertions of Lemma 4.2.4. We have p−1 a ∈ p−1 H ∩ D ⊂ H, and we assume the induction hypothesis for p−1 a. For every u ∈ A(H), we have pu ∈ H, and β(u) = β(pu) implies pu ∈ A(H). Moreover, if u ∈ A(H) ∩ pD, then p−1 u ∈ H, and again β(p−1 u) = β(u) implies p−1 u ∈ A(H). We may assume that p | u1 and p | u1 . Then y = (p−1 u1 )u2 · . . . · un and y = (p−1 u1 )u2 · . . . · un are factorizations of p−1 a satisfying β(y) = β(y ). By the induction hypothesis, −1 there exists a 2-chain y0 , . . . , yk in Z(p−1 a) ∩ β (β(y)) concatenating y and y . For i ∈ [0, k], we set yi = ui,1 · . . . · ui,n , where ui,ν ∈ A(H) for all ν ∈ [1, n], and in particular u0,1 = p−1 u1 , uk,1 = p−1 u1 , u0,ν = uν and uk,ν = uν for all ν ∈ [2, n]. For i ∈ [0, k], we set zi = (pui,1 )ui,2 · . . . · ui,n . Then z0 , . . . , zk is a 2-chain of −1 factorizations in Z(a) ∩ β (β(z)) concatenating z and z .

5

(Generalized) Block Monoids

In this section we show how the transfer principles for (T -)block monoids as explained in [25] follow from Theorem 4.1. We recall the deﬁnition of a T -block monoid as introduced in [15]. Definition 5.1. Let G be an additive abelian group, G0 ⊂ G a subset, T a monoid and ι : T → G a homomorphism. Let σ : F (G0 ) → G be the unique homomorphism satisfying σ(g) = g for all g ∈ G0 . Then we call B(G0 , T, ι) = {S t ∈ F(G0 )×T | σ(S) + ι(t) = 0 } the T -block monoid over G0 deﬁned by ι. If T = {1}, then B(G0 , T, ι) = B(G0 ), the (ordinary) block monoid of zero-sum sequences in G0 . Let B(G0 , T, ι) ⊂ F (G0 )×T be a T -block monoid as in the deﬁnition, and let ψ : F (G0 ) × T → G be deﬁned by ψ(St) = σ(S) + ι(t). It is easily checked that B(G0 , T, ι) ⊂ F (G0 )×T is saturated, and ψ induces a monomorphism

ψ : F (G0 )×T /B(G0 , T, ι) → G . If ψ is surjective, then B(G0 , T, ι) ⊂ F(G0 )×T is coﬁnal, and ψ is an isomorphism. Theorem 5.2. Let D be a reduced atomic monoid, P ⊂ D a set of primes and T ⊂ D a submonoid such that D = F (P )×T . Let H ⊂ D be a saturated submonoid, G = q(D/H), GP = [p]D/H p ∈ P ⊂ G and G1 = [u]D/H u ∈ A(D) ⊂ G .

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: D → F (GP )×T be the unique Let ι : T → G be deﬁned by ι(t) = [t]D/H , and let β | T = idT . homomorphism satisfying β(p) = [p]D/H for all p ∈ P and β is a transfer homomorphism. 1. β

−1 B(GP , T, ι) = H, and β = β | H: H → 2. β(H) = B(GP , T, ι), β B(GP , T, ι) is a transfer homomorphism satisfying c(H, β) ≤ 2 and t(H, u, β) ≤ D(G1 ) for all u ∈ A(H). Proof. For u, v ∈ D we set u ∼ v, if [u]D/H = [v]D/H . Then ∼ is a congruence relation on D, and if u, v ∈ D, [u]D/H = [v]D/H and u ∈ H, then [u]D/H = [v]D/H = [1]D/H and therefore v ∈ H. Hence we can apply Theorem 4.1 with P0 = ∅ and U = {1}. The special case F = H in Theorem 4.1 implies 1., and for the proof of 2. it remains to show that β(H) = B(GP , T, ι) and t(H, u, β) ≤ D(G1 ) for all u ∈ A(H). If a = p1 · . . . · pn t ∈ D, where n ∈ N0 , p1 , . . . , pn ∈ P and t ∈ T , then β(a) = [p1 ]D/H · . . . · [p]D/H t ∈ F(GP )×T and σ([p1 ]D/H · . . . · [pn ]D/H ) + ι(t) = [p1 ]D/H + . . . + [pn ]D/H + [t]D/H = [a]D/H = 0 ∈ G if and only if a ∈ H. Hence β(a) ∈ B(GP , T, ι) if and only if a ∈ H. To prove the estimate for t(H, u, β), we apply Proposition 3.7. Suppose that n ∈ N, u, u1 , . . . , un ∈ A(H), u | u1 · . . . · un and β(u) = β(u1 ). We must prove that we can renumber u1 , . . . , un in such a way that u | u1 · . . . · uD(G1 ) . We may suppose that u = p1 · . . . · pk t, where k ∈ N0 , p1 , . . . , pk ∈ P and t ∈ T . Then (as mentioned in Section 2) k + δ ≤ D(G1 ), where δ = 1, if t = 1 and δ = 0, if t = 1. If t = 1, we may renumber u1 , . . . , un so that u = p1 · . . . · pk | u1 · . . . · uk . If t = 1, then β(u) = β(u1 ) implies u1 = p1 · . . . · pk t, where pi ∈ P and [pi ]D/H = [pi ]D/H for all i ∈ [1, k]. Hence in this case we may renumber u2 , . . . , un so that p1 · . . . · pk | u1 · . . . · uk+1 and therefore u | u1 · . . . · uk+1 . The main application of Theorem 5.2 is the arithmetic of t-noetherian monoids, which we will sketch now using the notations and results of [26, Ch. 24]. Let H be a t-noetherian monoid whose complete integral closure admits a nonzero conductor, S the set of all strong t-maximal t-ideals, P = t-max(H) \ S, and

S=H\

P.

P ∈S

Then Hp is a discrete valuation monoid for all p ∈ P, H=

p∈t-max(H)

Hp =

Hp ∩ S −1 H ,

p∈P

and this intersection is of ﬁnite character. We set T = (S −1 H)red , and for p ∈ P, we denote by vp the valuation of q(H) associated with Hp . Then the map

ϕ : H → D = F (P)×T , deﬁned by ϕ(a) = (vp (a))p∈P , a(S −1 H)× ,

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is a divisor homomorphism, that means, ϕ(H) ⊂ D is a saturated submonoid, and ∼ ϕ induces an isomorphism ϕred : H → ϕ(H). We set G = D/H, GP = {[p]D/H | p ∈ P }, and we denote by ι : T → G the natural homomorphism. Then the map [p]D/H t , β : H → B(GP , T, ι) , deﬁned by β(a) = p∈P

is a transfer homomorphism for which the assertions of Theorem 5.2 hold. In general, only little is known about the arithmetic of the monoid T . If however T is well-known and GP is ﬁnite, then this transfer homomorphism can be used to obtain strong arithmetical results on H. This is the case if H is weakly Krull. Then T is a ﬁnite direct product of reduced ﬁnitely primary monoids, and strong arithmetical results were obtained in [19, Theorem 8.3]. In particular, this applies to weakly Krull Mori domains (and hence to one-dimensional Mori domains and to noetherian Cohen-Macaulay domains). If T = {1}, then Hred is a saturated submonoid of a free monoid F and thus H is a Krull monoid (see [11, Theorem 2.2]). In this case, we have the following stronger result (for simplicity, we formulate it for Hred instead of H). Theorem 5.3. Let H ⊂ F = F (P ) be a saturated submonoid, GP = p ∈ P and D = D(GP ).

[p]F/H

1. c(GP ) ≤ c(H) ≤ max c(GP ), 2} and c(H) ≤ D. 2. t(GP ) ≤ t(H) ≤ t(GP ) + D. 3. If d ∈ N, p1 , . . . , pd ∈ P and u = p1 · . . . · pd ∈ A(H), then U = [p1 ]F/H · . . . · [pd ]F/H ∈ A(G0 ), and if D ≥ 2, then 3 + (d − 1)(D − 1) d (D − 1) . t(GP , U ) ≤ t(H, u) ≤ max t(GP , U ), ≤1+ 2 2 4. We have the estimates 3 + (D − 1)2 D(D − 1) t(H) ≤ max t(GP ), , ≤1+ 2 2 and if GP = −GP ⊂ q(F/H) and H is not factorial, then t(H) ≥ D. Proof. We may assume that D ≥ 2, for otherwise H = F and the assertions are obvious. By Proposition 5.2, there is a transfer homomorphism β : H → B(GP ) satisfying c(H, β) ≤ 2 and t(H, u, β) ≤ D for all u ∈ A(H). Also, if γ ∈ N, p1 , . . . , pγ ∈ P and a = p1 · . . . · pγ ∈ H, then β(a) = [p1 ]F/H · . . . · [pγ ]F/H . 1. The ﬁrst series of inequalities follows by Theorem 3.4.5. (see also [18, Proposition 4.2]). By [18, Proposition 4.3], we have c(GP ) ≤ D.

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2. If u ∈ A(H), then U = β(u) ∈ A(GP ) by Proposition 3.2.2, and Theorem 3.4.1 implies t(GP , U ) ≤ t(H, u) ≤ t(GP , U ) + D. Taking the supremum over all u ∈ A(H), the assertion follows. 3. We just proved the ﬁrst inequality, and now we shall prove that 3 + (d − 1)(D − 1) d (D − 1) . a) t(H, u) ≤ max t(GP , U ), and b) t(H, u) ≤ 1+ 2 2 Then we apply b) to B(GP ) ⊂ F(GP ) instead of H ⊂ F and obtain t(GP , U ) ≤ 1 + d(D − 1)/2. From D ≥ 2 it follows that d (D − 1) 3 + (d − 1)(D − 1) ≤1+ , 2 2 which then completes the proof. a) Let a ∈ H be such that u | a and z ∈ Z(a). It is suﬃcient to prove that there exists some z ∈ Z(a) ∩ uZ(H) such that 3 + (d − 1)(D − 1) d(z, z ) ≤ t = max t(GP , U ), . 2 We proceed by induction on |z|. If z = 1, there is nothing to do. Thus suppose that z = v1 · . . . · vl , where l ∈ N and v1 , . . . , vl ∈ A(H). We may assume that for some n ∈ [1, l] we have u | v1 · . . . · vn (in F ), but u does not divide any proper subproduct of v1 · . . . · vn . If n < l, the induction hypothesis implies the existence of u2 , . . . , uk ∈ A(H) such that v1 · . . . · vn = uu2 · . . . · uk , and in Z(H) we have d(v1 · . . . · vn , uu2 · . . . · uk ) ≤ t. Then z = uu2 · . . . · uk vn+1 · . . . · vl ∈ Z(a) ∩ uZ(H) and d(z, z ) ≤ t. Hence we may suppose that n = l. Then l ≤ d, and if Vi = β(vi ) ∈ A(G0 ) for all i ∈ [1, l], then U | V1 · . . . · Vl . We may assume that there exists some m ∈ [1, l] such that U | V1 · . . . · Vm , but U does not divide any nonempty proper subproduct / P . Then of V1 · . . . · Vm . If u ∈ P , there is nothing to do. Thus suppose that u ∈ U = 0 and D ≥ |Vj | ≥ 2 for all j ∈ [1, m]. By the very deﬁnition of t(GP , U ), there exist U2 , . . . , Uk ∈ A(G0 ) such that V1 · . . . · Vm = U U2 · . . . · Uk , and in Z(G0 ) we have max{k, m} = d(V1 · . . . · Vm , U U2 · . . . · Uk ) ≤ t(GP , U ) ≤ t . Since β(u−1 a) = U2 · . . . · Uk Vm+1 · . . . · Vl and β : H → B(GP ) is a transfer homomorphism, there exist u2 , . . . , uk , wm+1 , . . . , wl ∈ A(H) such that β(ui ) = Ui for all i ∈ [2, k], β(wj ) = Vj for all j ∈ [m + 1, l] and u−1 a = u2 · . . . · uk wm+1 · . . . · wl . Then z = uu2 · . . . · uk wm+1 · . . . · wl ∈ Z(a) ∩ uZ(H) and d(z, z ) ≤ max{l, k + l − m} . If m = l, this implies d(z, z ) ≤ t. It remains to consider the case m < l. Since u does not divide any proper subproduct of v1 · . . . · vl , there is a factorization u = u u , where u , u ∈ F \ {1}, u | v1 · . . . · vm , u | vm+1 · . . . · vl , |u | ≥ m and |u | ≥ l − m. From the trivial estimate 2(k − 1) ≤ |U2 · . . . · Uk | = |U −1 V1 · . . . · Vm | ≤ mD − |U | ≤ |u |D − d

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133

|u |D − d + |u | max{l, k + l − m} ≤ max d, 1 + 2

we obtain

and 1+

|u |D − d 2 + |u |D − |u | + |u | + |u | = 2 2 2 + d(D − 1) − |u |(D − 2) 3 + (d − 1)(D − 1) ≤ ≤ . 2 2

b) We assume that a ∈ H, u ∈ A(H), u | a, and we estimate t(a, u). If d = 1, then t(a, u) = 0. Thus suppose that d ≥ 2 and z = u1 · . . . · ur ∈ Z(a), where r ∈ N and u1 , . . . , ur ∈ A(H). We must prove that there exists some z ∈ Z(a) ∩ uZ(H) such that d(D − 1) . d(z, z ) ≤ 1 + 2 We may assume that there exists some k ∈ [1, r] such that k ≤ d and u | u1 · . . . · uk (in F and thus also in H). Since p u for all p ∈ P ∩ H, we may also assume that u1 , . . . , uk ∈ / P ∩ H. If u1 · . . . · uk = uc, where c ∈ H, and if w ∈ Z(c), then z = uwuk+1 · . . . · ur ∈ Z(a), and d(z, z ) ≤ max{k, |w| + 1} ≤ max{d, |w| + 1}. Since |w| ≤

|c|F |u1 |F + . . . + |uk |F − |u|F kD − d d(D − 1) = ≤ ≤ , 2 2 2 2

we obtain the asserted bound for d(z, z ). 4. The estimate for t(H) follows immediately from 3. by taking the supremum over all u ∈ A(H). Assume now that H is not factorial, GP = −GP , and let N ∈ N be such that N < D. We shall prove that t(H) > N . By assumption, there exists some u ∈ A(H) such that u = p1 · . . . · pm , where p1 , . . . , pm ∈ P and m > N . Then there exist p1 , . . . , pm ∈ P such that [pj ]F/H = −[pj ]F/H and thus uj = pj pj ∈ A(H) for all j ∈ [1, m]. Since β is a transfer homomorphism, it follows that u = p1 · . . . · pm ∈ A(H) and Z(uu ) = {uu , u1 · . . . · um }. Since H is not factorial, we have t(H) ≥ 2, and therefore we may assume that D > N ≥ 2. Then m ≥ 3, and t(H) ≥ t(H, u) ≥ d(uu , u1 · . . . · um ) = m > N .

6

K+M-Domains

The most common examples of K+ M -domains are integral domains of the form K + XL[X] and K + XL[[X]], where K ⊂ L are integral domains. Their arithmetic has been investigated in several papers, see e. g. [7] or [2]. If K and L are ﬁelds, then the domains K +XL[X] and K +XL[[X]] are half-factorial. With our methods, we investigate more generally the case of a ring extension of the form R = K + m ⊂ L + m = D, where K ⊂ L are ﬁelds and m is a maximal ideal of D. Note that every ﬁnitely generated algebra R over an inﬁnite perfect ﬁeld with ρ(R) < ∞ is of this form (see [28, Theorem 6.2] where a transfer principle was used to investigate the elasticity of such domains).

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Proposition 6.1. Let R ⊂ D be integral domains and m ∈ max(R) such that D = RD× , D× ∩ R = R× and (R : D) = m. Then the injection j = (R• → D• ) is a transfer homomorphism satisfying c(a, j) ≤ 2

and

t(a, uR× , j) ≤ 2

for all

a ∈ R and u ∈ A(R) .

In particular, R is atomic [a BF-monoid, locally tame ] if and only if D has this property. Moreover, L(R) = L(D), ρ(R) = ρ(D), c(D) ≤ c(R) ≤ max{2, c(D)} ,

t(D) ≤ t(R) ≤ t(D) + 2 ,

and if D is factorial, then R is half-factorial. Proof. Since D = RD× and D× ∩ R = R× , j satisﬁes (T 1). For the proof of (T 2), let u ∈ R• and b, c ∈ D• be such that u = bc. We must prove that there exist v, w ∈ R• and ε, η ∈ D× such that u = vw, v = bε and w = cη. Since D = RD× , there exist b0 , c0 ∈ R and ε0 , η0 ∈ D× such that b = b0 ε0 and c = c0 η0 . Now we distinguish two cases. CASE 1: b0 ∈ m. Then we have ε0 η0 b0 ∈ m ⊂ R, and we set v = ε0 η0 b0 , w = c0 , ε = η0 and η = η0−1 . CASE 2: b0 ∈ / m. Since m ∈ max(R), there exists some d ∈ R such that b0 d ≡ 1 mod m, hence ε0 η0 c0 ≡ ε0 η0 b0 c0 d ≡ ud mod m and ε0 η0 c0 = ud + m ⊂ R. Now we set v = b0 , w = ε0 η0 c0 , ε = ε−1 0 and η = ε0 . We prove that jred satisﬁes the assumptions of Proposition 3.7 with W = 2. Thus suppose that n ∈ N≥2 , u, u1 , . . . , un ∈ A(R), a = u1 · . . . · un , u | a and u = u1 ε for some ε ∈ D× . We must prove that there exists some i ∈ [2, n] such that u | u1 ui . Let b ∈ R be such that a = ub. Then εb = u2 · . . . · un ∈ R. If b ∈ m, then ui ∈ m for some i ∈ [2, n]. Hence ε−1 ui ∈ m ⊂ R, and u1 ui = u(ε−1 ui ) implies u | u1 ui . If b ∈ / m, then there exists some c ∈ R such that bc ≡ 1 mod m and thus ε ≡ εbc ≡ u2 · . . . · un c mod m. Hence ε ∈ u2 · . . . · un c + m ⊂ R, and therefore ε ∈ R× , which implies u u1 in R. Now Proposition 3.7 implies c(a, j) ≤ 2 and t(a, uR× , j) ≤ 2 for all a ∈ R and u ∈ A(R). The remaining assertions follow by Proposition 3.2 and Theorem 3.4.

Proposition 6.2. Let R ⊂ D be an integral domain. Let K ⊂ L ⊂ D be subﬁelds, {0} = m ∈ max(D) such that D = L + m and R = K + m. Then R → D satisﬁes the assumptions of Proposition 6.1. Proof. We must show that D = RD× and D× ∩ R = R× . We obviously have L× R ⊂ D× R ⊂ D and R× ⊂ D× ∩ R. If a ∈ D, then a = u + m for some u ∈ L and m ∈ m. If u = 0, then a ∈ m ⊂ R. If u = 0, then u−1 a = 1+u−1m ∈ 1+m ⊂ R, and a = u(u−1 a) ∈ L× R. If x ∈ D× ∩ R, then x = u + m and x−1 = u + m , where u ∈ K × , u ∈ L and m, m ∈ m. Then 1 ∈ (uu + m) ∩ (K + m) implies uu ∈ K and therefore u ∈ K, x−1 ∈ R.

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7

135

C-Monoids and Congruence Monoids in N

In [21], we introduced C0 -monoids and AC-monoids as abstract tools for the arithmetical investigation of congruence monoids in Dedekind domains. In [27], W. Hassler used the concept of a Z-monoid as an abstract scheme for the arithmetic of noetherian domains satisfying some natural ﬁniteness conditions, and in [13], a special type of C0 -monoids (those which are also Z-monoids) are used to study monotone chains of factorizations in certain congruence monoids and ﬁnitely generated domains. The concept of a C-monoid was introduced in [23] as a common generalization of all notions of auxiliary monoids just mentioned. Here we show how our general transfer principle establishes a connection between all these very similar concepts. Congruence monoids will ﬁnally serve as characteristic examples to show how our formalism works. We start by recalling the deﬁnitions. Definition 7.1. A monoid H is called a C-monoid , if it is a submonoid of a factorial monoid F such that H ∩ F × = H × and C ∗ (H, F ) is ﬁnite. Then it follows by [23, Proposition 3.10] that there exists some α ∈ N and a subgroup V ⊂ F × such that H × ⊂ V , (F × : V ) | α, V (H \ H × ) ⊂ H

and q 2α F ∩ H = q α (q α F ∩ H) for all q ∈ F \ F × .

We refer to these properties by saying that H is deﬁned in F with exponent α and subgroup V ⊂ F × . If F = F × × F(P ) and x ∈ q(F ), we call suppP (x) = {p ∈ P | vp (x) = 0} the support of x in P . A subset E ⊂ P is called H-essential, if E = suppP (a) for some a ∈ H \ F × , and we set κ(H, F ) = max |E| E ⊂ P is a minimal H-essential subset . Note that κ(H, F ) only depends on F and not on the set P . By Theorem 7.2 below, H is a BF-monoid, and by [23, Theorem 5.1] we have ρ(H) < ∞ if and only if κ(H, F ) = 1. For Hilbert monoids, we shall make this criterion explicit in Theorem 7.8. A C0 -monoid is a C-monoid H deﬁned in a factorial monoid F possessing only ﬁnitely many pairwise nonassociated prime elements. A Z-monoid is a C-monoid H deﬁned in a factorial monoid F such that C(H, F ) is ﬁnite. We do not recall here the notion of an AC-monoid because it plays no role further on. We mention however that by [21, Theorem 5.4] every AC-monoid deﬁned in a factorial monoid is a C-monoid, and by [23, Theorem 4.3], every C0 -monoid is an AC-monoid deﬁned in a factorial monoid. If H is a C-monoid deﬁned in a factorial monoid F , then C(H, F ) is ﬁnite if and only if F × /H × is ﬁnite (see [27, Proposition 5.3] and [23, Lemma 3.3]). The following theorem generalizes [27, Theorem 5.4]. Theorem 7.2. Let F = F ××F(P ) be a factorial monoid and H a C-monoid deﬁned in F with subgroup V ⊂ F × . Let P0 ⊂ {p ∈ P | p−1 H ∩ F = H \ H × } be a subset, P = {[p]F H | p ∈ P \ P0 }, and F = F (P)×F × /V ,

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: F → F be the unique homowhere F (P ) is the free monoid with basis P . Let β F = 1 for all p ∈ P0 , and morphism satisfying β(p) = [p]H for all p ∈ P \ P0 , β(p) × β(e) = eV for all e ∈ F . Finally, we deﬁne = β(H) H

and

|H : H → H . β=β

is a C0 -monoid deﬁned in F , and F × = F × /V is ﬁnite. 1. H 2. β is a transfer homomorphism, c(H, β) ≤ 2 and t(a, uH × , β) ≤ ω(a, u) + 1 for all a ∈ H and u ∈ A(H). 3. H is locally tame, c(H) < ∞, and the Structure Theorem for Sets of Lengths holds for H. F). 4. κ(H, F ) = κ(H, Proof. 1. and 2. (H, F )-equivalence is a congruence relation on F , and if u, v ∈ F F, [u]F H = [v]H and u ∈ H, then v ∈ H. Hence we can apply Theorem 4.1 F) are ﬁnite, and with T = F × and U = V . Then F × = F × /V , P and C ∗ (H, × × H ∩ F = H . In particular, H is a C0 -monoid deﬁned in F . The estimates for c(H, β) and t(a, uH × , β) follow by Theorem 4.1.3 (note that H is a BF-monoid). has these properties, and by Theorem 3.4, the same 3. By [21, Theorem 5.9], H is true for H. 4. It is suﬃcient to prove the following assertions for a ﬁnite subset E ⊂ P \ P0 : a) If E is H-essential, then β(E) is H-essential. b) If β(E) is H-essential and |β(E)| = |E|, then E is H-essential. Indeed, suppose that this is done, and let E ⊂ P be a minimal H-essential subset such that |E| = κ(H, F ). Then E ∩ P0 = ∅ by Lemma 4.2.4, β(E) is H-essential ) is a minimal H-essential by a), and there exists a subset E ⊂ E such that β(E set, and |β(E )| = |E |. Then E is H-essential by b), hence E = E and κ(H, F ) = )| ≤ κ(H, F). Conversely, let E ⊂ P be a minimal H-essential subset such |β(E = β(E) = κ(H, F ). Then there exists a subset E ⊂ P \ P0 such that E that |E| = |E|. By b), E is H-essential, and there exists a minimal H-essential and |E| ) is H-essential )| = |E | and β(E ) ⊂ E. subset E ⊂ E. Now β(E by a), |β(E and therefore E = E and κ(H, F ) = |E| ≤ κ(H, F ). Hence β(E ) = E, a) If E ⊂ P is H-essential, then E = suppP (a) for some a ∈ H \ F × , and β(E) = suppPe β(a) is H-essential. 1 ), . . . , β(p r) b) Suppose that E = {p1 , . . . , pr } ⊂ P \ P0 is such that β(p such that are distinct and β(E) is H-essential. Then there exists some y ∈ H n1 nr × y = β(p1 ) · . . . · β(pr ) (eV ), where n1 , . . . , nr ∈ N and e ∈ F . Hence z = −1 (H), and therefore, by Theorem 4.1.1, z = ytc, where y ∈ H, pn1 1 · . . . · pnr r e ∈ β t ∈ V and c ∈ F(P0 ). It follows that c = 1, y = pn1 1 · . . . · pnr r et−1 ∈ H, and thus E = suppP (y) is H-essential.

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As already mentioned, congruence monoids in Dedekind domains and the arithmetic of ﬁnitely generated noetherian domains are the main instances for the application of Theorem 7.2. In [23] we study more generally congruence monoids in Krull domains. Here we discuss brieﬂy the most modest special case of congruence monoids: Hilbert monoids of natural numbers (see [21, Examples 3.3] and [6]). Definition 7.3. Let f ≥ 2 be a natural number. A subset H ⊂ N is called a congruence monoid or a Hilbert monoid deﬁned modulo f , if there is a nonempty, multiplicatively closed subset Λ ⊂ Z/f Z such that H = {a ∈ N | a + f Z ∈ Λ} ∪ {1}. Theorem 7.4. Let f ≥ 2 be a natural number and H ⊂ N a Hilbert monoid deﬁned modulo f . Let g ∈ N be squarefree such that H ⊂ Ng . Then g | f , H is a C-monoid deﬁned in Ng , and there is an epimorphism θ : (Z/f Z, ·) → C ∗ (H, Ng ), given by N θ(a + f Z) = [a]Hg . Proof. If p ∈ P and p f , then there exists some k ∈ N such that pk ≡ 1 mod f , and if 1 = a ∈ H, then also pk a ∈ H. Hence it follows that g | f . N N If a, b ∈ Ng \ {1} and a ≡ b mod f , then [a]Hg = [b]Hg by the very deﬁnition of the Hilbert monoid. Hence the canonical epimorphism Ng \ {1} → C ∗ (H, Ng ), N deﬁned by a → [a]Hg , induces an epimorphism Z/f Z → C ∗ (H, Ng ). Proposition 7.5. Let f ≥ 2 be a natural number and Λ ⊂ Z/f Z a nonempty multiplicatively closed subset. For a ∈ Z, set a = a + f Z ∈ Z/f Z, (Λ : a) = {c ∈ Z/f Z | a c ∈ Λ}, and consider the Hilbert monoid H = {a∈ N | a ∈ Λ} ∪ {1}. Let g ∈ N be squarefree such that H ⊂ Ng , and deﬁne P0 = H \ {1} . N

p ∈ Pg | p−1 H ∩ Ng =

N

1. For a, b ∈ Ng \ {1} we have [a]Hg = [b]Hg if and only if (Λ : a) = (Λ : b). 2. P0 = {p ∈ Pg \ H | (Λ : p) = Λ}. 3. P0 = ∅ if and only if 1 ∈ Λ. 4. If 1 ∈ Λ, then {p ∈ P | p ≡ 1 mod f } ⊂ {p ∈ Pg | (Λ : p) = Λ} ⊂ H, N {[p]Hg p ∈ Pg , (Λ : p) = Λ} = 1 , and if p ∈ Pf ∩ H, then (Λ : p) = Λ. Proof. 1. Obvious by the deﬁnitions. 2. If p ∈ P0 , then p−1 H ∩ Ng = H \ {1} and therefore p ∈ / H. For the proof of (Λ : p) = Λ, suppose that a ∈ Ng \ {1}. If a ∈ (Λ : p), then ap = a p ∈ Λ, hence ap ∈ H and a ∈ p−1 H ∩ Ng = H \ {1}, which implies a ∈ Λ. Conversely, if a ∈ Λ, then a ∈ H \ {1} = p−1 H ∩ Ng , hence pa ∈ H, p a ∈ Λ, and therefore a ∈ (Λ : p). Assume now that p ∈ Pg \ H and (Λ : p) = Λ. We must prove that p−1 H ∩ Ng = H \ {1}. If a ∈ p−1 H ∩ Ng , then pa ∈ H \ {1}, hence pa = p a ∈ Λ and thus

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a ∈ (Λ : p) = Λ. Since p ∈ / H, we have a = 1 and therefore a ∈ H \ {1}. Conversely, if a ∈ H \{1}, then a ∈ Λ = (Λ : p), hence pa = p a ∈ Λ, pa ∈ H and a ∈ p−1 H ∩Ng . / Λ and p ∈ P is such that p ≡ 1 mod f , then p ∈ / H and (Λ : p) = (Λ : 3. If 1 ∈ 1) = Λ. Hence p ∈ P0 . Assume now that 1 ∈ Λ and P0 = ∅. If p ∈ P0 , then 1 ∈ Λ = (Λ : p), hence p ∈ Λ and p ∈ H, a contradiction. 4. It suﬃces to prove the last assertion. Thus suppose that p ∈ Pf ∩ H. Then p ∈ Λ and thus pΛ ⊂ Λ. Hence (Λ : p) ⊃ Λ. Conversely, if a ∈ N \ {1} is such that a ∈ (Λ : p), then ap = a p ∈ Λ and thus ap ∈ H. Let n ∈ N be such that pn ≡ 1 mod f . Then apn = (ap)pn−1 ∈ H, and apn ≡ a mod f . Hence a ∈ H and a ∈ Λ. We proceed with two concrete examples. We use the notations introduced in Theorem 7.2, Theorem 7.4 and Proposition 7.5. Example 7.6. f = 3, Λ = {0, 1} ⊂ Z/3Z, H = {a ∈ N | a ≡ 0 or 1 mod 3}, g = 1. We ﬁnd (Λ : 0) = Z/3Z, (Λ : 1) = {0, 1} and (Λ : 2) = {0, 2}. Hence θ : (Z/3Z, ·) → C ∗ (H, N) is an isomorphism, P = { 7, 3, 5}, and 7 is a prime element of = { 3b 5c | a ∈ N0 , (b, c) ∈ T } ∼ H 7a = N0 ×T , where T = {(b, c) ∈ N20 | b ≥ 1 or (b = 0 , c ≡ 0 mod 2)} ⊂ (N20 , +) , and there is a transfer homomorphism β : H → H. If u0 = (1, 0), u1 = (1, 1) and v = (0, 2), then A(T ) = {u0 , u1 , v}, and = c(T ) = 3, 2u0 + v = 2u1 . From these relations it is easy to derive c(H) = t(T ) = 3. Hence we obtain t(T, u0 ) = t(T, u1 ) = t(T, v) = 3, and therefore t(H) c(H) = 3, and a detailed direct analysis shows that also t(H) = 3. Finally, we calculate the elasticity. Using Proposition 3.2.5, we obtain ρ(H) = ρ(T ). We use the semi-length function λ : T → N0 , deﬁned by λ(b, c) = 2b + c (see Section 2) and obtain ρ(H) ≤ 3/2. Hence ρ(H) = 3/2, since 2u0 + v = 2u1 . Example 7.7. f = 9, Λ = {0, 6} ⊂ Z/9Z, H = {a ∈ N | a ≡ 0 or 6 mod 9}, g = 1. We ﬁnd (Λ : 1) = (Λ : 4) = (Λ : 7) = {0, 6}, (Λ : 2) = (Λ : 5) = (Λ : 8) = {0, 3}, (Λ : 0) = Z/9Z, (Λ : 3) = {0, 2, 3, 5, 6, 8} and (Λ : 6) = {0, 1, 3, 4, 6, 7} . Hence θ(1) = θ(4) = θ(7) and θ(2) = θ(5) = θ(8), and C ∗ (H, N) = {θ(0), θ(1), θ(2), θ(3), θ(6)} is a semigroup consisting of 5 elements. Consequently, P = { 7, 2, 3}, and 7 is a prime element of = { H 7a 2b 3c | a ∈ N0 , (b, c) ∈ T } ∼ = N0 ×T , where T = {(b, c) ∈ N20 | c ≥ 2 or (c = 1 , b ≡ 1 mod 2)} ∪ {(0, 0)} ⊂ (N20 , +) , and there is a transfer homomorphism β : H → H.

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We deﬁne w = (0, 2), w = (0, 3), and for odd k ∈ N, we set uk = (k, 1) and v k = (k, 2). Then A(T ) = {w, w , uk , uk | k ∈ N odd }, and between these atoms the following relations hold for all odd k, l ∈ N: uk + ul = uk+l−1 + u1 , ul + v l = v k+l−1 + u1 = uk+l−1 + v 1 , v k + v l = v k+l−1 + v 1 uk + w = 2u1 + uk−2 and uk + w = 2u1 + v k−2 ,

v k + w = uk + w ,

v k + w = uk + 2w ,

if k ≥ 3 ,

3w = 2w .

From these relations we derive c(T ) = 3, and consequently, as in Example 7.6, also c(H) = 3. For odd k ∈ N, we set ak = (3k, 3k + 1), and we consider the factorizations ak = kw + u3k = (3k − 1)u1 + v 1 . They show that ρ(ak ) = (3k − 1)/(k + 1) and t(T, u3k ) ≥ k. Hence t(T ) = ∞, and consequently, as in Example 7.6, also t(H) = ∞. Moreover, 3k − 1 k ∈ N odd = 3. ρ(T ) ≥ sup k+1 Again as in Example 7.6, we use a semi-length function to obtain an upper bound for the elasticity. We deﬁne λ : T → N0 by λ(b, c) = c and obtain ρ(T ) ≤ 3. Hence ρ(T ) = 3 and it is easily seen that T does not have accepted elasticity. Therefore it follows that also ρ(H) = 3, and H does not have accepted elasticity. Theorem 7.8. Let H ⊂ N be a congruence monoid. Then the following statements are equivalent: (a) ρ(H) < ∞. (b) For every a ∈ H \ {1} there exists some p ∈ P and α ∈ N such that p | a and pα ∈ H. Proof. By Theorem 7.4, there exists a squarefree integer g ∈ N such that H is a C-monoid deﬁned in Ng , and therefore ρ(H) < ∞ if and only if κ(H, Ng ) = 1, that is, every minimal H-essential subset of Pg is a singleton (see [23, Theorem 5.1]). But this is just condition (b).

Bibliography [1] D. D. Anderson (ed.), Factorization in integral domains, Marcel Dekker, 1997. [2] D. F. Anderson and D. N. El Abidine, Factorization in integral domains. III., J. Pure Appl. Algebra 135 (1999), no. 2, 107–127. [3] D.D. Anderson and D.F. Anderson, Elasticity of factorizations in integral domains, J. Pure Appl. Algebra 80 (1992), 217–235. [4] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorizations in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19. [5] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorizations in integral domains II, J. Algebra 152 (1992), 78–93. [6] M. Banister, S. Chapman, J. Chiaka, and W. Myerson, On the arithmetic of arithmetical congruence monoids. [7] V. Barucci, L. Izelgue, and S. Kabbaj, Some factorization properties of A + XB[X]-domains, Commutative ring theory, Lecture Notes in Pure Appl. Math., Marcel Dekker 185 (1997), 69–78. [8] L. Carlitz, A characterization of algebraic number ﬁelds with class number two, Proc. AMS 11 (1960), 391–392. [9] S. T. Chapman and J. Coykendall, Half-factorial domains, a survey, Nonnoetherian commutative ring theory, Mathematics and its Applications, Kluwer Academic Publisher 520 (2000), 459–476. [10] S. T. Chapman, M. Freeze, and W. Smith, On generalized lengths of factorizations in Dedekind and Krull domains, Non-noetherian commutative ring theory, Mathematics and its Applications, Kluwer Academic Publisher 520 (2000), 117–137. [11] S.T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 73–112. [12] A. Foroutan, Monotone chains of factorizations, Int. J. of Commutative Rings 2 (2003). [13] A. Foroutan and A. Geroldinger, Monotone chains of factorizations in Cmonoids, these Proceedings. 140

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¨ [14] A. Geroldinger, Uber nicht-eindeutige Zerlegungen in irreduzible Elemente, Math. Z. 197 (1988), 505–529. [15] A. Geroldinger, T-block monoids and their arithmetical applications to certain integral domains, Comm. Algebra 22 (1994), 1603–1615. [16] A. Geroldinger, The catenary degree and tameness of factorizations in weakly Krull domains, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 113–153. [17] A. Geroldinger, Chains of factorizations and sets of lengths, J. Algebra 188 (1997), 331–363. [18] A. Geroldinger, Chains of factorizations in weakly Krull domains, Colloq. Math. 72 (1997), 53–81. [19] A. Geroldinger, A structure theorem for sets of lengths, Colloq. Math. 78 (1998), 225–259. [20] A. Geroldinger and F. Halter-Koch, Tamely generated ideals in ﬁnitary monoids, JP J. Algebra Number Theory Appl. 2 (2002), 205–239. [21] A. Geroldinger and F. Halter-Koch, Congruence monoids, Acta Arith. 112 (2004), 263–296. [22] A. Geroldinger and G. Lettl, Factorization problems in semigroups, Semigroup Forum 40 (1990), 23–38. [23] F. Halter-Koch, C-monoids and congruence monoids in Krull domains, these Proceedings. [24] F. Halter-Koch, Elasticity of factorizations in atomic monoids and integral domains, J. Th. Nomb. Bordeaux 7 (1995), 367–385. [25] F. Halter-Koch, Finitely generated monoids, ﬁnitely primary monoids and factorization properties of integral domains, Factorization in integral domains, Lecture Notes in Pure Appl. Math., Marcel Dekker 189 (1997), 73–112. [26] F. Halter-Koch, Ideal systems. An introduction to multiplicative ideal theory, Marcel Dekker, 1998. [27] W. Hassler, Factorization in ﬁnitely generated domains, J. Pure Appl. Algebra 186 (2004), 151–168. [28] F. Kainrath, Elasticity of ﬁnitely generated domains, Houston J. Math. 30 (2004). [29] W. Narkiewicz, Finite abelian groups and factorization problems, Colloq. Math. 42 (1979), 319–330. [30] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, 2nd ed., Springer, 1990. [31] W. Narkiewicz and J. Sliwa, Finite abelian groups and factorization problems II, Coll. Math. 46 (1982), 115–122.

Chapter 7

Cale Monoids, Cale Domains, and Cale Varieties by

Scott T. Chapman and Ulrich Krause

Introduction In an integral domain or monoid, two diﬀerent factorizations of an element into atoms often turn out to be essentially the same when lifted to powers. In this chapter, we analyze this fascinating phenomenon, which surprisingly occurs for a great many algebraic structures. These structures range from algebraic number rings over Diophantine monoids, to varieties and their deﬁning semigroup rings. Analytically we capture this phenomenon by the concept of a Cale representation, which is a kind of unique representation with respect to powers. Section 1 of this chapter presents the exact deﬁnitions concerning Cale concepts, as well as an array of rather diﬀerent examples which illustrate the lifting of nonunique factorization to unique Cale representation. In Section 2, we consider Cale representation with respect to extraction analysis, which is one of the key tools used in the arguments throughout this chapter. Section 3 considers some useful special elements suggested by the study of the Cale property and places these elements in context with respect to the class of Krull monoids. In Section 4 we present for monoids, as well as for domains, general conditions for the Cale representation to exist. We take mainly from a joint work of the current authors with Franz Halter-Koch [4]. In Section 5 we present particular results on Cale representation in algebraic orders, Diophantine monoids and semigroup rings. Much of this work is drawn from the papers [4, 18, 21, 22]. In Section 6 we investigate aﬃne toric varieties deﬁned via semigroup rings for Cale monoids. The main result (Theorem 6.5) gives a description of those varieties by only a few binomials stemming from Cale representation. The authors wish to thank Alfred Geroldinger and Florian Kainrath for very useful comments on an earlier draft of this chapter. 142

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1

143

Cale Representation: Deﬁnition, Examples, and Counterexamples

Throughout this chapter, by a monoid M we mean an abelian semigroup with 1 (usually written multiplicatively) which is cancellative. Denote by M × its group of units. M is reduced if M × = {1}. Definition 1.1. The monoid M is inside factorial with base Q ⊂ M \M × if for each x ∈ M \M × there exists a positive integer n(x) and nonnegative integers {t(q)}q∈Q such that (i) xn(x) = u q∈Q q t(q) where u ∈ M × and only ﬁnitely many of the t(q)’s are nonzero, and (ii) if xn(x) = u q∈Q q t(q) = v q∈Q q s(q) where v ∈ M × and each s(q) is a nonnegative integer, then u = v and t(q) = s(q) for each q. For each x ∈ M \M × we denote by m(x) the minimal value of n(x) from part (i) of the deﬁnition and by x(q) the uniquely determined t(q). The base Q is a Cale base f (q)x(q) ∈ N0 for (or a tame base) if for each q ∈ Q there exists a f (q) ∈ N such that m(x) × all x ∈ M \M . As with n(x), in a Cale base Q each f (q) can be chosen minimally, and we denote this value as e(q). An inside factorial monoid with a Cale base is a Cale monoid and (i) is referred to as a Cale representation of x. The above notions apply to an integral domain D if its multiplicative monoid M = D• = D\{0} has the corresponding properties. In particular, a Cale domain is a domain D such that D• is inside factorial with Cale base Q. In what follows, we illustrate the above concepts by an array of quite diﬀerent examples. In all these examples, the monoid considered is atomic, that is each nonunit is a product of ﬁnitely many atoms. An atom is an irreducible nonunit and we consider nonunique factorization into atoms which sometimes, but not always, can be explained by a Cale representation. Example 1.2. Algebraic number rings √ √ Let D = Z[ −5] be the principal order in Q( −5). The nonunique factorization into atoms in D √ √ 6 = 2 · 3 = (1 + −5) · (1 − −5) is commonly “explained” by the Kummer–Dedekind representation pp · qr = pq · pr into prime ideals with p = 2, 1 +

√ √ √ −5, q = 3, 1 + −5, and r = 3, 1 − −5,

which is unique up to the ordering of factors. Alternatively, by taking squares we have an explanation of this factorization by the following Cale representation √ √ √ √ 62 = 22 · (−2 + −5)(−2 − −5) = 2(−2 + −5) · 2(−2 − −5) which is unique up to ordering of factors and units.

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√ √ Whereas D is a Cale domain, the nonprincipal order R = Z[5 −1] in Q( √−1) fails to√be a Cale domain. This can be seen from the factorization 5 = (1 + 2 −1)(1 − 2 −1) into Gaussian primes for which no power belongs to R. More details on the Cale property in algebraic orders can be found in Section 5.1. Example 1.3. Diophantine monoids Consider the Diophantine monoid M = {x ∈ N30 | 2x1 + 5x2 = 3x3 } with addition (see [6]). One has the following nonunique factorization into atoms (3, 3, 7) = (1, 2, 4) + (2, 1, 3) = (3, 0, 2) + (0, 3, 5). Multiplying by 3, one obtains the following unique Cale representation 3 · (3, 3, 7) = (q1 + 2q2 ) + (2q1 + q2 ) = 3q1 + 3q2 with q1 = (3, 0, 2), q2 = (0, 3, 5). A Diophantine monoid, however, need not be a Cale monoid as the following example shows. Let M = {x ∈ N40 | x1 + x2 = x3 + x4 }. One has the following nonunique factorization into atoms (1, 1, 1, 1) = (1, 0, 1, 0) + (0, 1, 0, 1) = (1, 0, 0, 1) + (0, 1, 1, 0). It is easy to verify that (1, 0, 1, 0), (0, 1, 0, 1), (1, 0, 0, 1), and (0, 1, 1, 0) are the only 4 atoms of M . An elementary combinatorial argument can be used to show that it is impossible to combine the atoms so that a multiple of (1, 1, 1, 1) can be factored uniquely. Hence, the former nonunique factorization cannot be “explained ” by a Cale representation. More information on the Cale Property in Diophantine monoids can be found in Section 5.2. Example 1.4. Krull Monoids Examples 1.2 and 1.3 are members of the more general class of Krull Monoids. A monoid M is a Krull monoid if there exists a free Abelian monoid D and a homomorphism ∂ : M → D such that 1. x | y in M if and only if ∂(x) | ∂(y) in D, 2. every β ∈ D is the greatest common divisor of some set of elements in ∂(M ). The free basis elements of D are called the prime divisors of M and the quotient D/∂(M ) is called the divisor class group of M and denoted Cl(M ). If M is a Krull monoid and x is a nonunit of M , then there exist a unique set p1 , . . . , pk of prime divisors of M and unique positive integers n1 , . . . , nk such that ∂(x) = pn1 1 · · · pnk k .

(7.1)

Notice (7.1) implies that n1 p1 + · · · + nk pk = 0 in Cl(M ). Let M be a Krull monoid with torsion divisor class group Cl(M ) with set P = {pi | i ∈ A} of prime divisors and divisor theory ∂ : M → NP 0 . Select for each pi ∈ P with order ni in Cl(M ) a qi ∈ M with ∂(qi ) = pni i . If Q = {qi | i ∈ A}

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and x is a nonunit of M , then there exists a positive integer n such that xn factors uniquely (up to order) as a product of elements from Q. Hence, M is an inside factorial monoid. To see that M is a Cale monoid, choose pi ∈ P and suppose mj i p ∂(x) = pm i j =i j . Let w = lcm{nj }pj |∂(x) and set dj nj = w for each pj | ∂(x). n x(qi ) w i Then ∂(x) = (pni i )di mi j =i (pj j )dj mj and so m(x) = diwmi = m ni , completing the argument. To demonstrate how this relates to nonunique factorizations, suppose that M is a Krull monoid with Cl(M ) = Z3 . Let p1 , p1 , p2 and p2 be prime divisors of M where pi and pi lie in the divisor class ¯i (such a Krull monoid exists by [14]). If ∂(x) = p1 p2 , ∂(y) = p1 p2 , ∂(z) = p1 p2 and ∂(w) = p1 p2 , then in M v = xy = zw is a nonunique factorization. But notice that v 3 = (xy)3 = (zw)3 = abcd where ∂(a) = p31 , ∂(b) = p32 , ∂(c) = (p1 )3 and ∂(d) = (p2 )3 is the unique Cale representation of v. Example 1.5. Polynomial rings Let K be any ﬁeld. In the polynomial ring D = K[X, XY 2 , XY ] one has the nonunique factorization into atoms X 2 Y 2 = X · XY 2 = XY · XY. Taking squares we meet, nevertheless, a unique Cale representation (X 2 Y 2 )2 = XX · XY 2 XY 2 = XXY 2 · XXY 2 . Similarly, in the semigroup ring R = K[X 2 , X 3 ] one has the nonunique factorization into atoms X 6 = X 2 · X 2 · X 2 = X 3 · X 3. Interestingly enough, for K = R the domain R is not Cale but for K = F2 it is Cale (see [21, Remark 3.4]). Example 1.6. Trigonometric polynomials In the domain D = R[cos x, sin x] of trigonometric polynomials, one has the following nonunique factorization into atoms √ √ cos 2x = (cos x + sin x) · (cos − sin x) = ( 2 cos x + 1) · ( 2 cos x − 1). Taking squares, however, one gets the following unique Cale representation (cos 2x)2 =

1 1 1 1 q1 q2 · q3 q4 = q1 q3 · q2 q4 2 2 2 2

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with

where

q1 (x)

=

q2 (x)

=

q3 (x)

=

q4 (x)

=

√ 2(cos x − sin x) √ − 2(cos x − sin x) √ 2(cos x + sin x) √ − 2(cos x + sin x)

+

2

+

2

+

2

+

2

1 is a unit. Actually, a Cale base is given in [23] by 2 q(x) = a cos x + b sin x + 1 +

a2 + b 2 ; a, b ∈ R, a2 + b2 ≥ 4. 4

It is of interest to note here that [23] also implies that R[cos x, sin x] is a halffactorial domain (i.e., for a given nonunit f , the length of any irreducible factorization is constant). Note moreover that the domain Q[X, Y ] with X 2 + Y 2 = 1 is a unique factorization domain, whereas Q(i)[X, Y ] with X 2 + Y 2 = 1 is not a unique factorization domain ([26, p. 16], [23]). Example 1.7. Coordinate rings (see [7, 9, 28, 30]) The quadric cone given by Y 2 = XZ in C3 is an aﬃne toric variety with a rational double point singularity in 0. The coordinate ring of this variety is given by R = C[X, Y, Z] /

∼ C[X, XY, XY 2 ]. (Y 2 − XZ) =

By Example 1.5 the ring is not factorial but allows a Cale representation. The Neil parabola, however, given by Y 2 = X 3 in C3 is an aﬃne toric variety with a cusp singularity in 0. The coordinate ring is given by R = C[X, Y ] /

∼ C[X 2 , X 3 ]. (Y 2 − X 3 ) =

As in Example 1.5, R is not Cale. (Replacing C by F2 , however, yields a Cale domain.) Example 1.8. Natural monoids By a natural monoid we mean a submonoid of N0 under addition (a numerical monoid ) or a submonoid of N under multiplication. For the numerical monoid M = N0 \{1, 2, 4, 7} one has the following nonunique factorization into atoms 23 = 6 · 3 + 1 · 5 = 1 · 3 + 4 · 5. Multiplying by 3 one obtains, nevertheless, a unique Cale representation 3 · 23 = 18 · 3 + 5 · 3 = 3 · 3 + 20 · 3 where {3} acts as a Cale base. For the Hilbert monoid (see [16]) M = 4N0 + 1 (considered multiplicatively) one has the following nonunique factorization into atoms 53361 = 9 · 49 · 121 = 21 · 33 · 77,

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which, in terms of “ideal numbers” ∈ / M , has the unique representation 53361 = 3 · 3 · 7 · 7 · 11 · 11 = 3 · 7 · 3 · 11 · 7 · 11. Taking squares, one has the unique Cale representation 92 · 492 · 1212 = 9 · 49 · 9 · 121 · 49 · 121. A Cale base of the Hilbert monoid is given for p prime in N by p if p ≡ 1 mod 4 and by p2 if p ≡ 3 mod 4. We note that the Hilbert monoid is but a simple example of a more complex structure known as a congruence monoid (see again [16]). Example 1.9. Semigroup rings and power series rings Generalizing Examples 1.6 and 1.8, consider the semigroup ring D[S] for an arbitrary domain D and an arbitrary additive monoid S. In general D[S] shows nonunique factorization into atoms. More precisely, by the Gilmer–Parker Theorem (see [12, 13]), D[S] has unique factorization into atoms if and only if both D and S possess unique factorization into atoms. Nevertheless, one may have unique Cale representation in case neither D nor S have unique factorization into atoms. More information on the Cale Property in semigroup rings can be found in Section 5.3. For generalized power series rings D[[S]] in the sense of Ribenboim ([24]), one can have nonunique factorizations even if both D and S possess unique factorization. A classical example was given by Samuel [26, p. 9] where D[[X]] (that is, S = N0 ) is not a unique factorization domain for a certain two–dimensional unique factorization domain D.

2

Extraction Analysis and Valuations

For a factorial domain, “extracting” the highest power of a prime p from elements of the domain deﬁnes a (p–adic exponential) valuation on the quotient ﬁeld. Below we show for arbitrary inside factorial monoids that extracting maximally elements of a base from other elements obeys some nice rules. Moreover, we show for root-closed domains which are inside factorial that extracting base elements yields (exponential) valuations. First, we deﬁne the concepts of extraction analysis as introduced in [17] and review its related elementary properties (see [4], [19], and [20]). If M is a monoid then the function λ : M × M → [0, ∞] given by λ(x, y) = sup

m n

| m ∈ N0 , n ∈ N, xm | y n

is called the extraction degree on M . The monoid M is called an extraction monoid if for any x, y ∈ M \M × there exist m ∈ N0 , n ∈ N such that xm | y n and λ(x, y) = m n. An integral domain D is called an extraction domain if its multiplicative monoid D• = D\{0} is an extraction monoid. A review of elementary properties related to extraction, such as kλ(xk , xl ) = lλ(x, y) as well as others, can be found in the above references.

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Lemma 2.1. Let M be an inside factorial monoid with base Q. x(q) q is the Cale representation of x ∈ M \M × then λ(q, x) = (i) If xm(x) = u q∈Q x(q) m(x)

for all q ∈ Q.

(ii) For every q ∈ Q, the map λ(q, ·) : M → Q≥0 is a monoid homomorphism. (iii) M is an extraction monoid. (iv) Each q ∈ Q is almost irreducible (i.e., z|q implies z m = uq n with m, n ∈ N, u ∈ M × ). Each q ∈ Q is almost primary (i.e., q|xy implies that q|xn or q|y n for some n ∈ N). Proof. See [4, Lemma 2]. Remark 2.2. In general elements of a base need neither be irreducible nor primary, but are always almost primary (see Section 3). The following lemma shows for a root-closed inside factorial domain that the extraction of base elements yields exponential valuations. Lemma 2.3. For an extraction domain D with quotient ﬁeld F the following statements hold. (i) For x, y, z nonunits in D• λ(z, x + y) ≥

1 λ(z, x) · λ(z, y) ≥ min{λ(z, x), λ(z, y)} λ(z, x) + λ(z, y) 2

(ii) Suppose, D is root-closed (i.e., f ∈ F and f n ∈ D for n ∈ N implies f ∈ D). Then, for x, y, z nonunits in D• , λ(z, x + y) ≥ min{λ(z, x), λ(z, y)}. (iii) Suppose, D is root-closed and inside factorial with base Q. For q ∈ Q let vq (x) = λ(q, x) for x ∈ D• , x ∈ D× vq ( xy ) = vq (x) − vq (y) for x, y ∈ D•

and vq (x) = 0 for x ∈ D× . and vq (0) = ∞

Then vq : F → Q∪{∞} is an exponential valuation on F . That is, for f, g ∈ F we have vq (f · g) = vq (f ) + vq (g), vq (f + q) ≥ min{vq (f ), vq (g)}. If Q is a tame base, then there exists e(q) ∈ N such that e(q)vq is a discrete valuation. (iv) D is solid. That is, the root-closure D = {f ∈ F | f n ∈ D for some n ∈ N} is again an integral domain.

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m Proof. (i) Let λ(z, x) = m | xn and λ(z, y) = kl with z k | y l . For n with z nk ≤ ν ≤ nk + ml it follows that z mk | xνy nk+ml−ν . Similarly, for 0 ≤ ν ≤ nk it follows that z mk | xνy nk+ml−ν . From

(x + y)nk+ml =

nk+ml ν=0

nk + ml xνy nk+ml−ν ν

it follows that z mk | (x + y)nk+ml and, hence, λ(z, x + y) ≥

mk λ(z, x) · λ(z, y) 1 = ≥ min{λ(z, x), λ(z, y)}. nk + ml λ(z, x) + λ(z, y) 2

k (ii) For m, n, k, l in (i), suppose without restriction that m n ≤ l . It follows for 0 ≤ ν ≤ n that z m(n−ν)l · z knν | (xn−ν · yν)nl and, hence, z mnl | (xn−ν · yν)nl . Since D is root-closed this implies z m | xn−ν · yν for 0 ≤ ν ≤ n. Because of n n n−ν yν we obtain that z m | (x + y)n and, hence, (x + y)n = ν x ν=0

λ(z, x + y) ≥

m = min{λ(z, x), λ(z, y)}. n

(iii) By Lemma 2.1, λ(q, x) ∈ Q≥0 and λ(q, x · y) = λ(q, x) + λ(q, y) for q ∈ Q and x, y ∈ D• \D× . Therefore, λ(q, ·) can be extended to F with νq (f ·g) = νq (f )+νq (g) for f, g ∈ F . Furthermore, for f = ff12 , g = gg12 in F and q ∈ Q we have that vq (f + g) = vq (f1 g2 + g1 f2 ) − vq (f2 g2 ) and, by step (ii), vq (f + g) ≥ min{vq (f1 g2 ) − vq (f2 g2 ), vq (g1 f2 ) − vq (f2 g2 )} ≥ min{vq (f1 ) − vq (f2 ), vq (g1 ) − vq (g2 )} = min{vq (f ), vq (g)}. (iv) Since D is multiplicatively closed, it suﬃces to show that D is additively closed. Let f = ff12 , g = gg12 in D and, without loss, f n , g n ∈ D for some n ∈ N. For m ∈ N0 arbitrary let m = kn + i, 0 ≤ i ≤ n − 1. It follows that f2n−1 f m = (f2n−1 f i ) · (f n )k ∈ D for all m and, similarly, g2n−1 g m ∈ D for all m. For x = f2n−1 g2n−1 ∈ D and h = f + g ∈ F it follows for arbitrary l ∈ N that x · hl = l l n−1 f ν) · (g2n−1 g l−ν ) ∈ D. From xl−1 ∈ D and xhl ∈ D it follows from ν (f2 ν=0

(xh)l = xl−1 (xhl ) that xl−1 | (xh)l (in D) and, hence, λ(x, xh) ≥ l−1 l . Since l ∈ N is arbitrary, this implies λ(x, xh) ≥ 1 and since D is an extraction domain, there exists p ∈ N such that xp | (xh)p (in D). Thus, we arrive at (xh)p = xp y with y ∈ D and, hence, (f + g)p = hp = y ∈ D.

From the two lemmas we obtain in particular that every inside factorial integral domain is necessarily solid. In general, an integral domain need not be solid. For example, the integral domain D = R + X 2 R[X] is not solid because 1, X ∈ D but 1 + X ∈ D. Thus, this domain D cannot be inside factorial and, by Lemma 2.3 (iv), not even be an extraction domain (see also Example 1.5 in Section 2).

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Special Elements and the Cale Property in Krull Monoids

In Lemma 2.1 (iv), we deﬁned the notion of almost primary and almost irreducible. We expand this list with the following deﬁnition. Definition 3.1. A nonunit x of a monoid M is a strong atom if for every n ∈ N the only nonunit divisors of xn are associates of powers of x. Obviously, a prime element is necessarily a strong atom and in turn must be an atom. None of these implications can be reversed. If M is a Krull monoid with torsion divisor class group, then the strong atoms of M are merely the primary atoms of M . Primary atoms need not be prime as, for example, the element (3, 0, 2) in the ﬁrst case of Example 1.3. In general, a primary atom may not be a strong atom. For the numerical monoid N0 \{1}, the element 2 is clearly primary and an atom, but is not a strong atom since 23 = 32 . Thus, the notion of a strong atom lies strictly between that of an atom and a prime element. In a factorial monoid there is precisely one representation of a power of an atom, namely the trivial representation by itself. In an inside factorial monoid even the most simple Cale representation may be nontrivial. Definition 3.2. An element x ∈ M \{1} in a reduced Cale monoid M with base Q has a simple representation if xm = q n with q ∈ Q. If m, n ∈ N are minimally chosen, then x is said to be simple of type (m, n). Lemma 3.3. Let M be a reduced inside factorial monoid with base Q. 1. For x ∈ M \{1} the following properties are equivalent: (a) x is almost primary, (b) x is simple, (c) y|xn for some n ∈ N, y = 1 implies that y k = xl with k, l ≥ 1. 2. Strong atoms are almost primary and simple of type (k, 1). 3. An almost primary element has at most one strong atom as a divisor. 4. If x is a strong atom of M , then there is a unique positive integer w(x) such that xw(x) ∈ Q. Proof. 1. We show both (a) and (c) are equivalent to (b). Let xm = q∈q q x(q) be the Cale representation of x. (a) ⇔ (b) If x is almost primary, then x|q0n for n ∈ N. Hence, q0 ∈ Q implies x(q) > 0 if and only if q = q0 by uniqueness of the Cale representation. Conversely, xm = q0n and x|yz imply y m z m = q0n w, w ∈ M . Therefore, q0 must appear in the Cale representations of y or of z. Without loss, q0r |y s and, hence, q0nr |y ns . That is, x|y ns . (b) ⇔ (c) If xm = q0t and y|xn then q0tn = xmn = y m z with z ∈ M . Therefore, the Cale representation contains only q0 (that is y r = q0s ). This implies y rt = q0st = xms . Conversely, by the Cale representation of x, q0 |xm for some q0 ∈ Q and, by (c), q0k = xl . Uniqueness implies that x(q) = 0 for q = q0 .

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2. Obviously, a strong atom possesses property 1(c) and, hence, it is almost primary. Furthermore, it is simple and by 1, xm = q n . Since x is a strong atom we must have that q = xk . 3. Let x ∈ M \{1} be almost primary with strong atoms y, z as divisors. By 1 we must have y k = xl and z r = xs with k, l, r, s ∈ N. Therefore, y ks = xls = z rl and y, z being strong atoms implies that z = y m , y = z n and hence, z = z mn . Thus, m = n = 1 and y = z. 4. Since the only nonunit divisors of xn are powers of x, the Cale property implies that Q contains at least one element of the form xn . If xn and xm ∈ Q for n = m, then xnm = (xn )m = (xm )n yields a contradiction. Remark 3.4. 1. By Lemma 3.3 part 1, primary elements in an inside factorial domain are automatically almost irreducible (see also Lemma 2.1 (iv)). Furthermore, x is almost primary if and only if some power of x is almost primary, which holds if and only if all powers of x are almost primary. 2. By Lemma 3.3 part 1, simple elements are almost primary. In particular, the elements of the Cale base Q which are simple of type (1, 1) are all almost primary (compare with Lemma 2.1 (iv).) By property 3, the elements of Q are “weak atoms” in the sense that they cannot be decomposed into diﬀerent strong atoms. 3. For the notion of a strong atom see Krause [19] where these elements are called “strongly irreducible” and Stanley [27] where these elements are called “completely fundamental”. We characterize which subsets Q ⊆ M of a Krull monoid with torsion divisor class group can serve as the Cale base for M . To do this, we need to recall some deﬁnitions related to the primary elements of M . If q1 and q2 are primary elements of M , then deﬁne a relation q1 ∼ q2 if and only if q1 | q2n for some positive integer n. Obviously, ∼ is an equivalence on the set of all primary elements ℘ of M (compare to [15, Corollary 1.3]). Proposition 3.5. Let M be a Krull monoid with torsion divisor class group. Then Q ⊆ M is a base for the Cale monoid M if and only if Q consists of exactly one element chosen from each equivalence class of ℘/ ∼. Hence, Q must consist of primary elements. Proof. (⇒) We ﬁrst argue that the elements of Q are primary. Our previous observaw(q ) tion combined with Lemma 3.3 part 4 imply that Q ⊇ {qi i }i∈A and Q contains no w(qi ) other powers of elements of the form qi . Suppose Q = {qi }i∈A . Then (by relabelk i ing the prime divisors if necessary) Q contains an element x = i=1 pm where each i mi > 0 and k ≥ 2. Let m = lcm{w(q1 ), w(q2 ), . . . w(qk )} and n = lcm{n1 , . . . , nk } where m = di w(qi ) and n = ci ni for each i. Then xmn = (

k

i=1

i mn pm = i )

k i=1

((pni i )w(qi ) )di ci mi =

k i=1

w(qi ) di ci mi

(qi

)

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and using Lemma 3.3, we obtain a contradiction. Thus Q = {qi i }i∈A . Notice w(q ) w(q ) that if i = j then qi i ∼ qj j and that every equivalence class of ℘/ ∼ is represented in Q. (⇐) Suppose Q consists of exactly one element from each equivalence w(q ) class of ℘/ ∼. Then Q = {qi i }i∈A for some sequence {w(q1 ), w(q2 ), . . .} of posik i tive integers. Let x = i=1 pm (with relabeling of the prime divisors if necessary) i be an arbitrary element in M . Using m and n as deﬁned in the ﬁrst part of the k w(q ) proof, xmn = i=1 (qi i )di ci mi and clearly this representation is unique up to order. Using the property that an element of Q must be almost primary, we are able to characterize Krull monoids which are Cale monoids. We will require the following proposition. Proposition 3.6. Let M be a Krull monoid with divisor class group Cl(M ). Then x ∈ M is almost primary in M if and only if x is primary in M . Proof. (⇐) Follows by deﬁnition. (⇒) Let P represent the set of prime divisors of M . For p ∈ P, let | [p] | denote the order of [p] in Cl(M ). By [14, Satz 10A ii)] the primary elements of M are of the form x = pk with 1. | [p] |< ∞ and 2. | [p] | divides k. For an abelian group G and subset E ⊆ G, set E = {g1 + · · · + gs | gi ∈ E}. The proof hinges on the following result [2, Lemma 3.3]: if M is a Krull monoid with divisor class group Cl(M ) and associated set P of prime divisors, then for any p1 ∈ P, Cl(M ) = {[p] | p ∈ P and p = p1 }. Suppose x = pn1 1 pn2 2 · · · pnk k where the pi ’s are distinct prime divisors and each ni > 0. If | [p1 ] |= ∞, then k ≥ 2. We argue that such an x is not almost primary. By [2, Lemma 3.3], choose s1 , . . . , st ∈ P such that si = p1 for each i and [p1 ] = [s1 ] + · · · + [st ]. If y = sn1 1 · · · snt 1 pn2 2 · · · pnk k then y ∈ M and p1 is not a prime divisor of y. Using the same argument on the prime divisor p2 yields prime divisors r1 , . . . , rw (distinct n2 n3 p3 · · · pnk k then from p2 ) such that [p2 ] = [r1 ] + · · · + [rw ]. If z = pn1 1 r1n2 · · · rw again z ∈ M and p2 is not a prime divisor of z. Since x, y and z are in M , this n2 n3 p3 · · · pnk k ∈ M and xv = yz. Thus x | yz, implies that v = sn1 1 · · · snt 1 r1n2 · · · rw a but by construction, x divides no element of the form y b or z c . Hence, if x is divisible by a prime divisor whose image in the divisor class group is not torsion, mk 1 m2 where each then x cannot be almost primary. So suppose that x = pm 1 p2 · · · pk pi has torsion image in Cl(M ) and k ≥ 2. Using the notation of the Introduction, if w = lcm{n1 , . . . , nk } and ni di = w for each i, then x | q1m1 d1 · · · qkmk dk , but clearly xa divides no element of form (q1m1 d1 )b or (q2m2 d2 · · · qkmk dk )c . This completes the proof. Proposition 3.6 and the remarks following Deﬁnition 3.1 can be combined to show the following.

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Corollary 3.7. Let M be a Krull monoid with torsion divisor class group and suppose that x is a nonunit of M . The following statements are equivalent: 1. x is a strong atom. 2. x is an atom and primary. 3. x is an atom and almost primary. Example 3.8. While Proposition 3.6 shows that conditions 2 and 3 are equivalent in any Krull monoid, Corollary 3.7 is not true for all Krull monoids. Let M be a Krull monoid with Cl(M ) = Z such that the divisor class 1 contains the prime divisors p1 and p2 and the divisor class −1 contain prime divisors p3 and p4 . An example of such a Krull monoid is the set of solutions in nonnegative integers of the Diophantine equation x1 + x2 = x3 + x4 (see the latter part of Example 1.3). If x = p1 p3 in M , then clearly x is a strong atom since Cl(M ) is torsion-free. Setting y = p1 p4 and z = p2 p3 yields elements such that x | yz but x divides no power of y or z. Hence x is neither primary or almost primary. Theorem 3.9. For a Krull monoid M the following statements are equivalent. (1) M is a Cale monoid. (2) M is an inside factorial monoid. (3) Cl(M ) is a torsion group. Proof. (1) ⇒ (2) follows from Deﬁnition 1.1. For (2) ⇒ (3), suppose the divisor class group of M is not torsion. Suppose p1 is a prime divisor of M such that | [p1 ] |= ∞. There exists an x ∈ M with x = pn1 1 · · · pnk k where n1 > 0 and k ≥ 2. Then, for some m(x) ∈ N, xm(x) can be written uniquely as a product of elements in Q. By Proposition 3.6, Q contains no elements divisible by p1 , a contradiction. Hence Cl(M ) must be torsion. (3) ⇒ (1) follows from Example 1.4. Example 3.10. Lemma 2.1 part (iii) established that a Cale monoid is an extraction monoid. The converse of this statement is not true. It is easy to argue n(x,pi ) that any Krull monoid M is an extractive monoid. In fact, if x = pi ∈P pi n(y,pi ) and y = pi ∈P pi where P is the set of prime divisors of M , then λ(x, y) = n(y,pi ) min{ n(x,p | for all pi ∈ P with n(x, pi ) > 0}. By Theorem 3.9, any Krull monoid i) with class group not a torsion group is an extraction monoid which is not a Cale monoid.

4

Characterizations of Cale Monoids and Cale Domains

In Section 3, we characterized Cale monoids M which are Krull monoids. In this section, we extend this characterization and formulate it also for integral domains. We open with a characterization based on extraction within a monoid M . Some deﬁnitions are required. We note that a monoid M is said to be of ﬁnite type if increasing chains of rad(x) = {y ∈ M | x | y n for some n ∈ N} ideals are stationary.

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If y ∈ rad(x) then we say that x is a component of y. The nonunit element x is extractive if λ(x, •) is additive. Finally, x is discrete if n(x)λ(x, •) ∈ N0 for some n(x) ∈ N. Theorem 4.1. [19, Theorem 2] A monoid M is a Cale monoid if and only if (i) M is an extraction monoid of ﬁnite type (ii) each nonunit of M has an extractive and discrete component. If M is a Cale monoid with base Q, then for each q ∈ Q deﬁne fq : F → N0 by fq (x) = e(q)x(q) m(x) (see Deﬁnition 1.1). Definition 4.2. Let M be a monoid. The root closure M of M in G is deﬁned as M = {w ∈ G | wk ∈ M for some k ≥ 1}. Further, if M is a Cale monoid with base Q, then set = { x ∈ G | fq ( x ) ≥ 0 for all q ∈ Q}. M y y . Lemma 4.3. If M is a Cale monoid with base Q, then M = M ⊆ M . Let x ∈ M (i.e., fq (x)−fq (y) ≥ 0 for all q ∈ Q) Proof. We ﬁrst argue that M y and hence n(q)(λ(q, x) − λ(q, y)) ≥ 0 for all q ∈ Q. Thus, λ(q, x) ≥ λ(q, y) for all x(q) y(q) ≥ m(y) for all q ∈ Q (i.e., m(y)x(q) ≥ m(x)y(q) q ∈ Q and by Lemma 2.1 (i), m(x) for all q ∈ Q). Thus xm(x)m(y) = q x(q)m(y) = q m(x)y(q) q x(q)m(y)−m(x)y(q) = y m(x)m(y) · z q∈Q

q∈Q

q∈Q

⊆ M . For the where z ∈ M . Hence (x/y)m(x)m(y) ∈ M (i.e., (x/y) ∈ M ) and M x k reverse containment, let w = y ∈ G with w ∈ M for some k ≥ 1. By noting k

k

for every nonunit x ∈ M that (xm(x) )km(x ) = ((xk )m(x ) )m(x) , we obtain that xk (q) k = kx(q) m(x) for every q ∈ Q. This implies that kfq (w) = fq (w ) ≥ 0 for all m(xk ) . q ∈ Q. Thus fq (w) ≥ 0 for all q ∈ Q and x ∈ M y

Theorem 4.4. For a monoid M with quotient group G the following statements are equivalent. 1) M is a Cale monoid with base Q. 2) The root closure M of M is a Krull monoid with torsion divisor class group. is contained in G. If w ∈ G, Proof. 1) ⇒ 2) Obviously the quotient group of M x , w is in the quotient group of then w = y with x and y in M . Since M ⊆ M ) = G. We have that M . By Lemma 4.3, M = M and hence quot(M ) = quot(M = {w ∈ G | fq (w) ≥ 0 for all q}. Finally, {q ∈ Q | fq (w) = 0} is ﬁnite for w ∈ G. M To see this, notice that by Lemma 2.1 (i) λ(q, x) =

x(q) m(x)

= 0 only for ﬁnitely many

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q, and hence n(q)(λ(q, x) − λ(q, y)) = 0 only for ﬁnitely many q ∈ Q. Thus, the set F = {fq | q ∈ Q} forms a deﬁning family for M and M is a Krull monoid. To complete the argument, let F = {fq ∈ F | fq is an essential state of M }, and for each fq ∈ F , let fˆq be the normalization of fq . If I represents the essential (I) normalized states of M and ϕ : M → N0 deﬁned by ϕfˆq (x) = fˆq (x), then Cl(M ) ∼ = (I) ˆ ˆ N0 /ϕ(M ). Let q be an element of Q such that fq ∈ I. If q ∈ Q, q = q and fq ∈ I, (I) then fˆq (q) = 0 and fˆq (q) = z > 0. Let eq be the element of N0 which is 1 in fˆq coordinate and zero elsewhere. Then z · eq = ϕ(q) which means that in the divisor class group, z[eq ] = 0. Thus [eq ] is a torsion element of Cl(M ). Since these elements generate the divisor class group, Cl(M ) is a torsion group. 2) ⇒ 1) Let M be a monoid whose root closure M is a Krull monoid with torsion divisor class group. By our remarks in the Introduction, M is a Cale monoid. Let Q be the Cale base for M given in the Introduction. By [11, Lemma 5.4], {1} = M × = × x(q ) M ∩ M . Let x ∈ M ⊆ M . Then xm(x) = u · qi ∈Q qi i , where u is a unit in M x(q )

and each qi i ∈ M . For each i ∈ A, let ni be a positive integer such that qini ∈ M and n0 be a positive integer such that 1 = un0 ∈ M . If n = lcm({n i }i∈A , n0 ), then set n = ni ti (for i ∈ A) and n = n0 t0 . Thus, xm(x)n = (un0 )t0 qi ∈Q (qini )x(qi )ti = ni x(qi )ti and clearly this representation is unique with respect to Q = qi ∈Q (qi ) ni {qi | i ∈ A}. Thus Q forms a base for M and hence M is inside factorial. Setting f (qini ) = f (qi )ni also yields that Q is tame. Hence, M is a Cale monoid. Corollary 4.5. Let M be a Cale monoid. If M is a monoid such that M ⊆ M ⊆ M , then M is Cale monoid. Proof. Since M is itself root closed ([10, p. 680]), M = M . By Theorem 4.4, M is a Krull monoid with torsion divisor class group. The converse of Theorem 4.4 completes the argument. Example 4.6. We can use Theorem 4.4 to construct an example of a Cale monoid H with base Q where Q does not consist entirely of primary elements. Let M be a Krull monoid with Cl(M ) = Z3 and countably many prime divisors in each nonzero divisor class (for instance, one could choose a reduced algebraic ring of integers with class number 3 modulo the monoid generated by its prime elements). By Theorem 4.4, M is a Cale monoid. Let a, b and c be distinct prime divisors of M taken from the divisor class ¯ 1. Consider the following atoms in M: x = a3 , y = b3 , z = c3 , 2 2 v = a b, u = ab , e = ac2 , f = b2 c. Let I represent the atoms of M . Let H be the monoid generated by I − {u}. Clearly H is contained in M . Notice that if w is in M and u does not divide w, then w is also in H. Further, u3 is in H (u3 = a3 (b3 )2 = xy 2 ). Hence, if w is in M , then either w is in H or w3 is in H. Thus M contains the root closure of H. Since M itself is root closed, the root closure of H equals M . Further, M and H have isomorphic quotient groups and hence H is a Cale monoid. Let {pi }i∈A be the complete set of prime divisors of M . By the proof of 2) implies 1) in Theorem 4.4, the set {(pi )3 }i∈A is a Cale base for H. We argue that x = a3 is almost primary, but not primary in H. To see this, we have that x divides

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v 2 z (indeed, (a2 b)2 c3 = (a3 )(ac2 )(b2 c) and v 2 z = xef ), but x does not divide v 2 (v 2 = a4 b2 = a3 (ab2 ) and u = ab2 is not in H). Clearly x does not divide any power of z. Thus x is not primary. By Lemma 2.1 part (iv), x is almost primary. Theorem 4.4 allows for the construction of some interesting integral domains which satisfy the Cale property. Example 4.7. Here is perhaps the simplest construction of a half-factorial domain not involving an algebraic number ring. Let K be any ﬁeld and A ⊆ K. By [3, Theorem 5.3], R = A + XK[X] is half-factorial if and only if A is a subﬁeld of K. Now, if K = F4 and A = F2 , then R = F2 + XF4 [X]. Notice that the quotient ﬁeld of R is isomorphic to F4 (X) and that the root closure of R is R = F4 [X]. Since R is a factorial domain, R is a Cale domain by Theorem 4.4. This example easily generalizes to the case where K = Fpn and A is any subﬁeld of K. Example 4.8. Let F be any ﬁnite ﬁeld and set R = F [[X 2 , X 3 ]] and A = F [X 2 , X 3 ]. The integral closure and root closure of both R and A are unique factorization domains (F [[X]] and F [X] respectively). By Theorem 4.4, both R and A are Cale domains. √ Example 4.9. Let R = Z[ −3]. Then R is a nonintegrally closed half-factorial √ 1+ −3 domain whose integral closure and root closure (A = Z[ 2 ]) is a unique factorization domain [32, p. 285]. By Theorem 4.4, √ R is a Cale domain. But, not in √ R that all algebraic √ orders are Cale monoids. Let R = Z[ 17]. One can argue λ(2, 1 + √17) = 1 (see [19, pp. 149–150]) but there is no n ∈ N with 2n | (1 + 17)n . Thus Z[ 17] is not an extraction monoid and is hence not a Cale monoid. From Theorem 4.4, the following characterization of Krull monoids is obtained. Corollary 4.10. A monoid M is a Krull monoid with torsion divisor class group if and only if (i) M is a Cale monoid. (ii) M is root closed. Proof. (⇒) If M is Krull with torsion divisor class group, then (i) holds by Theorem 3.9. Since all Krull monoids are root closed, (ii) holds. (⇐) Suppose M is a monoid with properties (i) and (ii). By Theorem 4.4, M is a Krull monoid with torsion divisor class group. Example 4.11. An example for Corollary 4.10 is given by discrete valuation domains, which can be characterized as those domains whose multiplicative monoids satisfy (i) and (ii) with a base Q consisting of exactly one element. Condition (i) cannot be replaced by requiring M to be inside factorial only. Let D be a valuation domain with value group Q. Since D is integrally closed, it must also be root closed. Hence, D• is inside factorial with (ii), but D• is not a Krull monoid (because D is not atomic, see [8, Proposition 2.3]). We close this section by examining the situation for integral domains. The basis of a Cale domain is essentially determined by its minimal prime ideal structure.

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Theorem 4.12. [4, Theorem 2] For a Cale domain D a subset Q ⊂ D is a Cale basis if and only if q → rad(q) is a bijection of Q onto the minimal prime ideals of D• . In particular, a Cale basis is essentially uniquely determined. ˜ represent the integral closure of D. The If D is an integral domain, then let D following characterizes when D is Cale. Theorem 4.13. [4, Theorem 7] An integral domain D is a Cale domain if and only if ˜ is a Krull domain, (i) D ˜ is a torsion group, and (ii) Cl(D) ˜ is a root extension (i.e., for all x ∈ D ˜ there is an n ∈ N such that (iii) D ⊂ D n x ∈ D). Theorem 4.13 also characterizes inside factorial integral domains if “rational generalized” is inserted prior to the Krull statement in (i). Under the Noetherian hypothesis, the result above leads to the following important observation. Corollary 4.14. Let D be a noetherian domain. Then D is a Cale domain if and only if D is inside factorial.

5

Cale Representation in Particular Integral Domains and Monoids

5.1

Cale Representation in Algebraic Orders

In this section, we review some implications of the Cale property when considered for algebraic orders. By the results of the previous section, if K is an algebraic number ﬁeld and D = OK the principal algebraic order in K, then D is a Cale domain. Proposition 3.5 implies that the primary atoms (up to equivalence) of D form a Cale basis. Moreover, any Cale basis of D consists of primary elements. Theorem 5.1. [18, Theorem] Let D be the principal order in an algebraic number ﬁeld K. Then Cl(D) ∼ = Zpk where p is a prime integer if and only if there exists a k Cale base Q of D such that for any atom x of D, xp factors into at most pk factors of Q. Theorem 5.1 has several nice interpretations for principal orders of small class number. Corollary 5.2. Let D be the principal algebraic order in an algebraic number ﬁeld K with class number h. 1. The following statements are equivalent. (a) h = 1. (b) Every atom of D is primary. (c) D is factorial.

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2. The following statements are equivalent. (a) h = 2. (b) The square of every atom of D is the product of two primary atoms of D. (c) D is a half-factorial domain. ˜ we have equivaFor arbitrary algebraic orders R ⊂ OK with integral closure R, lence of the Cale and inside factorial properties. Theorem 5.3. [4, Proposition 9] Let R ⊂ OK be an order in an algebraic number ﬁeld K. The following statements are equivalent. (a) R is inside factorial. ˜ is a root extension. (b) R ⊂ R ˜ → spec(R) is bijective. (c) The map spec(R) (d) R is a Cale domain. (e) R is an extraction domain. (f) The elasticity of R is ﬁnite. Theorem 5.3 has been partly extended in a recent work of Martine Picavet– L’Hermitte (for the concepts of an almost B´ezout domain and an almost factorial domain see [1] and [31], respectively). Theorem 5.4. [22, Theorem 2.1] Let R be a 1–dimensional noetherian domain with torsion class group. The following conditions are equivalent. (a) R is inside factorial. ˜ is a root extension. (b) R ⊂ R (c) R is an almost B´ezout domain. (d) R is an almost factorial domain. ˜ is a ﬁnitely generated R–module and R is residually ﬁnite, then If in addition R (a)–(d) above are equivalent to ˜ → spec(R) is bijective. (e) The map spec(R) √ Example 5.5. Let R = Z[f ω] be a quadratic order of the ﬁeld Q( d). As usual, √ √ ω = 12 (1 + d) if d ≡ 1 (mod 4) and ω = d if d ≡ 2, 3 (mod 4). Theorem 5.3 implies that R is a Cale domain if and only if ( D p ) = 1 for all primes p|f . For √ example, if d = −1 and f = 5, then R = Z[5 −1] is not a Cale domain because 5 is a product of Gaussian primes.

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159

Cale Representation in Diophantine Monoids

We focus in this section on the class of monoids introduced in Example 1.3. Let be a matrix with A ∈ Zm×n be a matrix with integer coeﬃcients, B ∈ Nm×n 0 nonnegative integer coeﬃcients, and c ∈ Rm×1 be a column vector with nonnegative + real entries. By a Diophantine monoid, we mean any additive monoid MA = {x ∈ Nn0 | Ax = 0} deﬁned by equations or MB,c = {x ∈ Nn0 | Bx c} ∪ {0} deﬁned by inequalities. From [21, Theorem 4.1 & Theorem 4.4] we obtain the following propositions. Proposition 5.6. Let A ∈ Zm×n be a matrix for which MA = 0. The following statements are equivalent. 1. MA is a Cale monoid. 2. Cl(MA ) is a torsion group. 3. MA is isomorphic to a monoid MA where each row of A has just one negative entry. be a matrix and c ∈ Rm×1 be a column vector Proposition 5.7. Let B ∈ Nm×n + 0 such that MB,c = 0. The following statements are equivalent. 1. MB,c is a Cale monoid. 2. MB,c = Nn0 . 3. If ci represents the ith entry of c, then ci > 0 implies that the ith row of B is positive. Example 5.8. Much of Proposition 5.6 can be demonstrated in the case where MA is a Diophantine monoid deﬁned by one equation. In other words, A = [ a 1 , a2 , · · · a n ] and hence MA = {x ∈ Nn0 | a1 x1 + · · · + an−1 xn−1 + an xn = 0}. In the case where exactly one of the ai ’s is negative, then MA is a Cale monoid with base Q = {q1 , . . . , qn−1 } which assuming without loss that an < 0 and ai ≥ 0 for 1 ≤ i ≤ n − 1, is given by qi =

an ai ei + en . gcd{ai , an } gcd{ai , an }

Notice that Q consists of all primary atoms of MA . On the other hand, if A consists of at least two positive and at least two negative elements, then Cl(MA ) ∼ = Z [5, Theorem 2.3] and Proposition 5.6 implies that MA is not a Cale monoid.

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Cale Representation in Semigroup Rings

As with the beautiful Theorem of Gilmer and Parker [12] concerning unique factorization in semigroup rings, the Cale property has a nice characterization when S is reduced and torsion-free. An integral domain D is called solid if the root-closure of its multiplicative monoid D• is additively closed or, equivalently, if D = D• ∪ {0} is a subdomain of the quotient ﬁeld of D (see Lemma 2.3 (iv)). Theorem 5.9. [21, Theorem 3.2] If D is an integral domain and S an additive reduced torsion-free monoid, then D[S] is a Cale domain if and only if 1. D is a Cale domain, 2. S is a Cale monoid, and 3. D[S] is solid. Corollary 5.10. [21, Corollary 3.3] Let D and S be as above with D integrally closed and S root-closed. Then D[S] is a Cale domain if and only if D is a Cale domain and S is a Cale monoid. In the case where D is factorial and S is a Diophantine monoid, then we can say the following. Theorem 5.11. [21, Theorem 4.1] Suppose that D is a factorial domain. 1. For a Diophantine monoid S given by equations, D[S] is a Cale domain if and only if S isomorphic to MA where each row of A has just one positive entry. 2. For a Diophantine monoid S given by inequalities where D[S] is solid, D[S] is a Cale domain if and only if ci > 0 implies that the ith row of B is positive. Example 5.12. Caution should be taken when considering Theorem 5.9. The hypothesis that D[S] be solid cannot be omitted. For example, let S = N0 \{1}. S is a numerical monoid and hence Cale. If D = R then D[S] is not solid, and therefore not a Cale domain. On the other hand, if D = F2 , then D[S] is solid, and hence a Cale domain (compare to Example 1.5).

6

Toric Ideals and Cale Varieties

In Section 1, Example 1.7, we mentioned two aﬃne toric varieties, the quadric cone and the Neil parabola. Whereas the coordinate ring of the former is a Cale domain, this is not true for the latter. Nevertheless, in both cases the coordinate ring is generated by a Cale monoid S and both varieties are determined as the zero sets of the respective toric ideals belonging to S. Those toric varieties we call Cale varieties (see Deﬁnition 6.1 below). In the following we shall investigate this new type of toric variety and show, in particular, that they can be described by special polynomial equations given by a Cale base of S. Let D be an arbitrary integral domain and let S be any additive monoid that is ﬁnitely generated by atoms g1 , . . . , gl . The R´edei map Φ : Nl0 → S deﬁned by Φ(a) =

l i=1

ai g i

(7.2)

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is a surjective homomorphism of additive monoids (this map appears in the proof of R´edei’s theorem that every ﬁnitely generated monoid is ﬁnitely presented [25]). The mapping Φ induces a surjective homomorphism of rings ra X a ) = ra X Φ(a) (7.3) Φ∗ : D[Nl0 ] → D[S] by Φ∗ ( where X a for a = (a1 , . . . , al ) denotes the monomial X1a1 X2a2 · · · Xlal . Therefore, D[S] ∼ = D[Nl0 ]/IS with IS = ker Φ∗ . The ideal IS is called the toric ideal of S for g1 , . . . , gl . Definition 6.1. The zero set of the toric ideal IS , V (IS ) = {x ∈ Dl | f (x) = 0 for all f ∈ IS }, is called the aﬃne toric variety deﬁned by S for g1 , . . . , gl . D[S] is called the coordinate ring with respect to S. For a Cale monoid S the aﬃne toric variety V (IS ) is called a Cale variety. Remark 6.2. 1. Usually, the toric ideal is considered for D a ﬁeld and S ⊂ Zd+ d or S ⊂ Z (see [28, Section 5.1], [29, Chapter 4]). The toric ideal is also called the presentation ideal [30, Chapter 7] and the kernel ideal [13, Section 7]. The latter is considered in a general framework and it is shown that IS is generated by the binomials X a − X b with a, b ∈ Nl0 , Φ(a) = Φ(b) ([13, Theorem 7.2]). Obviously, IS is already generated under the constraint on a, b ∈ Nl0 that min{ai , bi } = 0 for all i. If S is cancellative and torsion-free, then D[S] is an integral domain and the toric ideal is a prime ideal ([13, Theorem 8.1]). 2. The above deﬁnition of an aﬃne toric variety follows [29, Chapter 4] where, however, D is assumed to be a ﬁeld and S is assumed to be a monoid in Zd . A toric variety in the sense of [29, Chapter 4] (or even more in the sense of our Deﬁnition 6.1) need not be normal (see Theorem 6.8 below). When deﬁning an aﬃne toric variety in this generality one has to be careful. All notions in Deﬁnition 6.1 refer to a ﬁxed S which, in general, can neither be recognized from V (IS ) nor from D[S]. For what follows, let S be an additive Cale monoid which is torsion-free with S × = {0}. For S ﬁnitely generated we may choose a Cale base consisting of atoms. Let g1 , . . . , gl be a ﬁxed generating set of atoms and let Q = {g1 , . . . , gk } for k ≤ l be a Cale base of S. The Cale representation then yields mj g j =

k

cij gj for all k + 1 ≤ j ≤ l

(7.4)

i=1

where mj ∈ N is minimally chosen and the cij ∈ N0 are uniquely determined. Definition 6.3. Let αj = mj ej ∈ Nl0 , βj =

k i=1

cij ei ∈ Nl0 for k + 1 ≤ j ≤ l

(7.5)

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where eh is the hth unit vector in Nl0 . A binomial ±(X αj − X βj ) for k + 1 ≤ j ≤ l is called a base binomial for S in D[Nl0 ]. Let IQ ⊆ D[Nl0 ] be the ideal generated by all base binomials for S with Cale base Q = {g1 , . . . , gk }. Obviously, Φ(αj ) = mj gj and Φ(βj ) =

k

cij gi by deﬁnition of Φ. The Cale

i=1

representation in S is, therefore, equivalent to Φ(αj ) = Φ(βj ) for all k + 1 ≤ j ≤ l. It follows that Φ∗ (X αj −X βj ) = X Φ(αj ) −X Φ(βj ) = 0 and, hence, all base binomials for S belong to the toric ideal IS . We shall show that the base binomials “almost” generate IS in the sense that IQ and IS coincide if and only if IQ is a prime ideal. Although the base binomials may not generate the toric ideal, we will nevertheless show that the Cale variety of S is given by the common zero set of all base binomials for S. For this we need the following binomial lemma which may be looked at as a generalization of the “basic formula” in [9, equation(2.1)]. Lemma 6.4 (Binomial Lemma). Let R be a commutative ring and ui , vi ∈ R, ki ∈ N0 for 1 ≤ i ≤ n. Then there exist wi ∈ R, 1 ≤ i ≤ n such that n

uki i i=1

−

n i=1

viki

=

n

(ui − vi )wi .

(7.6)

i=1

Proof. By induction over n. For n = 1, the assertion uk11 − v1k1 = (u1 − v1 )w1 k1 is true for w1 = uk11 −j v1j−1 ∈ R. Suppose, the assertion is true for n and set U=

n i=1

uki i ,

V =

j=1 n i=1

viki . For u, v ∈ R and k ∈ N0 we have that U uk − V v k = U (uk − v k ) + v k (U − V ).

By assumption U − V =

n

(ui − vi )wi and, from n = 1, uk − v k = (u − v)w with

i=1

w ∈ R. Therefore

U uk − V v k =

n

(ui − vi )wi v k + (u − v)wU.

i=1

This proves the lemma. Now we are ready to prove the main result of this section, which states in particular that a Cale variety can be obtained as the zero set of base binomials. Theorem 6.5. Let D be an integral domain and S = g1 , . . . , gl be a ﬁnitely generated Cale monoid which is reduced. Suppose further that Q is a ﬁxed Cale Base of S. (i) There exists exactly one prime ideal P such that IQ ⊆ P ⊆ IS , namely P = IS . In particular, IQ = IS if and only if IQ is a prime ideal. (ii) The Cale variety for S equals the joint zero set of all base binomials for S.

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Proof. (1) In a ﬁrst step, we show for a binomial X a − X b ∈ IS that we have ma +

l

mj (aj βj + bj αj ) = mb +

j=k+1

l

mj (aj αj + bj βj ),

(7.7)

j=k+1

a b where m = lcm{mk+1 , . . . , ml } and mj = mm−1 j . Since X − X ∈ IS implies l l Φ(a) = Φ(b), by deﬁnition of Φ, it follows that aj g j = bj gj . Using (7.4) we j=1

j=1

obtain m

k

l

aj g j +

j=1

aj mj

k

cij gi = m

i=1

j=k+1

k

bj gj +

j=1

l

bj mj

j=k+1

k

cij gi .

i=1

Since Q = {g1 , . . . , gk } is a Cale base, this implies mai +

l

mj aj cij

l

= mbi +

j=k+1

mj bj cij for all 1 ≤ i ≤ k.

(7.8)

j=k+1

This shows that (7.7) holds for components with indices 1 ≤ i ≤ k. For a component with k + 1 ≤ i ≤ l equation (7.7) means that mai + mi (bi mi ) = mbi + mi (ai mi ), which holds trivially. (2) Next we show that for X a − X b ∈ IS it follows that (X ma − X mb )B (X ma − X mb )A

= =

X mb (A − B) ∈ IQ X ma (A − B) ∈ IQ

(7.9)

where A = X c and B = X d with c=

l

mj (aj αj

+ bj βj ) and d =

j=k+1

l

mj (aj βj + bj αj ).

j=k+1

From equation (7.7) we get X ma B = X mb A. Therefore, it suﬃces to show that A− B ∈ IQ . But this follows from the Binomial Lemma applied to the commutative ring R = D[Nl0 ]. Let n = 2(l − k) and set for 1 ≤ i ≤ l − k ui = X αk+i , vi = X βk+i , ul−k+i = vi , and vl−k+i = ui . Setting ki = ak+i mk+i and kl−k+i = bk+i mk+i for 1 ≤ i ≤ l − k, from Lemma 6.4 we obtain that A−B =

l

(X

αj

− X )fj +

j=k+1

with fj , gj ∈ D[Nl0 ]. Thus, A − B ∈ IQ .

βj

l j=k+1

(X βj − X αj )gj

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(3) To prove assertion (i) let P be a prime ideal in D[Nl0 ] such that IQ ⊆ P ⊆ IS . For a binomial X a − X b ∈ IS we show that it belongs to P , which proves (i) since IS is a prime ideal generated by these binomials. By step (2), (X ma − X mb )B ∈ IQ ⊆ P and, hence, X ma − X mb ∈ P or B ∈ P . The latter is impossible, because otherwise B ∈ IS (that is Φ∗ (B) = X Φ(d) would be the null polynomial). Therefore, X ma − X mb ∈ P . Now, if D has characteristic p = 0, it may happen that p divides m. Let m = pr q with r ≥ 0 and p q. This notation covers also the case p = 0 by setting pr = 1. For a = pr a and b = pr b, we have that q X ma − X mb = (X a − X b ) X a(q−i) X b(i−1) . i=1

From Φ(a) = Φ(b) it follows that Φ(a) = Φ(b) and, hence, q q a(q−i) b(i−1) X )= X Φ(a)(q−i)+Φ(b)(i−1) = qX Φ(a)(q−1) Φ ( X ∗

i=1

i=1

which cannot be the null polynomial. Therefore,

q

X a(q−i) X b(i−1) cannot be in P

i=1

and we must have that X a − X b ∈ P . In D[Nl0 ] it holds that r

r

r

X a − X b = (X a )p − (X b )p = (X a − X b )p

(which is valid also for p = 2), which implies that X a − X b ∈ P . (4) Concerning assertion (ii), from (i) we have that V (IS ) ⊆ V (IQ ). To show the reverse inclusion, let x ∈ V (IQ ). Since IS is generated by binomials X a − X b with Φ(a) = Φ(b), it suﬃces to show that xa = xb . From equation (7.9) we have that (xma − xmb ) · xc = 0 and (xma − xmb ) · xd = 0. If xc = 0 or xd = 0 then xma − xmb = 0 and, hence, xa = xb . We shall show that xc = 0 = xd implies that xa = 0 = xb . For this it is suﬃcient to show that xc = 0 implies xa = 0. By deﬁnition of c and d in step (2) together with (7.8) we have that ma + d = mb + c l l and ci = mj bj cij , di = mj aj cij for all 1 ≤ i ≤ k as well as cj = maj j=k+1

j=k+1

and dj = mbj for k + 1 ≤ j ≤ l. Suppose xc = 0, that is xi = 0 and ci > 0 for at least one 1 ≤ i ≤ l. If k + 1 ≤ i then ci = mai and xai i = 0, which implies xa = 0. If i ≤ k then by the above ai > 0 or di > 0. If ai > 0 then again xa = 0. If di > 0 then there exists k + 1 ≤ j ≤ l such that aj > 0 and cij > 0. Because of x ∈ V (IQ ) m c c c m we have that xj j − x11j · · · xkkj = 0. From xi ij = 0 we get xj j = 0 and, hence, aj xj = 0. Thus, xa = 0. This proves (ii). The homogeneity of varieties (or ideals) is reﬂected by the property of halffactoriality of monoids (which is of its own interest and has been intensely studied by many authors). An aﬃne variety V ⊆ Dn is homogeneous if for λ ∈ D\{0} and x = (x1 , . . . , xn ) ∈ V then λx = (λx1 , . . . , λxn ) ∈ V . A monoid is half-factorial if all representations of a nonunit by ﬁnitely many atoms have the same number of atoms. For a ﬁnitely generated half-factorial monoid S, this yields for the Red´ei map in equation (7.2) that Φ(a) = Φ(b) implies

l i=1

ai =

l i=1

bi .

(7.10)

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165

The following result shows that for Cale varieties homogeneity can be checked by the Cale base of the monoid. Corollary 6.6. Let D be an integral domain, let S be a ﬁnitely generated monoid (with ﬁxed generators), and let V = V (IS ). Suppose that char(D) = 0 and that V contains a point x with xi = 0 for all 1 ≤ i ≤ n. Then (i) V is homogeneous if and only if S is half-factorial. (ii) For S a Cale monoid, V is homogeneous if and only if in equation (7.4) one k cij for all k + 1 ≤ j ≤ l. has mj = i=1

Proof. (i) Since IS is generated by the binomials X a − X b , where Φ(a) = Φ(b), V (IS ) is homogeneous if and only if xa − xb = 0 implies that (λx)a − (λx)b for λ ∈ D\{0}. Because there is at least one x ∈ V with xa = 0, we have λa1 +···+al = λb1 +···+bl . Because D is an integral domain with char(D) = 0, the l l ai = bi . latter is equivalent to i=1

i=1

(ii) Suppose now, S is a Cale monoid. Let Q = {g1 , . . . , gk } be a Cale base according to equation (7.4). Obviously, if S is half-factorial, equation (7.4) imk cij for all k + 1 ≤ j ≤ l. Now, assume these equations hold. plies that mj = i=1

Then, of course, V (IQ ) is homogeneous because the base binomials generate IQ . From Theorem 6.5, part (ii), we obtain that V (IS ) = V (IQ ). Thus V (IS ) is also homogeneous. Remark 6.7. For char(D) = 0 neither (i) nor (ii) in Corollary 6.6 need to hold. Consider D = F2 and S the numerical monoid generated by 2 and 3 (see also Example 6.11 below). Then IS is generated by X13 − X22 and V (IS ) is homogeneous. The monoid S is a Cale monoid but not half-factorial. As remarked already, a Cale variety need not be normal. Thereby, V (IS ) is normal if D[S] is integrally closed. The following result derives criteria for normal Cale varieties from material in the previous section (recall by general assumption that S is cancellative, reduced, and torsion-free). Theorem 6.8. Let D = K be a ﬁeld and let S be a ﬁnitely generated monoid (with generators ﬁxed). The following statements are equivalent (a) The variety V (IS ) is normal and S is Cale. (b) S is Cale and root-closed. (c) S is isomorphic to a Diophantine monoid with a matrix having in each row just one positive entry or just one negative entry. (d) D[S] is a Cale domain and S is root-closed. Proof. (a) ⇒ (b): By assumption S is Cale and the coordinate ring K[Nl0 ]/IS ∼ = ¯ S¯ the rootK[S] is integrally closed. The integral closure of K[S] equals K[S], closure of S. Therefore, S¯ = S.

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(b) ⇒ (c): By [6, Corollary 3.3] a ﬁnitely generated monoid that is root-closed must be isomorphic to a Diophantine monoid. By [21, Theorem 4.1(i)] the latter must be isomorphic to a Diophantine monoid with a matrix as in (c). (c) ⇒ (d): By [21, Theorem 4.1(i)] K[S] is a Cale domain. As a Diophantine monoid S is root-closed. (d) ⇒ (a): By [21, Corollary 3.3], K[S] Cale domain and S root-closed implies ¯ = K[S]. that S is Cale. Furthermore, the integral closure of K[S] equals K[S] Therefore, D[S] is integrally closed. Remark 6.9. For aﬃne toric varieties that are normal see [29, Proposition 13.5]. To this the characterization may be added that V (IS ) is normal if and only if S is a Diophantine monoid. The general results obtained we shall illustrate by a couple of examples and counterexamples. Example 6.10. Consider for n ≥ 3 the additive monoid S = {x ∈ Nn0 | x1 + · · · + xn−1 = bxn }

(7.11)

for b ∈ N ﬁxed. Obviously, S is cancellative, torsion-free, and S × = {0}. Moreover, S is a Diophantine monoid, hence ﬁnitely generated, given by the matrix A = [1 ··· 1

− b ].

By [21, Theorem 4.1 (i)] S is a Cale monoid and, taking D to be a ﬁeld, by Theorem 6.8 (c) the variety V for S is a normal Cale variety. Actually, for this example a Cale base can be easily computed, Q = {qi }1≤i≤n−1 with qi = bei + en for 1 ≤ i ≤ n − 1. Namely, x = (x1 , . . . , xn ) ∈ S has as Cale representation with respect to Q in the sense of Deﬁnition 1.1 m(x)x =

n−1

x(qi )qi with m(x) = b, x(qi ) = xi .

(7.12)

i=1

According to Theorem 6.5 (i), the Cale variety V equals the joint zero set of all base binomials. To determine the latter we have to compute the Cale representation (7.12) for all atoms g not in the Cale base. This may be diﬃcult insofar as it is diﬃcult in general to screen all atoms. Below we calculate the atoms in special cases. Nevertheless, from equation (7.12) we can conclude that for any atom g = (g1 , . . . , gn−1 , gn ) ∈ S which is not in Q the base binomial must be of the form g

n−1 Xgb − X1g1 · · · Xn−1 .

(7.13)

For example, since g = ei + (b − 1)ej + en , 1 ≤ i, j ≤ n − 1, is an atom not in Q for i = j, particular base binomials are given by Xgb − Xi Xjb−1 . For b = 2 these exhaust all base binomials and the variety for S is homogeneous and an intersection of quadric cones with just one quadric cone for n = 3 (Example 1.7). For b ≥ 3, however, other types of base binomials occur (see b = 3 below).

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By Theorem 6.5 (ii) the Cale variety of S is the joint zero set of all binomials given by equation (7.13) — though the toric ideal is not generated by those binomials. Actually, the number of generators may be much bigger than the number of base binomials. The Cale variety for the Diophantine monoid given by equation (7.11) turns out to be the bth Veronese embedding of projective space P n−2 . The latter [29, Chapter 14] is the toric variety V (IT ) belonging to the monoid T generated by the set A of all with component sum equal g ∈ Nn−1 0 to b. Consider the mapping ϕ : T → S deﬁned for x = cg g by ϕ(x) = (x, cg ). ϕ is a well-deﬁned monoid homomorphism g∈A

g∈A

that is injective and has the property n−1

n−1

i=1

i=1

g∈A

ϕi (x) =

xi =

cg

n−1

i=1

g∈A

gi = b

cg = bϕ(x)n .

This shows that ϕ : T → S is a monoid isomorphism. In particular, consider for b = 3, n = 3 the twisted cubic curve which is deﬁned by A = {(3, 0), (2, 1), (1, 2), (0, 3)} and has a toric ideal generated by binomials Y1 Y3 − Y22 , Y1 Y4 − Y2 Y3 , and Y2 Y4 − Y32 ([29, Example 1.2 (a)]). By the above, the monoid T generated by A is isomorphic to the monoid S = {x ∈ N30 | x1 + x2 = 3x3 }. The Cale base Q = {q1 , q2 } consists of q1 = 3e1 + e3 = (3, 0, 1) and q2 = 3e2 + e3 = (0, 3, 1). There are two more atoms g3 = (1, 2, 1) and g4 = (2, 1, 1) for which the Cale representation (7.12) is 3g3 = q1 + 2q2 and 3g4 = 2q1 + q2 . Therefore, the base binomials not in Q are according to equation (7.13) given by X33 − X1 X22 and X43 − X12 X2 , setting X3 = Xg3 , X4 = Xg4 . Thus, by Corollary 6.6 (ii) the variety for S is homogeneous. It can easily be seen by identifying X1 = Y1 , X3 = Y3 , X2 = Y4 , X4 = Y2 that the ideal IQ generated by binomials in X’s as above is contained in the ideal IT generated by the binomials in Y ’s and that IQ and IT do not coincide. Nevertheless, from Theorem 6.5 we have that V (IQ ) = V (IT ). According to [29, Theorem 3.3] the toric variety V (IT ) “is cut out by its circuits” which in the case of T are the generators of IT , with the exception of Y1 Y4 − Y2 Y3 , together with the two base binomials above taken in terms of the Y ’s. By Theorem 6.5 the latter are suﬃcient to cut out variety V (IT ). By the way, the monoid S = {x ∈ N30 | x1 + x2 = 3x3 } is isomorphic also to the monoid T = {x ∈ N30 | 2x1 + 5x2 = 3x3 } by ϕ : S → T, ϕ(x) = (x1 , x2 , x2 + 2x3 ). The monoid T is discussed in [21] where it is argued that V (IT ) is the joint zero set of X33 − X1 X22

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and X32 − X2 X4 ([21, last line]). This is implied by Theorem 6.5 (ii), because in D for x33 = x1 x22 the equation x23 = x2 x4 holds if and only if x34 = x21 x2 . It is, however, not the case that the toric ideal IT is generated by X33 − X1 X22 and X32 − X2 X4 . In general, the number of generators for the toric ideal IS is much higher than the number of base binomials which describe the variety V (IS ) in the sense of Theorem 6.5 (ii). Consider for example the monoid S given by equation (7.11) for b = 2. As seen already, the base binomials are Xg2 − Xi Xj for atom g = ei + (b − 1)ej + en , 1 ≤ i, j ≤ n − 1 and i = j. The Cale base Q consists of k = n − 1 atoms. and the total number of atoms is l = The number of base binomials is n−1 2 n−1 n−1 n k + 2 = (n − 1) + 2 = 2 . For the case n = 4 and for 1 ≤ i ≤ 3 let Xi correspond to qi in the Cale base and let Xij correspond to g = ei +ej +e4 . The base 2 2 2 − X1 X2 , X13 − X1 X3 , X23 − X2 X3 and the remaining binomials are given by X12 binomials to generate IS are X1 X23 − X12 X13 , X12 X3 − X13 X23 , X13 X2 − X12 X23 . See also [29, p. 142] where this minimal system of generators has been computed for the corresponding Veronese embedding. Example 6.11. We present simple examples of Cale and non-Cale aﬃne toric varieties. In the latter case, we exhibit Cale varieties which are not normal or have a coordinate ring which may or may not be a Cale domain (depending on the ﬁeld). Consider a numerical semigroup S, that is the additive monoid S = g1 N0 + · · · + gl N0 with 1 < g1 < · · · < gl pairwise relative prime and l ≥ 2. We may assume the gi to be atoms in S. Because of gcd{g1 , . . . , gl } = 1 it follows that for every n ∈ N0 there exists k ∈ N such that kn ∈ S. Therefore, S¯ = N0 and from Theorem 4.4 we obtain that S is a Cale monoid. Actually, Q = {q} with q = g1 can serve as a Cale base and the Cale representation of x ∈ S with respect to Q is given by m(x)x = x(q)q with m(x) = q and x(q) = x. Thus, the variety V for S is a Cale variety. Since the gj with 2 ≤ j ≤ l are exactly the atoms not in the Cale base, the base binomials are given by g

Xjq − X1 j for 2 ≤ j ≤ l.

(7.14)

The zero set of such a base binomial is a generalized Neil parabola. By Theorem 6.5 (ii) the variety V for S is the intersection of the generalized Neil parabolas given by equation (7.14). Since q = g1 = gj for 2 ≤ j ≤ l it follows that V is never homogeneous (see also Corollary 6.6 (ii)). To illustrate Theorem 6.8, suppose now that D = K is a ﬁeld. Because of S¯ = N0 but S N0 , S is not root-closed. By Theorem 6.8, therefore, V is not normal. Concerning Theorem 6.8 (d) the question arises if the coordinate ring of V could still be a Cale domain. This depends on the ﬁeld K. For simplicity consider the case of the common Neil parabola, that is l = 2, g1 = 2, g2 = 3 and the base binomial X22 − X13 . For K = C and K = R, K[S] is not solid and, hence, cannot be a Cale domain by Lemmas 2.1 and 2.3. For K = F2 , however, K[S] is solid and, hence, a Cale domain (see Section 5.3 and [21, Remark 3.4]). Geometrically, for K = C or K = R the singularity of V in 0 is cusp. For K = F2 , however, x22 − x31 = (x2 + x1 )(x2 − x1 ) and, hence, V has a double point singularity in 0 (as has the quadric cone the coordinate of which is a Cale domain). In the “standard toric varieties”, D is an algebraically closed ﬁeld and the variety for D is normal. For those varieties, to be Cale means that the coordinate ring is

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a Cale domain. The above discussion shows that this may fail for varieties not being normal. Finally, a simple example of a standard variety which is not a Cale variety is given by C[S] for S the Diophantine monoid given by just one equation, S = {x ∈ N40 : x1 + x2 = x3 + x4 }. Obviously, S is root-closed and by Theorem 6.8 the variety of S cannot be a Cale variety and its coordinate ring cannot be a Cale domain.

Bibliography [1] D.D. Anderson and M. Zafrullah, Almost B´ezout domains, J. Algebra 142 (1991), 285–309. [2] D.D. Anderson, S. T. Chapman, F. Halter-Koch and M. Zafrullah, Criteria for unique factorization in integral domains, J. Pure Appl. Algebra 127 (1998), 205–218. [3] D.D. Anderson, D.F. Anderson and M. Zafrullah, Rings between D[X] and K[X], Houston J. Math. 17(1991), 109–129. [4] S.T. Chapman, F. Halter-Koch and U. Krause, Inside factorial monoids and integral domains, J. Algebra 252 (2002), 350–375. [5] S.T. Chapman, U. Krause and E. Oeljeklaus, Monoids determined by a homogenous linear Diophantine equation and the half-factorial property, J. Pure Appl. Algebra 151 (2000), 107–133. [6] S.T. Chapman, U. Krause and E. Oeljeklaus, On diophantine monoids and their class groups, Pac. J. Math. 207 (2002), 125–147. [7] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, New York, Springer Verlag, 1992. [8] J. Coykendall, D. Dobbs and B. Mullins, On integral domains with no atoms, Comm. Algebra 27 (1999), 5813–5831. [9] D. Eisenbud and B. Sturmfels, Binomial ideals, Duke J. Math 84 (1996), 1–45. [10] A. Geroldinger, On the structure and arithmetic of ﬁnitely primary monoids, Czech. Math. J. 46 (1996), 677–695. [11] A. Geroldinger and F. Halter-Koch, Arithmetical theory of monoid homomorphisms, Semigroup Forum 48 (1994), 333–362. [12] R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Mich. Math. J. 21 (1974), 65–86. [13] R. Gilmer, Commutative Semigroup Rings, The University of Chicago Press, Chicago, 1984. [14] F. Halter-Koch, Halbgruppen mit Divisorentheorie, Expo. Math 8 (1990), 27– 66. 170

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[15] F. Halter-Koch, Divisor theories with primary elements and weakly Krull domains, Bol. Un. Mat. Ital. B 9 (1995), 417–441. [16] F. Halter-Koch, Ideal Systems, Marcel Dekker, New York, 1998. [17] U. Krause, Eindeutige Faktorisierung ohne ideale Elemente, Abh Braunschweigische Wiss Ges 33 (1982), 169–177. [18] U. Krause, A characterization of algebraic number ﬁelds with cyclic class group of prime power order, Math. Zeitschrift 186 (1984), 143–148. [19] U. Krause, Semigroups that are factorial from inside or from outside, Lattices, Semigroups and Universal Algebra, Plenum Press, New York, 1990, 147–161. [20] U. Krause, A general principle of marked extraction, Advances in Mathematical Systems Theory: A Volume in Honor of Diederich Hinrichsen, Birkh¨ auser, Boston, 2001, 169–183. [21] U. Krause, Semigroup rings that are inside factorial and their Cale representation, Rings, Modules, Algebras, and Abelian Groups, Lecture Notes in Pure and Appl. Math., Marcel Dekker 236 (2004), 353–363. [22] M. Picavet–L’Hermitte, Cale bases in algebraic order, Annales Math´ematiques Blaise Pascal 10 (2003), 117–131. [23] G. Picavet and M. Picavet–L’Hermitte, Trigonometric polynomial rings, Commutative Ring Theory and Applications, Lecture Notes in Pure and Appl. Math., Marcel Dekker 231 (2003), 419–433. [24] P. Ribenboim, Generalized power series rings, Lattices, Semigroups and Universal Algebra, New York, Plenum Press, 1990, 271–277. [25] J.C. Rosales and P. A. Garc´ıa–S´ anchez, Finitely Generated Commutative Monoids, New York, Nova Science Publishers, 1999. [26] P. Samuel, On unique factorization domains, Ill. Jour. Math. 5 (1961), 1–17. [27] R. Stanley, Combinatorics and Commutative Algebra, Birkhauser, Boston, 1983. [28] B. Sturmfels, Equations deﬁning toric varieties, Proc. Sympos. Pure Math. 62 (1997), 437–449. [29] B. Sturmfels, Gr¨ obner Bases and Convex Polytopes, American Mathematical Society, Providence, 1996. [30] R. H. Villarreal, Monomial Algebras, New York, Marcel Dekker Inc, 2001. [31] M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), 29–62. [32] A. Zaks, Half-factorial domains, Israel J. of Math. 37(1980), 281–302.

Chapter 8

Weakly Krull Inside Factorial Domains Daniel D. Anderson, Muhammed Zafrullah, and Gyu Whan Chang by

1

Introduction

Chapman, Halter-Koch, and Krause [8] introduced the notion of an inside factorial monoid and integral domain. Throughout we will conﬁne ourselves to the integral domain case, but the interested reader may easily supply deﬁnitions and proofs for the monoid case. An integral domain D is inside factorial if there exists a divisor homomorphism ϕ: F → D∗ = D − {0} where F is a factorial monoid and for each x ∈ D∗ , there exists an n ≥ 1 with xn ∈ ϕ(F ). They showed [8, Proposition 4] that an inside factorial domain may be deﬁned in terms of a Cale basis and it will be this characterization that we use for the deﬁnition of an inside factorial domain. A subset Q ⊆ D∗ is a Cale basis for D if Q = {uqα1 · · · qαn | u ∈ U (D), qαi ∈ Q} is a factorial monoid with primes Q and for each d ∈ D∗ there exists an n ≥ 1 with dn ∈ Q. Here U (D) is the group of units of D. A domain D is inside factorial if and only if D has a Cale basis. They showed [8, Theorem 4] that D is inside factorial if and only if D, the integral closure of D, is a generalized Krull domain with torsion t-class group Clt (D), for each P ∈ X (1) (D), the valuation domain DP has value group order-isomorphic to a subgroup of (Q, +) (we say that D is rational), and D ⊆ D is a root extension (i.e., for each x ∈ D, there exists an n ≥ 1 with xn ∈ D). Recall that an integral domain D is weakly Krull [4] if D = P ∈X (1) (D) DP where the intersection is locally ﬁnite. While an integrally closed inside factorial domain is weakly Krull, an inside factorial domain need not be weakly Krull (see Example 3.2 below). The purpose of this chapter is threefold. First we show (Theorem 3.1) that an inside factorial domain has a unique minimal root extension which is weakly Krull (and necessarily inside factorial). Second, we give (Theorem 3.3) a number of 172

Weakly Krull Inside Factorial Domains

173

characterizations of weakly Krull inside factorial domains. For example, we show that for an inside factorial domain the following conditions are equivalent: (1) D is weakly Krull, (2) D is an almost GCD domain, (3) t-dim D = 1, and (4) for each element q of a Cale basis for D, (q) is primary. Third, Theorems 3.1 and 3.3 are then used to give a new proof that if D is inside factorial, then D is a rational generalized Krull domain with torsion t-class group and D ⊆ D is a root extension.

2

Preliminaries

In this section we review some results on Cale bases, AGCD domains, and root extensions. Let D be an inside factorial domain with Cale basis Q. Now by deﬁnition the elements map Q → X (1) (D) given by of Q are primes in the monoid Q and the q → (q) is a bijection [8, Theorem 2]. (In fact, if Q ⊆ D∗ with bijection as above, then Q is a Cale basis.) However, the elements of Q need not be irreducible nor primary as elements of D. Also, recall that an integral domain D is an almost GCD domain (AGCD domain) [10] if for x, y ∈ D∗ , there exists an n ≥ 1 with xn D ∩ y n D, or equivalently (xn , y n )v , principal. An AGCD domain D has torsion t-class group [5, Theorem 3.4], D ⊆ D is a root extension [10, Theorem 3.1], and D is a PVMD with torsion t-class group [10, Theorem 3.4 and Corollary 3.8]. Now a PVMD with torsion tclass group is an AGCD domain [10, Theorem 3.9], but if D is an integral domain with D ⊆ D a root extension and D a PVMD with torsion t-class group, or equivalently, D is an AGCD domain, D need not be an AGCD domain. The following example is [3, Theorem 3.1] (also see Example 3.2 of this chapter). Let K L be a purely inseparable ﬁeld extension and let D = K + (X)L[X] where X is a set of indeterminates with |X| > 1. Then D D = L [X] is a root extension and L[X] is a UFD (and hence an AGCD domain), but D is not an AGCD domain. Note that if |X| = 1, D is an AGCD domain. The following results on root extensions will be frequently used. (1) [5, Theorem 2.1], [8, Proposition 5] Let R ⊆ S be a root extension of commutative rings. The map Spec(S) → Spec(R) given by Q → Q √ ∩ R is an order isomorphism and homeomorphism with inverse given by P → P = {s ∈ S | sn ∈ P for some n ≥ 1}. Moreover, RP ⊆ SQ is a root extension. (2) [8, Proposition 5] Let D ⊆ S be a root extension of integral domains. Then D is inside factorial if and only if S is inside factorial. Of course, in the case where D is inside factorial and S ⊆ K, the quotient ﬁeld of D, D ⊆ S is a root extension if and only if D ⊆ S is integral. We have remarked that an inside factorial domain need not be weakly Krull. The previously mentioned example D = K +(X)L[X] where K L is purely inseparable and |X| > 1 is such an example, see Example 3.2 below. In [6, Proposition 2.4] we showed that the following conditions are equivalent for an inside factorial domain D with Cale basis Q: (1) for each q ∈ Q, (q) is a maximal t-ideal, (2) D is weakly Krull, (3) D is an AGCD domain, (4) each q ∈ Q is a primary element, and (5) distinct elements of Q are v-coprime. See Theorem 3.3 below where these and several more equivalences are given.

174

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Weakly Krull Inside Factorial Domains

We begin this section by showing that an inside factorial domain has a unique minimal root extension which is weakly Krull. We then give an example of an inside factorial domain that is not weakly Krull and give a number of characterizations of weakly Krull inside factorial domains. Theorem 3.1. Let D be an inside factorial domain and let D# = P ∈X (1) (D) DP . (1 ) D ⊆ D# is a root extension and hence D# is inside factorial. (2 ) D# is weakly Krull. (3 ) Let S be an integral domain containing D such that S is weakly Krull and D ⊆ S is a root extension. Then D# ⊆ S. (4 ) Suppose that S is an integral domain with D# ⊆ S a root extension. Then S is a weakly Krull inside factorial domain. Proof. Here D is an inside factorialdomain with quotient ﬁeld K. Let Q = {qα } be a Cale basis for D and let Pα = (qα ). So Q → X (1) (D) given by qα → Pα is a bijection. (1) Let 0 = x ∈ D# . Write x = a/b where a, b ∈ D∗ . Since suitable powers of a and b lie in Q, we can choose n ≥ 1 with xn = qαn11 · · · qαnss where ∈ U (D) and each ni ∈ Z∗ . Suppose that some ni < 0. With a change of notation, if necessary, we can assume that n1 , · · · , ni < 0 and ni+1 , · · · , ns > 0. Now xn ∈ DPαi , so qαn11 · · · qαnss = r/t where r ∈ D and t ∈ D − Pαi . Then ni+1 1 i tqαi+1 · · · qαnss = rqα−n · · · qα−n ∈ Pαi . But no factor on the left hand side lies in 1 i Pαi , a contradiction. Hence each ni > 0, so xn ∈ D. Since D ⊆ D# is a root extension, D# is inside factorial. (2) For N ∈ X (1) (D# ), let S = D# − N and P = N ∩ D. Since D ⊆ D# is a root # extension, P ∈ X (1) (D). Now D# ⊆ DP gives DN ⊆ (DP )S = DP where the # , so equality follows since ht PP = 1 and PP ∩ S = ∅. But certainly DP ⊆ DN # # . Thus D# ⊆ N ∈X (1) (D# ) DN = P ∈X (1) (D) DP = D# and hence DP = DN # . Let 0= x ∈ K. Then x is a unit in almost all the D# = N ∈X (1) (D# ) DN DPα ’s. This follows since 0 = a ∈ D is in Pα = (qα ) if and only if an ∈ Q has qα as one of its factors and thus a is in only ﬁnitely many Pα . Since each # # DN is equal to a unique DPα , x is a unit in almost all the DN ’s. Hence D# is weakly Krull. (3) Suppose that D ⊆ S is a root extension and that S = N ∈X (1) (S) SN . For N ∈ X (1) (S), P = N ∩ D ∈ X (1) (D); so DP ⊆ SN . Then D# = P ∈X (1) (D) DP ⊆ N ∈X (1) (S) SN = S. extension. (4) Suppose that D# ⊆ S is a root extension. Then D ⊆ S is also a root # # Hence Q is a Cale basis for D and S. Since D is weakly Krull, qα D# ∈ √ We claim that qα S is qα SX (1) (D# ) gives that qα D# is qα D# -primary. √ primary. Suppose that xy ∈ qα S and x∈ / qα S. Choose n ≥ 1 with xn , y n ∈ # n n # n / qα D# , so for some m ≥ 1, (y n )m ∈ D and x y ∈ qα D . Then x ∈ # qα D ⊆ qα S. Thus S has a Cale basis Q = {qα } with each qα S primary. By the remarks preceding Theorem 3.1, S is a weakly Krull inside factorial domain.

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Of course, for D inside factorial, D = D# if and only if D is weakly Krull. Note that by [8, Proposition 5(f)] D is tamely inside factorial if and only if D# is. We next give an example of an inside factorial domain that is not weakly Krull. Example 3.2. Let D = K + (X)L[X] where K ⊆ L is a purely inseparable ﬁeld extension and X is a nonempty set of indeterminates over L. Then D ⊆ D = L[X] is a root extension since K ⊆ L is purely inseparable and D being factorial is inside factorial. Thus D is inside factorial. Alternatively, if we take Q to be a complete set of nonassociate prime elements of K[X], then Q is a Cale basis for D and hence D is inside factorial. Suppose that |X| = 1. Then dim D = 1, so D = D# and hence D is weakly factorial. Suppose that |X| > 1. We claim that D# = L |X|. Let P be a height-one prime ideal of D; so P = f L[X] ∩ D where f ∈ L[X] is irreducible. Choose g ∈ D − P with g(0)=0. ( If |X| = 1 and f = X, we can’t choose such a g. But suppose |X| > 1. If f ∈ X, take any g ∈ X − {f } while if f ∈ / X, take g = X ∈ X.) Now for a ∈ L, a = ag/g ∈ DP . Hence L ⊆ D# , so D# = L[X]. We next give a number of conditions equivalent to D being a weakly Krull inside factorial domain. But ﬁrst we need some more deﬁnitions. An integral domain D is almost weakly factorial [4] if some power of each nonunit of D is a product of primary elements. By [4, Theorem 3.4], D is almost weakly factorial if and only if D is weakly Krull and Clt (D) is torsion. An integral domain D is a generalized weakly factorial domain [7] if each nonzero prime ideal of D contains a nonzero primary element. Thus an almost weakly factorial domain is a generalized weakly factorial domain. Two elements x and y of an integral domain D are power associates if n m there exist n, m ≥ 1 and a unit ∈ U (D) with x = y . Given a nonzero prime ideal P of D, b is a base for P if P = (b). Thus if (b) is P -primary, b is a base for P . Finally, t-dim D = 1 if every prime t-ideal of D is a maximal t-ideal. Theorem 3.3. For an integral domain D the following conditions are equivalent. (1 ) D is inside factorial and weakly Krull. (2 ) D is inside factorial and AGCD. (3 ) D is inside factorial and t-dim D = 1. √ (4 ) D is inside factorial with a Cale basis Q such that qD is a maximal t-ideal for each q ∈ Q. √ (5 ) D is inside factorial and for each Cale basis Q for D, qD is a maximal t-ideal for each q ∈ Q. (6 ) D is inside factorial with a Cale basis Q such that each q ∈ Q is primary. (7 ) D is inside factorial and for each Cale basis Q for D and each q ∈ Q, q is primary. (8 ) D is inside factorial with a Cale basis Q such that distinct elements of Q are v-coprime. (9 ) D is inside factorial and for each Cale basis Q for D distinct elements of Q are v-coprime. (10 ) D is almost weakly factorial and two nonzero primary elements of D are either v-coprime or power associates. (11 ) D is a weakly Krull AGCD domain and two nonzero primary elements of D are either v-coprime or power associates.

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(12 ) D is a generalized weakly factorial domain such that for each height-one prime ideal P of D, every pair of base elements of P are power associates and for each nonzero x ∈ P , there is an n ≥ 1 with xn = db where b is a base for P and d∈ / P. Proof. The equivalence of (1), (2), and (4)–(9) √ is given by [6, Proposition 2.4]. (3)⇐⇒(4) For D inside factorial, X (1) (D) = { qD | q ∈ Q} for any Cale basis Q for D. Since a height-one prime ideal is a t-ideal, the equivalence follows. (6)=⇒(10) Let x be a nonzero nonunit of D. Then some power xn ∈ Q. So xn is a product of primary elements and hence D is almost factorial. Suppose that q1 and q2 weakly are nonzero primary elements of D. If (q ) = (q2 ), then (q1 , q2 )v = D (for say 1 by (q1 ) = (q2 ) = 2 ) are maximal t-ideals). So suppose (6)=⇒(3) (q1 ) and (q (q) where q ∈ Q. Then q1n1 = 1 q m1 and q2n2 = 2 q m2 for some n1 , n2 , m1 , m2 ≥ 1 and units 1 , 2 . But then q1 and q2 are power associates. (10)=⇒(6) Since D is almost weakly factorial, D is weakly Krull and each height-one prime Pα contains a Pα -primary element qα . Let Q = {qα }. We claim that Q is a Cale basis for D. If 1 qαn11 · · · qαnss = 2 qαm11 · · · qαmss where 1 , 2 are units and ni , mi ≥ 0, then qαnii DPαi = 1 qαn11 · · · qαnss DPαi = 2 qαm11 · · · qαmss DPαi = qαmii DPαi ; so ni = mi . Hence Q is a factorial monoid. Let x be a nonzero nonunit of D. Then some power xn is a product of primary elements. But each primary element in this factorization is power associate to some qα . Thus raising xn to an appropriate power gives that some power of x is in Q. (11)=⇒(10) It suﬃces to show that an AGCD weakly Krull domain is almost weakly factorial. But we have already remarked that an AGCD domain has torsion t-class group and that a domain is almost weakly factorial if and only if it is a weakly Krull domain with torsion t-class group. (6)=⇒(11) Since (6)=⇒(1), D is weakly Krull and since (6)=⇒(2), D is an AGCD domain. And by (6)=⇒(10) two nonzero primary elements are either v-coprime or power associates. (10)=⇒(12) Certainly an almost weakly factorial domain is a generalized weakly factorial domain. Now an almost weakly factorial domain is weakly Krull and in (x) = P , P a height-one prime, is a weakly Krull domain an element x with primary (for (x) = x P ∈X (1) (D) DP = P ∈X (1) (D) xDP = xDP ∩ D). Thus two bases for a height-one prime ideal P are P -primary and hence by hypothesis are power associates. Finally, let 0 = x ∈ P . Since D is almost weakly factorial, some xn is a product of primary elements, say xn = q1 · · · qi qi+1 · · · qm where q1 , · · · , qi ∈ P (with necessarily i ≥ 1) and / P . Now (q1 · · · qi ) is still P -primary qi+1 , · · · , qm ∈ (for D is weakly Krull and (q1 · · · qi ) = P ). Take b = q1 · · · qi and d = qi+1 · · · qn ; so xn = db where d ∈ / P and b is a base for P . (12)=⇒(1) A generalized weakly factorial domain is weakly Krull [7, Corollary 2.3]. For each P ∈ X (1) (D) choose a base element xP for P. We claim that Q = {xP } is a Cale basis for D; and hence D is inside factorial. As in the proof of (10)=⇒(6), Q is a factorial monoid. Let x be a nonzero nonunit of D. Let P1 , · · · , Pm be the height-one prime ideals of D containing x. By hypothesis, there is an n ≥ 1 so that xn = db where b is a base for P1 and d ∈ / P1 . Now b and xP1 are power associates, say br = uxsP1 where r, s ≥ 1 / P1 . Now P2 , · · · , Pm are and u is a unit. Then xnr = dr uxsP1 = x1 xsP1 where x1 ∈ the height-one prime ideals containing x1 . By induction, we have xt1 = βxsP22 · · · xsPmm where β is a unit and t, s2 , · · · , sm ≥ 1. Hence xtnr ∈ Q.

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Miscellaneous Results

We have already remarked that an integral domain D is inside factorial if and only if D is a rational generalized Krull domain with torsion t-class group and D ⊆ D is a root extension [8]. We begin this section by giving an alternative proof of the implication (=⇒) based on Theorems 3.1 and 3.3. We then discuss when certain extensions of an inside factorial domain are again inside factorial. Theorem 4.1. [8, Theorem 4] An integral domain D is inside factorial if and only if D is a rational generalized Krull domain with torsion t-class group and D ⊆ D is a root extension. Proof. (=⇒) Suppose that D is inside factorial. By Theorem 3.1, D ⊆ D# is a root extension and D# is a weakly Krull inside factorial domain. By Theorem 3.3 D# is an AGCD domain. Thus we can apply results from [10] concerning AGCD domains. By [10, Theorem 3.1], D# ⊆ D# = D is a root extension. Hence D ⊆ D is a root extension. By [10, Theorem 3.4 and Corollary 3.8] D is a PVMD with torsion t-class group. But by Theorem 3.1 again, D is weakly Krull. Hence D is a generalized Krull domain. It remains to show that for each P ∈ X (1) (D), the value group of D P is isomorphic to a subgroup of (Q, +). Let Q be a Cale basis for D with P = (p) where p ∈ Q. For 0 = x ∈ K, there is a natural number n with xn = pn0 q1n1 · · · qsns where n0 , · · · , ns ∈ Z, ∈ U (D) and q1 , · · · , qs ∈ Q. But q1 , · · · , qs are units in DP , so we can write xn = pn0 where is a unit in DP . If v: K ∗ → (R, +) is the valuation for DP , then nv(x) = v(xn ) = v( pn0 ) = n0 v(p). Hence v(x) = nn0 v(p). Thus im v is order-isomorphic to (Q, +). (⇐=) [8, Theorem 4] We end by considering extensions of inside factorial domains. Let D be inside factorial. Then each overring E of D contained in D is again inside factorial (for D ⊆ E is a root extension). Moreover, by Theorem 3.1 if D is weakly Krull, so is E. Next suppose that S is a multiplicatively closed subset of D. Then D ⊆ D a root extension gives that DS ⊆ DS = DS is a root extension, DS is a rational generalized Krull domain, and DS has torsion t-class group since D does [2, Theorem 4.4]. Thus DS is inside factorial. Alternatively, observe that if Q is a Cale basis for D, then (1) {q ∈ Q | qD S = DS } is a Cale basis for DS . Moreover, if ∅ = Λ ⊆ X (D), then R = P ∈Λ DP is a weakly Krull inside factorial domain. For if we set S = √ / Λ , then (D# )S = = P ∈Λ DP = R. q ∈ Q | qD ∈ P ∈X (1) (D) DP S

We next show that D[X] is inside factorial if and only if D is inside factorial and D[X] ⊆ D[X] is a root extension. Also see [9, Theorem 3.2]. Now D[X] is inside factorial if and only if D[X] ⊆ D[X] is a root extension and D[X] is a rational generalized Krull domain with torsion t-class group. But D[X] is a rational generalized Krull domain with torsion t-class group if and only if D is (for D is a generalized Krull domain if and only if D[X] is, the natural map Clt (D) → Clt (D[X]) is an isomorphism, and for N ∈ X (1) (D[X]), D[X]N is a DVR if N ∩D = 0 and if N ∩ D = 0, then D[X]N = DN ∩D (X) has the same value group as DN ∩D ) and D[X] ⊆ D[X] a root extension forces D ⊆ D to be a root extension. However, D ⊆ D√ a root extension need not imply the D[X] ⊆ D[X] is a root extension. Let D = Z[ 5]. Then D D is a root extension and D is an AGCD (= weakly Krull) inside factorial domain, but D[X] D[X] is not a root extension [1, Example 3.6].

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Hence D[X] is not inside factorial. According to [1, Theorem 3.4], for an integral domain D, D[X] is an AGCD domain if and only if D is an AGCD domain and D[X] ⊆ D[X] is a root extension. Hence if D[X] is inside factorial, D[X] is AGCD (= weakly Krull) if and only if D is.

Bibliography [1] D. D. Anderson, T. Dumitrescu, and M. Zafrullah, Almost splitting sets and AGCD domains, Comm. Algebra 32 (2004), 147–158. [2] D. D. Anderson, E. G. Houston, and M. Zafrullah, t-linked extensions, the tclass group, and Nagata’s Theorem, J. Pure Appl. Algebra 86 (1993), 109–124. [3] D. D. Anderson, K. R. Knopp, and R. L. Lewin, Almost B´ezout domains, II, J. Algebra 167 (1994), 547–556. [4] D. D. Anderson, J. L. Mott, and M. Zafrullah, Finite character representations for integral domains, Boll. Un. Mat. Ital. B (7 ) 6 (1992), 613–630. [5] D. D. Anderson and M. Zafrullah, Almost B´ezout domains, J. Algebra 142 (1991), 285–309. [6] D. D. Anderson and M. Zafrullah, A note on almost GCD monoids, Semigroup Forum 24 (2004), 811–828. [7] D. F. Anderson, G. W. Chang, and J. Park, Generalized weakly factorial domains, Houston J. Math 29 (2003), 1–13. [8] S. Chapman, F. Halter-Koch, and U. Krause, Inside factorial monoids and integral domains, J. Algebra 252 (2002), 350–375. [9] U. Krause, Semigroup rings that are inside factorial and their Cale representation, Rings, Modules, Algebras, and Abelian Groups (A. Facchini, E. Houston, and L. Salce, eds.), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 236 (2004), 353–363. [10] M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), 29–62.

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Chapter 9

The m-Complement of a Multiplicative Set by

David F. Anderson and Gyu Whan Chang Abstract Let D be an integral domain. For any ∅ = S ⊆ D \ {0}, we deﬁne the mcomplement for S (in D) to be ND (S) = {0 = x ∈ D|xD ∩ sD = xsD for all s ∈ S}. In this chapter, we investigate the relationship between S and ND (S).

1

Introduction

Let D be an integral domain with quotient ﬁeld qf (D), nonzero elements D∗ = D\{0}, and group of units U (D). For any ∅ = S ⊆ D∗ , we deﬁne the m-complement for S (in D) to be ND (S) = {x ∈ D∗ |xD ∩ sD = xsD for all s ∈ S}. Then ND (S) is a saturated multiplicative subset of D. When the context is clear, we write N (S) for ND (S). As in [1], a saturated multiplicative subset S of D is called a splitting set if SN (S) = D∗ . It is well-known, and easily veriﬁed, that if S is a splitting set, then N (S) is also a splitting set, S ∩ N (S) = U (D), N (N (S)) = S, and D = DS ∩ DN (S) . As an example, U (D) is a splitting set with N (U (D)) = D∗ , and thus D∗ is a splitting set with N (D∗ ) = U (D). Also, if S is a splitting set, then Cl(D) = Cl(DS ) ⊕ Cl(DN (S) ) [1, Corollary 3.8], where Cl(D) is the t-class group of D (see [6]). In [7], we investigated when Cl(D) = Cl(DS ) ⊕ Cl(DN (S) ) in the case that SN (S) = D \ P for some prime ideal P of D. For other properties and applications of splitting sets, see [1], [2], [3], [5], and [9]. The m-complement of S has also been studied in [3], where they used the notation S ⊥ for N (S). In this chapter, we investigate the relationship between S and N (S). We give some basic properties for N (S) and compute NR (S) for certain ring extensions R of D. Several of these results recover as special cases earlier results when S is a splitting set. As usual, Iv = (I −1 )−1 and It = ∪{(a1 , . . . , an )v |0 = (a1 , . . . , an ) ⊆ I} for a nonzero fractional ideal I of D, and I is a t-(resp., v-)ideal of D if It = I (resp., Iv = I). Note that for x, s ∈ D∗ , xD ∩ sD = xsD if and only if (x, s)v = D; so ND (S) = {x ∈ D∗ |(x, s)v = D for all s ∈ S}. We will often use the fact that 180

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(x, s)v = D if and only if (x, s) is contained in no prime t-ideal of D. Any undeﬁned notation is standard, see [10].

2

Basic Properties of the m-Complement

In this section, we give some basic properties of the m-complement and several examples. Our ﬁrst lemma is implicit in [1] and shows that N (S) is a saturated multiplicative subset of D (cf.[3, Proposition 2.4] and [5, page 2210]). Moreover, N (S) = N (S) = N (S), where S = {ux1 · · · xn |u ∈ U (D) and xi ∈ S} is the multiplicative subset of D generated by S and the saturation of a multiplicative subset T of D is T = {x ∈ D|xy ∈ T for some y ∈ D}. We may thus assume that S is a (saturated) multiplicative subset of D. The proofs of these results are routine, and hence are omitted. Lemma 2.1. Let D be an integral domain. Then the following statements are equivalent for all x, y, z ∈ D∗ . 1. xD ∩ zD = xzD and yD ∩ zD = yzD. 2. xyD ∩ zD = xyzD. Proposition 2.2. Let D be an integral domain and ∅ = S ⊆ D∗ . Then N (S) is a saturated multiplicative subset of D. Moreover, N (S) = N (S) = N (S). Let Σ be the set of all saturated multiplicative subsets of D and P(D∗ ) the power set of D∗ . By Proposition 2.2, we have a mapping N : P(D∗ ) \ {∅} → Σ which restricts to a mapping Σ → Σ. We next list several properties of the N operator. These properties all follow directly from deﬁnitions and Lemma 2.1, and they will be used without further reference ((1), (2), (3), and (4) have been observed in [3, Proposition 2.4], and part (8) is a special case of [5, Proposition 1.1]). Proposition 2.3. Let D be an integral domain, S, S1 , and S2 nonempty subsets of D∗ , and {Sα } a family of nonempty subsets of D∗ . 1. If S1 ⊆ S2 , then N (S2 ) ⊆ N (S1 ). 2. S ∩ N (S) ⊆ U (D). (Equality holds if U (D) ⊆ S.) 3. S ⊆ N (N (S)). 4. N (S) = N (N (N (S))). 5. N (∪Sα ) = N (∪Sα ) = ∩N (Sα ). 6. N (S1 S2 ) = N (S1 ) ∩ N (S2 ). 7. If S1 ∩ S2 = ∅, then N (S1 )N (S2 ) ⊆ N (S1 ∩ S2 ). 8. D = DS ∩ DN (S) . We next give several examples to show that the inclusions in Proposition 2.3 may be proper.

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Example 2.4. (a) Let p be a prime element of an integral domain D and S = p = {upn |u ∈ U (D) and n ≥ 0}. Then N (S) = D \ pD and SN (S) = D \ Q, where Q = ∩pn D. More generally, for p1 , . . . , pm prime elements of D and S = p1 , . . . , pm = {upk11 · · · pkmm |u ∈ U (D) and each ki ≥ 0}, N (S) = D \ (∪pi D), and SN (S) = D \ (∪Qi ), where Qi = ∩(pni D). In particular, S is a splitting set if and only if each Qi = {0}, i.e., each pi D has height-one (cf. [2, Proposition 1.6]). For an inﬁnite set {pα |α ∈ A} of nonassociate prime elements of D, let S = {pα }. Then N (S) = D \ (∪pα D) and SN (S) = D \ [(∪Qα ∪ (∪JB )], where Qα = ∩pnα D and JB = ∩{pβ D|β ∈ B} for each countably inﬁnite B ⊆ A. Thus S is a splitting set if and only if each Qα and each JB is {0} (cf. [2, Proposition 1.6]). (b) Let R = Z + XQ[X]. Then S = Z∗ is a saturated multiplicative subset generated by prime elements for both Z and R with NZ (S) = U (Z) and NR (S) = {a + Xf |a ∈ U (Z) and f ∈ Q[X]} (cf. Proposition 3.3(2)). However, X ∈ NR (NR (S)); so the inclusion in Proposition 2.3(3) may be proper even when S is a saturated multiplicative subset generated by prime elements (for another example, see [3, Example 2.5]). (c) The inclusion in Proposition 2.3(7) may be proper. For example, let S1 = {1, 2}, S2 = {1, 4}, and D = Z. Then N (S1 ∩ S2 ) = N ({1}) = Z∗ , while N (S1 )N (S2 ) = N (S1 ) = Z \ 2Z by part (a) above. For another example, let S be a nonsplitting multiplicative subset of an integral domain D such that S = N (N (S)). Then S ∩ N (S) = U (D), and hence N (N (S))N (S) = SN (S) D∗ = N (S ∩ N (S)). However, if S1 and S2 are saturated multiplicative subsets of a UFD, then N (S1 )N (S2 ) = N (S1 ∩ S2 ). We next give a necessary and suﬃcient condition to have S = N (N (S)) for each saturated multiplicative subset S of D. First we recall a few deﬁnitions. A nonzero nonunit x ∈ D is said to be primary if xD is a primary ideal. As in [8], D is called a generalized weakly factorial domain (GWFD) if every nonzero prime ideal of D contains a primary element. It is known that if D is not a ﬁeld, then D is a GWFD if and only if t-dim(D) = 1 and each height-one prime ideal of D is the radical of a principal ideal [8, Theorem 2.2] (note that t-dim(D) = 1 means that every prime t-ideal of D has height-one). In [4, Theorem], it was shown that each saturated multiplicative subset S of D is a splitting set (and hence S = N (N (S))) if and only if D is a weakly factorial domain (i.e., each nonzero nonunit of D is a product of primary elements). Proposition 2.5. An integral domain D is a GWFD if and only if S = N (N (S)) for each saturated multiplicative subset S of D. Proof. (⇒) Let S be a saturated multiplicative subset of a GWFD D. Then S = D \ ∪P√α , where the Pα ’s are height-one primes of D since t-dimD = 1. Let each Pα = xα D, and let T = {xα }. Then S ⊆ N (T ) since (xα , s)v = D for all xα and s ∈ S. Also, N (T ) ⊆ S; for if z ∈ N (T ), then (z, xα )v = D for all xα , and hence z ∈ D \ ∪Pα = S. Thus S = N (T ), and hence N (N (S)) = N (N (N (T ))) = N (T ) = S. (⇐) We ﬁrst show that t-dim(D) = 1. For this, it suﬃces to show that every prime ideal minimal over a principal ideal is a maximal t-ideal. Let 0 = x ∈ D, and let P be a prime ideal of D minimal over xD (hence P is a t-ideal). Then S = D \ P is a saturated multiplicative subset of D, and hence S = N (N (S)).

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Clearly N (S) U (D) because S = D∗ . Let a ∈ N (S) \ U (D). Then a ∈ P and for each b ∈ S, √ (a, b)v = D, which implies √ that P is a maximal t-ideal. Next we show that P = aD. Assume that P = aD. Then there is another prime ideal Q of D minimal over aD, and thus for c ∈ Q \ P , D = (a, c)v ⊆ Qt . But this contradicts the fact that √ Q is a t-ideal (note that Q is minimal over aD, and thus a t-ideal). Hence P = aD, and thus D is a GWFD [8, Theorem 2.2]. Remark 2.6. (a) The proof of Proposition 2.5 shows that D is a GWFD if and only if S = N (N (S)) for each S = D \ P , where P is a prime t-ideal of D. (b) Let P be a prime t-ideal of an integral domain D and S = D \ P . Then either N (N (S)) = S or N (N (S)) = D∗ . To see this, suppose that S N (N (S)). Then choose a ∈ N (N (S)) \ S; so a ∈ P . Then (a, b)v = D for all b ∈ N (S). Thus N (S) ⊆ D \ P = S, and hence N (S) = U (D) by Proposition 2.3(2). Thus N (N (S)) = D∗ . We end this section with a generalization of Proposition 2.5. Proposition 2.7. Let P be a prime t-ideal of an integral domain D and S = D \ P . Then N (N (S)) = S if and only if P is a maximal t-ideal and there is an x ∈ D such that P is the unique maximal t-ideal containing x. Proof. (⇒) Suppose that N (N (S)) = S. We ﬁrst show that P is a maximal t-ideal of D. Assume that P is not a maximal t-ideal, and let P Q be a prime t-ideal of D. Choose a ∈ Q \ P . Then a ∈ S and (a, b)v ⊆ Q for all b ∈ P ; so N (S) = U (D) since N (S) \ U (D) ⊆ P by Proposition 2.3(2). Hence N (N (S)) = D∗ . Thus if N (N (S)) = S, then P is a maximal t-ideal. We next show that there is an x ∈ D such that P is the unique maximal t-ideal containing x. Since N (N (S)) = S = D∗ because P is nonzero, N (S) = U (D), and hence N (S) ∩ P = ∅ by Proposition 2.3(2). Thus for any x ∈ N (S) ∩ P , P is the unique maximal t-ideal containing x. (⇐) Since P is the unique maximal t-ideal of D containing x, (x, s)v = D for all s ∈ S; so x ∈ N (S). Thus as x is a nonunit of D, x ∈ N (N (S)) by Proposition 2.3(2), and hence N (N (S)) = S by Remark 2.6 (b).

3

The Relationship Between NA (S) and NB (S)

In this section, we investigate the relationship between NA (S) and NB (S), where A ⊆ B is an extension of integral domains and ∅ = S ⊆ A∗ . In general, there is no containment between the two sets. For example, let A = Z ⊆ B = Z[X] and S = U (Z). Then NA (S) = Z∗ and NB (S) = Z[X]∗ ; so NB (S) NA (S). Conversely, let A = Q[X, Y ] ⊆ B = Q[X, Y /X] and S = X. Then Y ∈ NA (S), but Y ∈ NB (S). Thus NA (S) NB (S). Let A ⊆ B be an extension of integral domains and ∅ = S ⊆ A∗ . Then NA (S) ⊆ NB (S) if and only if xB ∩ sB = xsB for all s ∈ S whenever xA ∩ sA = xsA for x ∈ A and all s ∈ S. In particular, NA (S) ⊆ NB (S) when B is LCM-stable over A (recall from [13] that B is LCM-stable over A if (xA ∩ yA)B = xB ∩ yB for all x, y ∈ A). A ﬂat extension of integral domains is clearly LCM-stable. An extension A ⊆ B of integral domains is said to be R2 -stable if aA ∩ bA = cA with a, b, c ∈ A implies aB ∩ bB = cB. It is clear that an LCM-stable extension is R2 -stable and

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that if A ⊆ B is an R2 -stable extension, then NA (S) ⊆ NB (S) for any ∅ = S ⊆ A∗ . But NA (S) ⊆ NB (S) does not imply that A ⊆ B is an R2 -stable extension (for example, let S = U (B) ∩ A). We have already observed that if A ⊆ B is a ﬂat extension of integral domains and ∅ = S ⊆ A∗ , then NA (S) ⊆ NB (S). If in addition, A ⊆ B is faithfully ﬂat, then NB (S) ∩ A = NA (S). This need not hold for a ﬂat extension. For example, let A = Z ⊆ B = Z[ 12 ] and S = 2. Then AS = B, NB (S) = B ∗ , and NA (S) = Z \ 2Z by Example 2.4(a); so NA (S) NB (S) ∩ A. Let A ⊆ B be an R2 -stable extension of integral domains with B an overring of A and S a saturated multiplicative subset of A. In [9, Theorem 1], the authors showed that if S is a splitting (resp., lcm splitting) set in A, then the saturation of S in B is a splitting (resp., lcm splitting) set in B. (Recall that a splitting set S of D is said to be an lcm splitting set if sD ∩ dD is principal for all s ∈ S and d ∈ D, equivalently, for each s1 , s2 ∈ S, s1 D ∩ s2 D = sD for some s ∈ D [1, Proposition 2.4].) The proof of [9, Theorem 1] also proves the following. Proposition 3.1. Let B be an overring of an integral domain A, S a saturated multiplicative subset of A, and S the saturation of S in B. If NA (S) ⊆ NB (S) and S is a splitting (resp., lcm splitting) set in A, then S is a splitting (resp., lcm splitting) set in B and S = {w ss12 |s1 , s2 ∈ S with ss12 ∈ B and w ∈ U (B)}. Corollary 3.2. (cf. [9, Proposition 1]) Let B be an overring of an integral domain A and p a prime element of A such that ∩pn A = {0} and pB B. Then p is a prime element of B if and only if B ∩ pApA is a maximal t-ideal of B. Proof. (⇒) Assume that p is a prime element of B. Then ApA ⊆ BpB , and thus ApA = BpB because ApA is a local PID. Hence pB = B ∩ pApA , and thus B ∩ pApA is a maximal t-ideal of B (cf. [11, Proposition 1.3]). (⇐) Assume that B ∩ pApA is a maximal t-ideal of B. Let S = {upn |u ∈ U (A) and n ≥ 0}; then NA (S) ⊆ NB (S). For if t ∈ NA (S), then t ∈ B ∩ pApA , and hence for all upn ∈ S, ((t, upn )B)v = B because B ∩ pApA is the unique prime t-ideal of B containing upn . By Proposition 3.1, S = {wpn |w ∈ U (B) and n ≥ 0} and S is a splitting set in B. Thus one can easily verify that p is also prime in B (cf. [2, Corollary 1.4(c)]). We next consider the case of an integral domain of the form T = K + M , where M is a nonzero maximal ideal of T and K is a subﬁeld of T . For D a subring of K, R = D + M is then a subring of T with qf (R) = qf (T ). This construction has proved very useful for constructing examples. In this case, we can relate ND (S) and NR (S) when ∅ = S ⊆ D∗ (cf. [7, Lemma 3.1] for Corollary 3.5). One can easily verify that ND (S) ⊆ NR (S) and NR (S) ∩ D = ND (S); this also follows since D ⊆ R is a faithfully ﬂat extension. Proposition 3.3. Let T = K + M and R = D + M be integral domains, where M is a nonzero maximal ideal of T , K is a subﬁeld of T , and D is a subring of K; and let ∅ = S ⊆ D∗ . 1. If S ⊆ U (D), then NR (S) = R∗ and SNR (S) = R∗ . 2. If S U (D), then NR (S) = ND (S) + M and SNR (S) = SND (S) + M .

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Proof. (1) This is clear. (2) We show that NR (S) = ND (S) + M ; the second equality is then clear since SM = M . “ ⊆ ” : Let d ∈ S \ U (D). We ﬁrst show that NR (S) ⊆ R \ M . Suppose that m ∈ NR (S) ∩ M . Then mR ⊆ M ⊆ dR. Thus mR = mR ∩ dR = mdR yields dR = R, and hence d ∈ U (R) ∩ D = U (D), a contradiction. Now let z ∈ NR (S). Then z = x + m for some x ∈ D∗ and m ∈ M by above. We show that x ∈ ND (S). Let s ∈ S. Then zR ∩ sR = zsR yields xD ∩ sD = xsD, and hence x ∈ ND (S). Thus z = x + m ∈ ND (S) + M . “ ⊇ ” : Let z = x+m, where x ∈ ND (S) and m ∈ M . Suppose that z(d1 +m1 ) = s(d2 + m2 ) for some d1 , d2 ∈ D, s ∈ S, and m1 , m2 ∈ M . Then xd1 = sd2 , and hence d1 = sd3 for some d3 ∈ D since x ∈ ND (S). Thus z(d1 +m1 ) = z(sd3 +m1 ) = zs(d3 + ms1 ) with ms1 ∈ M . Hence zR ∩ sR = zsR, and thus z ∈ NR (S). (For this inclusion, we did not need that S contains a nonunit of D.) Corollary 3.4. Let T = K + M and R = D + M be integral domains, where M is a nonzero maximal ideal of T , K is a subﬁeld of T , and D is a subring of K; and let ∅ = S ⊆ D∗ . Then S is a splitting set in R if and only if S = U (R) (so S = U (D) = U (R)). Proof. (⇒) Suppose that S is a splitting set of R. Then S ⊆ U (D) by Proposition 3.3. Since S is saturated in R, we must have S = U (R) = U (D). (⇐) U (R) is always a splitting set. Corollary 3.5. Let T = K + M and R = D + M be integral domains, where M is a nonzero maximal ideal of T , K is a subﬁeld of T , and D is a subring of K; and let S be a saturated multiplicative subset of D∗ which contains a nonunit. Then S is a splitting set of D if and only if SNR (S) = R \ M . Proof. This follows directly from Proposition 3.3(2) since D∗ + M = R \ M . Let X be an indeterminate over an integral domain D. As usual, for f ∈ qf (D)[X], Af is the fractional ideal of D generated by the coeﬃcients of f . One can easily verify that ND (S) ⊆ ND[X] (S) and ND[X] (S) ∩ D = ND (S) for each ∅ = S ⊆ D∗ ; this also follows since D ⊆ D[X] is a faithfully ﬂat extension. We next explicitly determine ND[X] (S). Proposition 3.6. Let D be an integral domain, X an indeterminate over D, and ∅ = S ⊆ D∗ . Then ND[X] (S) = {0 = f ∈ D[X]|(Af , s)v = D for all s ∈ S}. Proof. “ ⊆ ” : Let f ∈ ND[X] (S). Then (f, s)v = D[X] for all s ∈ S, and thus (Af , s)v = D because D[X] = (f, s)v ⊆ ((Af , s)D[X])v = (Af , s)v [X] ⊆ D[X] by [12, Proposition 2.2]. “ ⊇ ” : Let 0 = g ∈ D[X] such that (Ag , s)v = D for all s ∈ S. Suppose that (g, s)v D[X] for some s ∈ S, and let Q be a maximal t-ideal of D[X] containing (g, s)v . Then Q = (Q ∩ D)[X] and Q ∩ D is a t-ideal of D [11, Proposition 1.1], and hence D = (Ag , s)v ⊆ Q ∩ D, a contradiction. Thus (g, s)v = D[X] for all s ∈ S, and hence g ∈ ND[X] (S).

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Corollary 3.7. ([5, Theorem 2.2]) Let D be an integral domain, X an indeterminate over D, and ∅ = S ⊆ D∗ . Then S is a splitting set in D[X] if and only if S is an lcm splitting set in D. Moreover, S is an lcm splitting set in D[X] if and only if S is a splitting set in D[X]. Proof. (⇒) Suppose that S is a splitting set in D[X]. Let a, b ∈ S and f = a + bX. Since S is a splitting set in D[X], by Proposition 3.6, there are an s ∈ S and g ∈ D[X] with (Ag , s )v = D for all s ∈ S such that f = sg, and so s|a and s|b in D. Since a, b ∈ S and S is saturated, we have as , sb ∈ S, and thus D = (Ag , as )v = ( as , sb )v ; so (a, b)v = sD. Thus S is an lcm splitting set in D [1, Proposition 2.4]. (⇐) Suppose that S is an lcm splitting set in D. Let 0 = f = a0 + a1 X + · · · + an X n ∈ D[X]. Since S is a splitting set in D, each ai = si ti for some si ∈ S and ti ∈ D with (ti , s )v = D for all s ∈ S. Also, since S is an lcm splitting set, (s0 , . . . , sn )v = sD for some s ∈ S (cf. [1, Proposition 2.4]). Let si = ssi for si ∈ D and let f = sg for g ∈ D[X]. Then (Ag , s )v = (s0 t0 , . . . , sn tn , s )v = ((s0 , . . . , sn , s )(t0 , . . . , tn , 1))v = (s0 , . . . , sn , s )v = D [1, Lemma 3.1], and thus g ∈ ND[X] (S) by Proposition 3.6. This implies that D[X] \ {0} = SND[X] (S), and so S is a splitting set in D[X]. The “moreover” statement follows from [1, Proposition 2.4] since s1 D ∩ s2 D = sD with s, s1 , s2 ∈ D if and only if s1 D[X] ∩ s2 D[X] = sD[X]. Acknowledgement The research on this chapter was started while the second author visited the University of Tennessee at Knoxville during 2001-2002. He would like to thank the University of Tennessee for its ﬁnancial support and hospitality.

Bibliography [1] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), 17–37. [2] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), 78–93. [3] D.D. Anderson, T. Dumitrescu, and M. Zafrullah, Almost splitting sets and AGCD domains, Comm. Algebra 32 (2004), 147–158. [4] D.D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), 907–913. [5] D.D. Anderson and M. Zafrullah, Splitting sets in integral domains, Proc. Amer. Math. Soc. 129 (2001), 2209–2217. [6] D.F. Anderson, The class group and local class group of an integral domain, in Non-Noetherian Ring Theory, Math. Appl., Kluwer Acad. Publ., Dordrecht 520 (2000), 33–55. [7] D.F. Anderson and G.W. Chang, The class group of integral domains, J. Algebra 264 (2003), 535–552. [8] D.F. Anderson, G.W. Chang, and J. Park, Genealized weakly factorial domains, Houston J. Math. 29 (2003), 1–13. [9] T. Dumitrescu and M. Zafrullah, Lcm-splitting sets in some ring extensions, Proc. Amer. Math. Soc. 130 (2002), 1639–1644. [10] R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972. [11] E. Houston and M. Zafrullah, On t-invertibility, II, Comm. Algebra 17 (1989), 1955–1969. [12] B.G. Kang, Pr¨ ufer v-multiplication domains and the ring R[X]Nv , J. Algebra 123 (1989), 151–170. [13] H. Uda, LCM-stableness in ring extensions, Hiroshima Math. J. 13 (1983), 357–377.

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Chapter 10

Some Remarks on Inﬁnite Products by

Jim Coykendall Abstract In this chapter we will look at some examples of inﬁnite product representations of elements of a commutative ring. Some obstructions to a good general theory of inﬁnite products are explored.

1

Introduction

In the spirit of the recent explosion of interest in the theory of factorization, we wish to consider the possibility of the notion of inﬁnite products. Certainly, notions of inﬁnite products (and sums) have proved quite useful in applied mathematics and analysis as well as some branches of number theory. The usefulness of the notion of inﬁnite products makes considering the possibility of inﬁnite products a “moral imperative” in the setting of general commutative rings. In [3] the present author encountered an application of inﬁnite products in the realm of formal power series. Although the inﬁnite product considered in [3] was mostly a formal notational convention, a similar notion of an inﬁnite product for formal power series is studied slightly more extensively by B. Kang and M. Park in [4]. The context in which the authors of [4] and [3] utilized inﬁnite products did not demand a full formal development of this notion (in the chapter [4] the behavior of the prime spectrum of a power series ring over nondiscrete valuation domains is investigated, and in [3] the dimension theory and SFT stability of power series rings over SFT rings are investigated). The aim of this chapter is to illustrate, via some examples, some possible implications of inﬁnite products and we will show various obstructions to the development of a “clean” general theory. This work will mostly focus on examples, some proposed axioms for general inﬁnite products, some ﬁrst consequences of such a theory, and directions for further study. 188

Some Remarks on Inﬁnite Products

2

189

A Motivating Example

The paper [1] was a milestone in the study of dimension theory of formal power series rings. A classical result in the study of Krull dimension behavior in commutative rings is that if R is a ring (commutative with identity) such that dim(R) = n then n + 1 ≤ dim(R[x]) ≤ 2n + 1. In [1] a near-Noetherian property termed the SFTproperty (for “strong ﬁnite type”) was introduced. We brieﬂy recall that an ideal I ⊆ R is SFT if there is a ﬁnitely generated ideal B ⊆ I and a ﬁxed integer N such that xN ∈ B for all x ∈ I. A ring, R, is SFT if all its (prime) ideals have the SFT property. As it turns out, this property is central in the dimension theory of formal power series rings. In [1] it is shown that the SFT property is necessary for the dimension of dim(R[[x]]) to be ﬁnite. In [3] the problem of determining if the SFT property (in addition to the property that dim(R) < ∞) is suﬃcient for dim(R[[x]]) to be ﬁnite as well as the question as to whether the SFT property is stable with respect to the adjunction of a power series variable (i.e. if R is SFT then is R[[x]] SFT?) are explored. These questions are both answered in the negative in [3]. In showing that the SFT property is not necessarily preserved in power series extensions, the notion of inﬁnite products arises in a natural fashion (it should also be noted that the investigation of the prime spectra of power series rings over nondiscrete valuation domains in [4] gives rise to the notion of an inﬁnite product in a distinct, but similiar context). As a starting point we will look at a slightly adjusted version of an example that appears in [3]. Example 2.1. Let V be a one-dimensional nondiscrete valuation domain with value group R and residue ﬁeld F2 . For convenience of computation we will write V := F2 [x; R+ ]M where F2 [x; R+ ] is the monoid ring over F2 with respect to R and M ⊆ F2 [x; R+ ] is the maximal ideal generated by the set {xα }α∈R+ . Let {αi }∞ i=0 be a countable collection of elements from R+ . We deﬁne P∞ ∞ x( i=0 αi ) , if ∞ αi converges; αi x = i=0 ∞ 0, if i=0 αi diverges. i=0 It is easy to verify that this deﬁnition is a well-deﬁned notion of an inﬁnite product of any countable collection of powers of the element x ∈ V . We brieﬂy remark here that we are being a bit sloppy about associativity, and indeed, a more formal notion of associativity would require the notion of “inﬁnite” regrouping of parentheses. What we mean by default is that given the inﬁnite product in R ∞

πi

i=0

there is a well-deﬁned, deterministic way to associate an element in R. In this ∞ ∞ example, we associate with i=0 xα , the real, positive term-series i i=0 αi . We will deal with associativity a bit more carefully later, but we must ﬁrst make a deﬁnition.

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Definition 2.2. Let V be a valuation domain with value group G and let x ∈ V . A + αi subset {αi }∞ is a well-deﬁned i=0 ⊆ G is called admissible (with respect to x) if x element of V for all i. The motivation for this deﬁnition can be found in the previous example. Indeed, the element x is such that xα ∈ V for all α ∈ R+ , but if u is a unit in V , then there is no guarantee of the existence of an arbitrary element of the form (ux)α . With this in mind we make a ﬁnal preliminary deﬁnition. Definition 2.3. Let V be a valuation domain with value group G and maximal ideal M. We say that the element x ∈ M is a pseudo-uniformizer if xg is deﬁned for all g ∈ G+ . Basically, this deﬁnition says that if x is a pseudo-uniformizer, then all subsets of G+ are admissible. In constructive examples of valuation domains, pseudouniformizers abound (e.g., in Example 2.1, the element x is a pseudo-uniformizer). We now pause to underscore a couple of observations about the behavior of the inﬁnite product in the more general setting of (1−dimensional) valuation domains. We point out here that the “inﬁnite product” to which the following proposition refers is the natural generalization of the inﬁnite product that appeared in Example 2.1. That is to say, we declare (for a pseudo-uniformizer x and a subset {αi }∞ i=0 ⊆ G+ ) that

∞ i=0

xαi

⎧ P∞ ( αi ) ⎪ ⎨x i=0 , = 0, ⎪ ⎩ undeﬁned,

∞ if i=0 αi converges to an element of G+ ; if ∞ i=0 αi = ∞; in all other cases.

Proposition 2.4. Let V be a one-dimensional valuation domain with value group G ⊆ R, valuation v and maximal ideal M. If x∈ M is a pseudo-uniformizer, then ∞ the following properties of the inﬁnite product i=0 xαi hold. αi + is deﬁned for all subsets {αi }∞ if and a) The inﬁnite product ∞ i=0 ⊆ G i=0 x only if G is isomorphic to either Z or R. ∞ ∞ b) If σ is any permutation of N {0} then i=0 xαi = i=0 xασ(i) . ∞ ∞ ∞ ∞ c) i=0 xβi divides i=0 xαi if and only if i=0 βi ≤ i=0 αi . ∞ + d ) i=0 xαi = 0 for all subsets {αi }∞ i=0 ⊆ G if and only if V is discrete. e) If G ∼ = R, then any element of M may be represented as a nontrivial inﬁnite product. Proof. We ﬁrst ∞show a) and d) in tandem. If G is isomorphic to Z , then consider the product i=0 xαi . The set {αi }∞ i=0 contains a least element since it is a subset of N and without loss of generality, we will let α0 be aleast element. Clearly since αi α diverges and hence ∞ = 0. (It should also every αi ≥ α0 , the series ∞ i i=0 i=0 x + ∼ be noted that since G = Z every subset of G is admissible with respect to y for any element y ∈ M.) If G ∼ = R then since x is a pseudo-uniformizer, then xα exists + for all α ∈ R . In particular, the product

Some Remarks on Inﬁnite Products

191

∞

xαi

i=0

∞

P∞

always exists; it is 0 if the series i=0 αi diverges and is x( i=0 αi ) if the series converges. We now complete the proof Assume that V is nondiscrete but that every of d). αi + x is 0 for all {αi }∞ inﬁnite product of the form ∞ i=0 ⊆ G . We note that since i=0 + x is a pseudo-uniformizer, every subset of G is admissible. Note that since V is nondiscrete, we can inductively choose ξn ∈ M such that v(ξn ) ≤

v(x) . 2n

We now let α0 = v(x) and inductively αn = v(ξn ) for all n ≥ 1. Consider the inﬁnite product ∞

xαi .

i=0

Note that since αn ≤

v(x) 2n ,

we have that

∞

∞

i=0

αi ≤ v(x)

∞

2−n = 2v(x).

i=0

Hence i= αiconverges (say, to β). One of two things occurs at this juncture. If αi αi = xβ = 0, and if β ∈ / G+ then the inﬁnite product ∞ β ∈ G+ then ∞ i=0 x i=0 x is not deﬁned. In either case, the contradiction ﬁnishes the proof of d). To ﬁnish the proof of a), we note that if V is nondiscrete and G is a proper subgroup of R then it is well-known that the closure of G (as a subset of R with the standard topology) is all of R. Select an element b ∈ R \ G+ and an increasing + (again with respect to the ordering on R) sequence {an }∞ n=0 of elements of G such that {an } −→ b. Now deﬁne α0 = a0 and for all n ≥ 1, deﬁne αn = an − an−1 . Note that each αn is positive and since x is a pseudo-uniformizer, xαn is a well-deﬁned element of V for all n. Now consider the inﬁnite product ∞

xαn .

n=0

Since, by construction, the inﬁnite sum converges to an element b ∈ / G, we have our desired contradiction. ∞ For part b), note that since each αi is positive, if the series i=0 αi converges, then it converges absolutely. Hence any rearrangement of the series converges to the same sum. More precisely, if σ is a permutation of N {0} then

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∞

αi =

∞

i=0

ασ(i)

i=0

and hence the inﬁnite products coincide. If the series diverges, then the same argument applies. βi αi divides ∞ (we are tacitly asFor part c) we ﬁrst assume that ∞ i=0 x i=0 x suming here that both inﬁnite products exist). So there is a w ∈ V such that ∞ ∞ w i=0 xβi = i=0 xαi . Using the deﬁnition of the inﬁnite product and computing values we get that v(w) +

∞

βi =

i=0

∞

αi

i=0

(note that if the right sum diverges, and if the left diverges then so ∞ we are done ∞ does the right). This gives that i=0 βi ≤ i=0 αi (in either case, convergence or divergence). ∞ ∞ ∞ Conversely, that i=0 βi ≤ i=0 αi . Note that if i=0 βi diverges, suppose ∞ then so does i=0 αi and we will discard this case as degenerate (we will declare ∞ that 0 divides 0 for brevity). Assume that i=0 βi converges and that its sum is b. ∞ If i=0 αi diverges, then we are done, so we will assume that this sum converges to a. Since b ≤ a (and xb and xa are deﬁned since x is a pseudo-uniformizer) then xb must divide xa and we are done. Finally, for part e) assume that m ∈ M. If m = 0, then we can represent m ∼ as the inﬁnite product ∞ i=0 x. If m = 0 then since G = R, select a positive term ∞ ∞ v(m) inﬁnite series i=0 αi such that i=0 αi = v(x) and note that ∞

xαi = x(

P∞

i=0

αi )

= a ∈ V.

i=0

So v(a) = v(m), and hence the proof is complete.

m a

is a unit in V . So we have m = ( m a)

∞ i=0

xαi and

Given an inﬁnite product in a commutative ring R with 1, one can extend this product in a nontrivial way to the power series ring R[[x]] as follows. Example 2.5. Suppose that {ri }∞ i=0 is a subset of a commutative ring with identity, R, such that for any subset of S ⊆ N {0} the inﬁnite product

ri

i∈S

is an element of R. Then the inﬁnite product ∞ (ri + si xi+1 ) i=0

where each si ∈ R is an element of R[[x]].

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193

By inﬁnite product, we produce a power series via the following algorithm. The constant term is the inﬁnite product of the ri ’s (which is well-deﬁned by assumption). The coeﬃcient of xn is determined via: ((si1 si2 · · · sik )( rj )). i1 +i2 +···+ik =n

j =i1 ,i2 ,··· ,ik

Note that any term in the sum exists since subproducts of the form i∈S ri exist. Also the sums are ﬁnite since the powers of the polynomial terms are increasing. This example is in the spirit of the previous example where inﬁnite products over valuation domains are considered. In the previous example (because of absolute convergence of convergent positive term series) every subproduct of a well-deﬁned inﬁnite product exists. Also note that this product can be done in more generality. In particular, the terms in the successive polynomials merely have to satisfy the property that for any ﬁxed degree, n, there are only ﬁnitely many polynomials of the form r + sxn in the inﬁnite product.

3

Hazards

In the previous sections the approach to the inﬁnite product was more informal. Basically, in all of the results, an algorithm was described to deﬁne an inﬁnite product representation of some elements of a commutative ring with identity. In this section we will look more closely at some of the obstructions to a general notion of inﬁnite products. In particular, for inﬁnite products, some problems with associativity and (“inﬁnite”) commutativity naturally arise. We will begin by deﬁning axioms for an inﬁnite product. All products here will be countable. Axioms for the infinite product. Let {ri }∞ i=0 be a subset of a commutative ring, R, and assume that the inﬁnite ∞ product i=0 ri exists. ∞ 1. If any ri = 0 then i=0 ri = 0. ∞ 2. If R has an identity, then i=0 1 = 1. 3. If {n1 , n2 , · · · } and {m1 , m2 · · · } are two partitions of N {0}, then (

n1

i=0

ri )(

n2

i=n1 +1

ri ) · · · = (

m1

i=0

ri )(

m2

ri ) · · · .

i=m1 +1

∞ 4. Any ﬁnite rearrangement of i=0 ri preserves the product; that is,if σ is a ∞ permutation of N {0} that ﬁxes all but ﬁnitely many ri ’s, then i=0 ri = ∞ i=0 rσ(i) . We note the third axiom is associativity in the inﬁnite case. One would also like for any subproduct to exist and for the last axiom to extend to arbitrary permutations (certainly one would wish for this when studying commutative rings), but we will presently see some problems with this.

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Example 3.1. Let V be a 1−dimensional valuation domain with value group R from Example 2.1. We will show that it is impossible to extend the “natural” inﬁnite product to the quotient ﬁeld, K, of V without incurring some strange diﬃculties. Indeed consider the following set of elements of K 1

1

{x, x(− 2 ) , x( 3 ) , · · · , x((−1)

n−1 1 n)

, · · · }.

Extending the deﬁnition of inﬁnite product to K gives that xln(2) ∈ V . However, the inﬁnite series ∞

(−1)i−1

i=1

∞ i=1

x((−1)

i−1 1 i)

=

1 i

is conditionally convergent. Hence, given any real number α (including α = ±∞), there is a rearrangement of the series that converges to α (one can also rearrange the series so as to diverge without making it approach ±∞). Hence there are rearrangements of this product that take on diﬀerent values (as well as being undeﬁned). The previous example shows that one cannot hope to extend this inﬁnite product in a meaningful way to powers of x in the quotient ﬁeld. The key was that inﬁnite products of powers of x in V can be rearranged in any fashion since any inﬁnite series taken from the positive elements of R either converges absolutely or diverges. This example can be extended to nondiscrete valuation domains in general. We now relay an example that is perhaps even more disturbing and shows that the algorithm for deﬁning an inﬁnite product must be carefully deﬁned. Example 3.2. Consider V to be the valuation domain from Example 2.1. We consider the inﬁnite product (deﬁned as in Example 2.5) in V [[t]]: 1

1

n

1

(x + t)(x 2 + t2 )(x 4 + t4 ) · · · (x 2n + t2 ) · · · . As we saw in Example 2.5, this is a well-deﬁned element of V [[t]] (as written, with the algorithm supplied there). But note that n

1

1

n

(x 2n + t2 ) = (x 22n + t)2 since char(V ) = 2. If we write the inﬁnite product 1

1

1

n

(x + t)(x 4 + t)2 (x 16 + t)4 · · · (x 22n + t)2 · · · then we see that this is not associative. In fact, grouping them in the original way gives a well-deﬁned power series, but taking the new factors “one at a time” is not even deﬁned since every term in the product has a linear term. We now produce a couple of general “obstruction” results. Theorem 3.3. Let R be a domain. If {p1 , p2 , · · · } are (not necessarily distinct) primes of R such that the inﬁnite product ∞ i=1

pi = a ∈ R

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195

then there is a localization of T ⊇ R in which there is an inﬁnite product of units that is a nonunit. ∞ Proof. Let i=1 pi = a ∈ R and consider the multiplicative set, S, generated by the set of primes P := {p1 , p2 , · · · }. Note that the ideal (a) S = ∅. Indeed, if mk 1 ra = pm then there are two cases to consider. If there exists p ∈ P distinct 1 · · · pk mk 1 which is a contradiction. If from pi , 1 ≤ i ≤ k then p divides a but not pm 1 · · · pk p1 , p2 , · · · , pk is an exhaustive list of distinct primes from P , then at least one of them (say p1 without loss of generality) must occur inﬁnitely often as a divisor of mk 1 +1 1 divides a, but not pm a. In particular, pm 1 1 · · · pk . In the localization RS , the ideal (a) survives (so a is a nonunit), but note that in RS each pi is a unit. Hence the nonunit a is an inﬁnite product of units. We now point out a positive result. Proposition 3.4. Any Boolean ring admits an arbitrary, nontrivial commutative inﬁnite product. Proof. We ﬁrst note that for an arbitrary index set, I, the ring F2 i∈I

admits a nontrivial inﬁnite product via ∞

(j)

{ai }i∈I = {bi }i∈I

j=0 (j)

where {ai }i∈I ∈

F2 and 1 if the ith term of {ai }(j) = 1 for all j bi = 0 otherwise. i∈I

The fact that any Boolean ring admits a nontrivial inﬁnite product follows from the fact that any Boolean ring is a subring of a direct product of the form i∈I F2 [2]. Proposition 3.5. With the exception of F2 , no ﬁeld admits a nontrivial general commutative inﬁnite product. Proof. Certainly F2 admits an arbitrary commutative inﬁnite product (as a special case of the previous result). So assume that K = F2 . If K admits an arbitrary inﬁnite product, then we select an x ∈ K such that x is neither 0 nor 1. Consider the product ∞

x := y

i=i

and note that xy = y. Since K is a ﬁeld and x = 1, y must be zero. In particular, the inﬁnite product of any nonzero, nonidentity element is 0. We now consider the inﬁnite product

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∞

x−1 := z.

i=i

Since we are assuming that the product is arbitrarily commutative (i.e., any rearrangement preserves an inﬁnite product), then yz = 1 by construction, since ∞ j=0 1 = 1. But y = 0 and this is our contradiction. The main theme to glean from these examples and results is that the hope of creating a general inﬁnite product must be handled with care. Even for the fairly straightforward examples that were produced for 1−dimensional valuation domains, we saw that extending the notion of inﬁnite products to the quotient ﬁeld induced some noncommutative (albeit, “inﬁnite” noncommutative) behavior when we allowed inﬁnite rearrangements of inﬁnite products. Care must also be taken to avoid problems with associativity as was pointed out in Example 3.2. Of course, in practice, inﬁnite products have been used in an ad hoc way to produce results like the ones in [4] and [3], but the focus in those papers was not to create a general inﬁnite product, but to use speciﬁc inﬁnite product representations as a tool to prove theorems. The question remains as to the nature of general inﬁnite products. In particular, it would be nice to classify all rings with a general commutative inﬁnite product (of course one could also include nil rings, which are a distinct class of rings from the Boolean rings). It would also be interesting to quantify the nature of the natural noncommutativity that arises when one considers the inﬁnite product deﬁned earlier for the quotient ﬁeld of a nondiscrete valuation domain. Finally, it would be good to understand the closure of a domain with respect to an inﬁnite product. More precisely, if V is a 1−dimensional nondiscrete valuation domain with value group a proper subgroup of R, then is the closure of V with respect to our deﬁned inﬁnite products the larger valuation domain with value group R? The answer is “yes” for the constructed Example 2.1. Additionally in this spirit, what is the structure of the domain obtained by adjoining all welldeﬁned inﬁnite products to the ring V [[t]] from Example 2.5? We believe that the answers to these questions could be quite interesting and provoke interesting further directions in factorization theory.

Bibliography [1] J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973), 299–304. [2] N. Bourbaki, Algebra I. Chapters 1–3, Translated from the French, Reprint of the 1989 English translation, Springer, Berlin, 1998. [3] J. Coykendall, The SFT property does not imply ﬁnite dimension for power series rings, J. Algebra 256 (2002), no. 1, 85–96. [4] B. G. Kang and M. H. Park, A localization of a power series ring over a valuation domain, J. Pure Appl. Algebra 140(2) (1999), 107–124.

197

Chapter 11

Rings with Prime Nilradical by

Ayman Badawi and Thomas G. Lucas Abstract A commutative ring R is said to be a φ-ring if its nilradical N il(R) is both prime and divided, the latter meaning N il(R) is comparable with each principal ideal of R. Special types include φ-Noetherian (also known as nonnilNoetherian), φ-Mori, φ-chained and φ-Pr¨ ufer. A ring R is φ-Noetherian if N il(R) is a divided prime and each ideal that properly contains N il(R) is ﬁnitely generated. If R is a φ-Noetherian ring and X1 , X2 , . . . , Xn are indeterminates, then an ideal I of R[X1 , X2 , . . . , Xn ] which contains a nonnil element of R is ﬁnitely generated. Also, for a ring R where N il(R) is a nonzero prime ideal with N il(R)2 = (0), there is a ring A whose nilradical N il(A) is a divided prime such that R embeds naturally in A with R/N il(R) isomorphic to A/N il(A), RNil(R) isomorphic to ANil(A) , and the corresponding total quotient rings, T (R) and T (A), are such that T (R) ⊂ T (A) and A ∩ T (R) = R + N il(T (R)).

1

Introduction

We assume throughout that all rings are commutative with 1 = 0. For such a ring R, we let Z(R) denote the set of zero divisors of R and N il(R) denote the nilradical. We say that N il(R) is divided if it compares with each principal ideal of R (see [9] and [2]). If N il(R) is both divided and a prime ideal, we say that R is a φ-ring. For convenience we let H denote the class of all φ-rings. In [2], [3], [4], [5], and [6] the ﬁrst-named author investigated this class of rings. In [2] and [4] he introduced the concepts of φ-pseudo-valuation rings and φ-chained rings. Also, D.F. Anderson and the ﬁrst-named author made further investigation on the class H in [1] and introduced the concepts of φ-Pr¨ ufer rings and φ-B´ezout rings. See Section 4 for deﬁnitions of these speciﬁc types of φ-rings. The “φ” in the name refers to the canonical map φ : T (R) → RN il(R) from T (R), the total quotient ring of R, to R localized at N il(R) which maps a fraction a/b ∈ T (R) to its image in RN il(R) . 198

Rings with Prime Nilradical

199

An ideal I of a ring R is said to be a nonnil ideal if it is not contained in N il(R). Recall from [5] that a ring R is called a nonnil-Noetherian ring if every nonnil ideal of R is ﬁnitely generated. To establish more consistency in nomenclature, we will refer to such rings as φ-Noetherian rings. In the ﬁrst section of this paper, we will show that many of the properties of Noetherian domains are valid for the nonnil ideals of a φ-Noetherian ring. In Section 3, we study polynomial rings with one or several variables over φ-Noetherian rings. For example, we show that if R is a φ-Noetherian ring and X1 , X2 , . . . , Xn are indeterminates, then an ideal I of R[X1 , X2 , . . . , Xn ] which contains a nonnil element of R is ﬁnitely generated. On the other hand, if N il(R) is not ﬁnitely generated, then X1 R[X1 , X2 , . . . , Xn ]+N il(R[X1 , X2 , . . . , Xn ]) is not ﬁnitely generated. Most, but not all, of our nondomain examples of φ-Noetherian rings are provided by the idealization construction R(+)B arising from a ring R and an R-module B as in Huckaba’s book [13, Chapter VI]. We recall this construction. For a ring R, let B be an R-module. Then the idealization of B over R is the ring R(+)B obtained from taking the product R × B and deﬁning addition and multiplication of elements (r, b) and (s, c) in R×B by (r, b)+(s, c) = (r+s, b+c) and (r, b)(s, c) = (rs, sb + rc). Under these deﬁnitions R(+)B becomes a commutative ring with identity. If R is reduced, N il(R(+)B) = (0)(+)B. If R is an integral domain, N il(R(+)B) is prime.

2

Properties of φ-Noetherian Rings

We start with the following characterization of φ-Noetherian rings. It is a combination of several results from [5]. Theorem 2.1. ([5, Corollary 2.3 and Theorems 2.2, 2.4 and 2.6]) Let R ∈ H. Then the following are equivalent. 1. R is a φ-Noetherian ring. 2. R/N il(R) is a Noetherian domain. 3. φ(R)/N il(φ(R)) is a Noetherian domain. 4. φ(R) is a φ-Noetherian ring. 5. Each nonnil prime ideal of R is ﬁnitely generated. In the following result, we show that a φ-Noetherian ring is related to a pullback of a Noetherian domain. Theorem 2.2. Let R ∈ H. Then R is a φ-Noetherian ring if and only if φ(R) is ring-isomorphic to a ring A obtained from the following pullback diagram: A −−−−→ ⏐ ⏐

S ⏐ ⏐

T −−−−→ T /M

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where T is a zero-dimensional quasilocal ring containing A with maximal ideal M , S = A/M is a Noetherian subring of T /M , the vertical arrows are the usual inclusion maps, and the horizontal arrows are the usual surjective maps. Proof. Suppose φ(R) is ring-isomorphic to a ring A obtained from the given diagram. By Theorem 2.1, it suﬃces to show that φ(R) is φ-Noetherian. Since T is a zero-dimensional quasilocal ring and M is a prime ideal of both T and A, A ∈ H with N il(A) = Z(A) = M . Since S = A/M is a Noetherian domain, A is a φ-Noetherian ring by Theorem 2.1. Thus R is a φ-Noetherian ring. Conversely, suppose that R is a φ-Noetherian ring. Then, letting T = RN il(R) , M = N il(RN il(R) ), and A = φ(R) yields the desired pullback diagram. A rather nice property of φ-Noetherian rings is that homomorphic images are either Noetherian or φ-Noetherian. Generally speaking this is an exclusive “or”, the exception being when the image is either an integral domain or a local Artinian ring. Lemma 2.3. Let R ∈ H. Then N il(R) is ﬁnitely generated if and only if R is either an integral domain or a local Artinian ring with nonzero maximal ideal, N il(R). In particular, if R is Noetherian and not an integral domain, then it is a local Artinian ring with maximal ideal N il(R) = (0). Proof. Obviously, N il(R) is ﬁnitely generated if R is either an integral domain or a local Artinian ring. Thus it suﬃces to prove that if N il(R) is ﬁnitely generated and not the maximal ideal of R, then R is an integral domain. Let M be a maximal ideal of R. Since N il(R) is a divided prime ideal of R, N il(R)RM = N il(RM ) is a divided prime of RM . Hence RM is also in H and N il(RM ) is ﬁnitely generated. Since N il(RM ) is a divided prime and properly contained in M RM , M N il(RM ) = N il(RM ). Thus N il(RM ) = (0) by Nakayama’s Lemma. As this happens for each maximal ideal of R, N il(R) must be (0) and we have that R is an integral domain. Proposition 2.4. Let R ∈ H be a φ-Noetherian ring and let I = R be an ideal of R. If I ⊂ N il(R), then R/I is a φ-Noetherian ring. If I ⊂ N il(R), then N il(R) ⊂ I and R/I is a Noetherian ring. Moreover, if N il(R) ⊂ I, then R/I is both Noetherian and φ-Noetherian if and only if I is either a prime ideal or a primary ideal whose radical is a maximal ideal. Proof. Suppose that I ⊂ N il(R). Then N il(R/I) = N il(R)/I is a divided prime ideal of R/I. Hence, R/I ⊂ H. Now, let Q be a nonnil prime ideal of R/I. Then Q = P/I for some prime ideal P ⊂ N il(R) of R. Since P is ﬁnitely generated, Q is ﬁnitely generated. Thus R/I is a φ-Noetherian ring by Theorem 2.1. Now, suppose that I ⊂ N il(R). Then N il(R) ⊂ I since N il(R) is divided. Let Q be a prime ideal of R/I. Then Q = P/I for some nonnil prime ideal P of R such that I ⊂ P . Hence Q is ﬁnitely generated since P is ﬁnitely generated. Thus R/I is a Noetherian ring since each prime ideal is ﬁnitely generated. The third statement follows from Lemma 2.3.

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Corollary 2.5. Let R ∈ H be a φ-Noetherian ring. Then a homomorphic image of R is either a φ-Noetherian ring or a Noetherian ring. An almost Dedekind domain that is not Dedekind is a ring that is locally Noetherian but not Noetherian. If a domain is locally Noetherian and each nonzero element is contained in at most ﬁnitely many maximal ideals, then the domain is Noetherian (see, for example, [14, Exercise # 10, page 73]). A similar statement holds for rings in H. Proposition 2.6. Let R ∈ H and suppose that RM is φ-Noetherian for every maximal ideal M of R, and that each nonnil nonunit of R lies in only a ﬁnite number of maximal ideals of R, then R is a φ-Noetherian ring. Proof. Set D = R/N il(R) (observe that D is an integral domain). If N il(R) is a maximal ideal of R, then D is Noetherian (being a ﬁeld), and hence R is φNoetherian by Theorem 2.1. Thus assume that N il(R) is not a maximal ideal of R. Let J be a maximal ideal of D. Then J = M/N il(R) for some maximal ideal M of R. Since RM ∈ H is φ-Noetherian by hypothesis and DJ is ring-isomorphic to RM /N il(R)RM , we conclude that DJ is Noetherian. Since each nonnil nonunit of R lies in only a ﬁnite number of maximal ideals of R by hypothesis, we conclude that every nonzero nonunit of D lies in only a ﬁnite number of maximal ideals of D. Thus, D is Noetherian by [14, Exercise # 10, page 73]. Hence R is φ-Noetherian by Theorem 2.1. Note that as long as N il(R) is prime and locally divided (i.e., N il(R)RM is a divided prime of RM ), N il(R) is divided. The key to the proof is that for each nonnil element r ∈ R, the conductor of r into N il(R) is N il(R) since N il(R) is prime. Thus no nonnil element can be transformed into a nilpotent element under localization. Hence for each maximal ideal M , rRM will contain N il(R)RM = N il(RM ). For a given nilpotent n ∈ N il(R) and maximal ideal M , there are elements s, t ∈ R with t not in M such that sr = tn. It follows that the ideal (r :R n) = R; i.e., n ∈ rR. Thus the hypothesis in Proposition 2.6 could be superﬁcially weakened to having N il(R) prime and each RM a φ-Noetherian ring (thus a φ-ring) with each nonnil element in only ﬁnitely many maximal ideals. However, the assumption that N il(R) is prime cannot be eliminated. For example, let V be a two dimensional valuation domain with principal maximal ideal M and height one prime P . Choose any proper P -primary ideal J and let R = V /J ⊕ V /J. Then the nilradical of R is P/J ⊕ P/J which is neither prime nor divided – simply consider the idempotents (1, 0) and (0, 1) and nilpotent multiples of each. On the other hand, M/J ⊕V /J and V /J ⊕ M/J are the only maximal ideals of R. Localizing at either yields the ring V /J which is a φ-Noetherian ring. Thus R is not φ-Noetherian, but it is locally φ-Noetherian with only ﬁnitely many maximal ideals. Note that if J does not contain P 2 , then V /J is deﬁnitely not formed by the idealization of a module over an integral domain. On the other hand, if we take V = K + YK[[Y]] + ZK((Y))[[Z]] and J = Z2 K((Y))[[Z]], then V /J is isomorphic to the idealization of K((Y)) over K[[Y]].

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Our next result shows that a φ-Noetherian ring will satisfy the conclusion of the Principal Ideal Theorem (and the Generalized Principal Ideal Theorem). Theorem 2.7. Let R ∈ H be a φ-Noetherian ring and let P be a prime ideal. If P is minimal over an ideal generated by n or fewer elements, then the height of P is less than or equal to n. In particular, each prime minimal over a nonnil element of R has height one. Proof. The ring D = R/N il(R) is a Noetherian domain by Theorem 2.1. Assume P is minimal over the ideal I = (a1 , a2 , . . . , an ). If I ⊂ N il(R), there is nothing to prove since we would have P = N il(R), the prime of height 0. Thus we may assume I is not nilpotent. Since N il(R) is divided, I properly contains N il(R). Thus I/N il(R) can be generated by n (or fewer) elements. Since D is Noetherian, the height of P/N il(R) is less than or equal to n by the Generalized Principal Ideal Theorem ([14, Theorem 152]). Thus P has height less than or equal to n. Other statements about primes of Noetherian rings that can be easily adapted to statements about primes of φ-Noetherian rings include the following. We leave the proofs to the reader. Proposition 2.8. [14, Theorem 145] Let R ∈ H satisfy the ascending chain condition on radical ideals. If R has an inﬁnite number of prime ideals of height 1, then their intersection is N il(R). Proposition 2.9. [14, Theorem 153] Let R ∈ H be a φ-Noetherian ring and P be a nonnil prime ideal of R of height n. Then there exist nonnil elements a1 , . . . , an in R such that P is minimal over the ideal (a1 , . . . , an ) of R, and for any i (1 ≤ i ≤ n), every (nonnil) prime ideal of R minimal over (a1 , . . . , ai ) has height i. Proposition 2.10. [14, Theorem 154] Let R ∈ H be a φ-Noetherian ring and let I be an ideal of R generated by n elements with I = R. If P is a prime ideal containing I with P/I of height k, then the height of P is less than or equal to n + k.

3

Polynomial Rings over φ-Noetherian Rings

By the Hilbert Basis Theorem, a polynomial ring in ﬁnitely many indeterminates over a Noetherian ring is Noetherian. The analogous statement cannot be made for a (nontrivial) φ-Noetherian R since a nonzero nilpotent element of R cannot be a multiple of an indeterminate. Also, if N il(R) is not ﬁnitely generated, then the ideal XR[X] + N il(R)R[X] is not ﬁnitely generated. On the other hand, a local Artinian ring that is not a ﬁeld is a φ-Noetherian ring. So it is possible for a polynomial ring over a φ-Noetherian ring to be Noetherian. We start this section with dimension related statements. Our ﬁrst consideration involves the Jaﬀard property. We will show that if R is an n-dimensional φ-Noetherian ring, then R[X1 , X2 , . . . , Xm ] has dimension n + m for each m > 0.

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Recall that if R is a Noetherian ring, P is a height n prime of R and Q is a prime of R[X] that contracts to P but properly contains P R[X], then P R[X] has height n and Q has height n + 1 [14, Theorem 149]. Proposition 3.1. Let R ∈ H be a φ-Noetherian ring and let P be a height n prime of R. If Q is a prime of R[X] that contracts to P but properly contains P R[X], then P R[X] has height n and Q has height n + 1. Proof. Since N il(R) is the minimal prime of R, N il(R[X]) = N il(R)R[X] is the minimal prime of R[X]. Thus we may shift the setting to D = R/N il(R) and D[X] (identiﬁed with R[X]/N il(R[X])). By Theorem 2.1, D is a Noetherian domain. Moreover, P/N il(R) is a height n prime of D and Q/N il(R[X]) is a prime of D[X] that contracts to P/N il(R) and properly contains (P/N il(R))D[X]. Thus the height of (P/N il(R))D[X] and P R[X] is n and the height of Q/N il(R[X]) and Q is n + 1. Similar height restrictions exist for the primes of R[X1 , X2 , . . . , Xm ]. Proposition 3.2. Let R ∈ H be a φ-Noetherian ring and let P be a height n prime of R. If Q is a prime of R[X1 , . . . , Xm ] that contracts to P but properly contains P R[X1 , . . . , Xm ], then P R[X1 , . . . , Xm ] has height n and Q has height at most n + m. Moreover the prime P R[X1 , . . . , Xm ] + (X1 , . . . , Xm )R[X1 , . . . , Xm ] has height n + m. Proof. Let D = R/N il(R). Then D is Noetherian as is each of the rings D[X1 , . . . , Xk ] ≡ R[X1 , . . . , Xk ]/N il(R)[X1 , . . . , Xk ]. Repeated applications of [14, Theorem 149] shows that the image of Q in D[X1 , . . . , Xm ] has height at most n + m. Hence the same restriction holds for the height of Q. Corollary 3.3. If R is a ﬁnite dimensional φ-Noetherian ring of dimension n, then dim(R[X1 , . . . , Xm ]) = n + m for each integer m > 0. As noted above, if N il(R) is not ﬁnitely generated, then XR[X] + N il(R)[X] is not ﬁnitely generated. In Examples 3.6 and 3.7, we will show how to construct a φ-Noetherian ring R that is not Noetherian, but where R[X] has ﬁnitely generated primes that contract to N il(R). In our next result we show that each ideal of R[X] that contracts to a nonnil ideal of R is ﬁnitely generated. Proposition 3.4. Let R ∈ H be a φ-Noetherian ring. If I is an ideal of R[X1 , X2 , . . . , for which I ∩ R is not contained in N il(R), then I is a ﬁnitely generated ideal of R[X1 , X2 , . . . , Xn ]. Xn ]

Proof. The key to the proof is that if I ∩ R is not contained in N il(R), then any single nonnil element in this intersection is enough to generate the nilradical of R[X1 , . . . , Xn ]. Since R/N il(R) is Noetherian, (I/N il(R))[X1 , . . . , Xn ] is ﬁnitely generated. Let {f1 , f2 , . . . , fm } ⊂ I generate the image of I modulo N il(R)[X1 , . . . , Xn ]. To get a ﬁnite set of generators for I, simply add any single nonnil element r of I ∩ R to the set {f1 , f2 , . . . , fm }. Since rN il(R) = N il(R), the set {r, f1 , . . . , fm } is a ﬁnite set of generators for I.

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Since three distinct comparable primes of R[X] cannot contract to the same prime of R, a consequence of Proposition 3.4 is that the search for primes of R[X] that are not ﬁnitely generated can be restricted to those of height one. A similar statement can be made for primes of R[X1 , X2 , . . . , Xn ]. Corollary 3.5. Let R ∈ H be a φ-Noetherian ring and let P be a prime of R[X1 , X2 , . . . , Xn ]. If P has height greater than n, then P is ﬁnitely generated. The ring in our next example shows that the converse of Proposition 3.4 does not hold even for prime ideals. Example 3.6. Let R = D(+)L be the idealization of L = K((Y))/D over D = K[[Y]]. Then R is a quasilocal φ-Noetherian ring with nilradical N il(R) isomorphic to L. Consider the polynomial g(X) = 1 − YX. Since the coeﬃcients of g generate D as an ideal and g is irreducible, P = gD[X] is a height one principal prime of D[X] with P ∩D = (0). Each nonzero element of L can be written in the form d/y n where n is a positive integer, y denotes the image of Y in L and d = d0 +d1 Y+· · ·+dn−1 Yn−1 with d0 = 0. Given such an element, let f (X) = 1 + y X + · · · + y n−1 Xn−1 ∈ L[X]. Then g(X)(df (X)/y n ) = d/y n since dy n /y n = 0 in L. It follows that g(X)R[X] is a height one principal prime of R[X] that contracts to N il(R). Since the domain D in the above example is a DVR, each prime of D[X] that contracts to (0) is principal and generated by an irreducible polynomial whose coeﬃcients generate D as an ideal. Let h(X) be such a polynomial. If h(X) has the form 1 − YXk(X) for some polynomial k(X) ∈ D[X], then the corresponding prime Q = hD[X] will be such that QR[X] = hR[X] is a height one principal prime of R that contracts to N il(R). The basic scheme of the proof is the same as for g(X) = 1 − YX. Given d = d0 + d1 Y + · · ·+ dn−1 Yn−1 with d0 = 0 and n positive, simply replace f (X) by the polynomial k(X) = 1+y Xk(X)+(y Xk(X))2 +· · ·+(y Xk(X))n−1 . Then, as above, h(X)(dk(X)/y n ) = d/y n . It follows that (h(X), 0) generates a height one principal prime of R[X]. On the other hand, the irreducible polynomial j(X) = Y − X also generates a height one prime of D[X], but it does not generate a height one principal prime of R[X] since its leading coeﬃcient is a unit. A slightly more complicated argument shows that if r(X) is an irreducible polynomial whose coeﬃcients generate D as an ideal, then rR[X] is prime if and only if r(X) is of the form u − YXt(X) for some unit u of D and some nonzero polynomial t(X) ∈ D[X]. Let d(X) ∈ L[X] be such that r(X)d(X) is a nonzero element of L[X]. Since D is a valuation domain, there is a largest positive integer m such that some coeﬃcient of d(X) is of the form v/y m where v is (the image of) a unit of D. Let q be the largest integer such that q X has a coeﬃcient of this form. Also let p be the largest integer such that rp , the coeﬃcient on Xp , is a unit of D. In the product r(X)d(X) the coeﬃcient on Xp+q has the form r0 dp+q + · · · rp−1 dq+1 + rp dq + rp+1 dq−1 + · · · rp+q d0 . Since the exponents on Y in r0 , . . . , rp−1 are nonnegative and those on y in dp+q , . . . , dq+1 are larger than −m, the resulting powers on y in the corresponding products are all strictly larger than −m. For i > p, the coeﬃcient on Y in ri is positive and that on y in the corresponding dj is greater than or equal to −m. Thus the coeﬃcient on Xp+q is

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a unit multiple of y −m . It follows that a necessary condition for rR[X] to contract to N il(R) is that r(X) have the form u − YXt(X) for some unit u of D and some nonzero polynomial t(X) ∈ D[X]. There are non-Noetherian φ-Noetherian rings with height one principal primes that contract to the nilradical but are not generated by a polynomial whose constant term is a unit nor do the coeﬃcients generate the entire ring as an ideal. Example 3.7. Let D = K[Y, Z] and let L = K(Y, Z)/DP where P is the principal prime YD. Then the ring R(+)L is a φ-Noetherian ring. Since D is a UFD, each height one prime of D[X] that contracts to (0) is principal and generated by an irreducible polynomial whose coeﬃcients generate an ideal of D whose inverse is D. Let Q = g(X)D[X] be such a prime and suppose that g(X) = a − YXt(X) where a ∈ D\YD (and t(X) ∈ D[X] is nonzero). Since DP is a DVR with maximal ideal YDP , each nonzero element of L can be written as a quotient d/y n for some d ∈ DP \P DP and some positive integer n. As in the previous example, let n be a positive integer and let f (X) = an−1 + an−2 y Xt(X) + · · · y n−1 Xn−1 t(X)n−1 and d/y n ∈ L with d ∈ DP \P DP . Since a is a unit of DP , a−n d/y n is an element of L. Hence we have (a − YX)(f (X)a−n d/y n ) = d/y n ∈ g(X)R[X]. It follows that g(X)R[X] is a principal height one prime of R[X] that contracts to N il(R). The argument given above characterizing the prinicipal height one primes of the polynomial ring in Example 3.6 can be adapted to this ring. Let r(X)D[X] be a height one prime of D[X] that contracts to (0) where r(X) is an irreducible polynomial whose coeﬃcients generate an ideal of D with inverse equal to D. Let d(X) ∈ L[X] be such that r(X)d(X) is a nonzero element of L[X]. As above let p be the largest integer where the coeﬃcient on Xp in r(X) is not in YD and let m and q be such that m is the largest integer such that some coeﬃcient of d(X) is of the form dk /y m for some unit dk of D and let q be the largest value of k where this maximum occurs. Similar analysis of the coeﬃcient on Xp+q shows that this coeﬃcient is of the form e/y m for some unit e of DP . It follows that the height one principal primes of R[X] that contract to N il(R) are of the form r(X)R[X] where r(X) = r0 − YXk(X) is an irreducible polynomial with r0 ∈ D\YD and k(X) nonzero. Moreover, each such polynomial generates such a prime. We end this section with the following problem. Problem. Let R ∈ H be a φ-Noetherian ring. Characterize the ﬁnitely generated height one primes of R[X].

4

Embedding R into a φ-Ring

We begin this section with a question. Question. Let R be a ring whose nilradical N il(R) is prime. Is it possible to embed R into a φ-ring A with nilradical N il(A) in such a way that all of the following hold? (I) AN il(A) is isomorphic to RN il(R) , (II) A/N il(A) is isomorphic to R/N il(R), and R embeds in A in such a way that

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(III) Z(R) ⊂ Z(A), (IV) T (R) ⊂ T (A), (V) A ∩ T (R) = R + N il(T (R)), and (VI) A = R + N il(A). For all but (I), it is easy to answer “Yes” if R is formed by taking the idealization of a module over an integral domain. For a D-module B, we let ZD (B) = {r ∈ D | rb = 0 for some nonzero b ∈ B}. If D is an integral domain, then for R = D(+)B, Z(R) = {(r, c) | r ∈ ZD (B), c ∈ B} [13, Theorem 25.3]. Lemma 4.1. Let D be an integral domain and let B be a D-module. Then there is a D-module C containing B which is a divisible D-module such that ZD (C) = ZD (B). Proof. Let S = D\ZD (B) and let U (DS ) denote the units of DS . Then BS is both a D-module and a DS -module and B naturally embeds as a D-submodule of BS . Also, each nonunit of DS is a zero divisor on BS , so ZD (BS ) = ZD (B) and ZDS (BS ) = ZD (B)S = DS \U (DS ). By [10, Proposition VII.1.4], there is a divisible DS -module C containing BS . Since each element of S is a unit of DS , ZD (C) = ZD (B). We also have that C is a divisible D-module. An immediate consequence is the following theorem. Theorem 4.2. Let R = D(+)B where D is an integral domain and B is a Dmodule. Then there is a divisible D-module C such that A = D(+)C is a φ-ring for which (i) A/N il(A) is isomorphic to R/N il(R), (ii) Z(R) ⊂ Z(A), (iii) T (R) ⊂ T (A), (iv ) A ∩ T (R) = R + N il(T (R)), and (v ) A = R + N il(A). Proof. As in Lemma 4.1, let S = D\ZD (B) and let C be a divisible DS -module that contains BS . Set A = D(+)C. Then N il(R) = (0)(+)B and N il(A) = (0)(+)C. Thus R/N il(R) = D = A/N il(A) and A = R + N il(A). As in the proof of Lemma 4.1, we have ZD (B) = ZD (C) and ZDS (BS ) = ZDS (C) = DS \U (DS ) where U (DS ) is the set of units of DS . With regard to zero divisors, we have Z(R) = ZD (B) × B and Z(A) = ZD (C) × C. From this it is easy to see that T (R) can be identiﬁed with DS (+)BS and T (A) with DS (+)C. Thus A ∩ T (R) = R(+)BS . To see that N il(A) is divided, consider (0, c) ∈ N il(A) and (r, b) ∈ A\N il(A). Since C is a divisible D-module, there is an element f ∈ C such that rf = c. Clearly, (r, b)(0, f ) = (0, c) and therefore N il(A) is divided.

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What is missing in the conclusions in Theorem 4.2 is the statement about the localizations RN il(R) and AN il(A) . In some sense, these were not simply ignored. The module C could be too large (since C ⊕ V is divisible for each vector space V over the quotient ﬁeld of D). Rather than showing that it is possible to cut C down to a divisible module C for which A = D(+)C would satisfy all six of the desired properties, we will establish the existence of a φ-ring satisfying all six under the somewhat more general assumption that the square of the nilradical is the zero ideal (no matter whether the original ring R is formed by idealization or not). In Chapter VII of [10], L. Fuchs and L. Salce present a useful technique for constructing a divisible module of projective dimension one over a given integral domain. They then use this module to generate a divisible module containing a speciﬁc D-module B. We make use of the general notion of their construction to establish the following theorem. Theorem 4.3. Let R be a ring with nilradical N il(R). If N il(R) is prime and N il(R)2 = (0), then there is a ring A with nilradical N il(A) such that (i) N il(A) is a divided prime of A, (ii) AN il(A) is isomorphic to RN il(R) , (iii) A/N il(A) is isomorphic to R/N il(R), and R embeds in A in such a way that (iv) Z(R) ⊂ Z(A), (v) T (R) ⊂ T (A) and (vi) A ∩ T (R) = R + N il(T (R)), and (vii) A = R + N il(A). Proof. Assume N il(R) is a prime ideal of R with N il(R)2 = (0). Then N il(T (R)) is a common prime ideal of T (R) and R + N il(T (R)) whose square is zero. Note that there are natural isomorphisms between R/N il(R) and [R + N il(T (R))]/N il(T (R)) and between the localizations RN il(R) and [R + N il(T (R))]N il(T (R)) . Thus we may assume N il(R) = N il(T (R)). We start by constructing a ring A such that (i) N il(A ) is a divided prime of A , (ii) AN il(A ) is isomorphic to T (R)N il(R) , (iii) A /N il(A ) is isomorphic to T (R)/N il(R), and T (R) embeds in A (and thus in T (A )) in such a way that (iv) A can be identiﬁed with T (R) + N il(A ) and (v) the image of T (R) in AN il(A ) is the same as the image of A . Given such a ring A , the ring A resulting from taking the pullback of R/N il(R) along N il(A ) will satisfy conditions (i)–(vii). Pictorially, A is the subring obtained from the following pullback diagram with i the inclusion map and σ the restriction of the isomorphism from T (R)/N il(R) to A /N il(A ). A −−−−→ R/N il(R) ⏐ ⏐ ⏐ ⏐σ i A −−−−→ A /N il(A ) Let S = R\Z(R), N = N il(R)\{0} and C = Z(R)\N . If C contains only 0 (equivalently, Z(R) = N il(R)), there is nothing to prove as N il(R) will be a divided prime (since we have assumed N il(R) = N il(T (R))). So we may assume Z(R) properly contains N il(R). Let F denote the subset of N × C N consisting of those elements of the form α = (α0 , α1 , α2 , . . . ) such that there is a positive integer nα where αj = 0 for each j ≤ nα and αk = 0 for each k > nα . Let E = T (R)[X ] where X = {Xα }

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is a set of indeterminates indexed over F . For each α ∈ F with nα > 1, let α−k = (α0 , α1 , . . . , αnα −k , 0, . . . ) for 1 ≤ k < nα . Let J be the ideal of E generated by (i) the products of the form Xα Xβ , (ii) the products of the form mXα for each m ∈ N il(R), (iii) the elements of the form αnα Xα − Xα−1 for those α with nα > 1, and (iv) the elements of the form α1 Xα − α0 for those α with nα = 1. By (i), the square of each Xα is in J. Also, J is contained in the ideal Q = N il(R) + X T (R)[X ], the set of polynomials with constant term a nilpotent of T (R) (and R). Since N il(R) is a prime ideal of T (R), Q is a prime ideal of E. As a power of each element of Q is contained in J, Q/J is the nilradical of A = E/J. Moreover, by the construction of J, J contains Q2 . Thus (Q/J)2 = (0) and we have that N il(A )2 = (0). We next show that J ∩ T (R) = (0) and no single Xα is in J. Note that the former implies that T (R) embeds naturally into A . Moreover, since N il(A ) = Q/J with Q = N il(R) + X T (R)[X ], we may view A as T (R) + N il(A ) and we have that there is a natural isomorphism between A /N il(A ) and T (R)/N il(R). Since J is contained in N il(R) + X T (R)[X ], J ∩ T (R) is contained in N il(R). By way of contradiction assume that some nonzero nilpotent of T (R) is contained in J. Based on the generating set described above, an arbitrary element of J can qγ (γnγ Xγ − be written as a ﬁnite sum of the form pα (α1 Xα − α0 ) Xβ + Xγ−1 ) X + r Xτ + yλ Xρ where nα = 1 for each α, the products µ σ Xβ , Xµ , Xτ and Xρ are ﬁnite with Xβ and Xµ arbitrary (including Xγ , at least two in each Xρ “empty products”), at least one term in each and pα , qγ , rσ , yλ ∈ T (R) with each rσ nilpotent. To have such a sum reduce Xβ are empty. To then to a nonzero nilpotent of T (R), it must be that some “cancel out” the terms pα α1 Xα , not all of the remaining sums can contain only products of two or more variables. The part of the expression that contributes nonzero constant terms and constant multiples of single variables has the form uα (α1 Xα − α0 ) + vα (α1 Xα − α0 )Xβ + zγ (γnγ Xγ − xγ−1 ) + wσ Xτ . All others involve only products of two or more indeterminates. Note that each wσ is nilpotent. If each uα is nilpotent, then there is no nonzero constant term since N il(R)2 = (0). So we may assume some uα is not nilpotent. For each such nonnil uα , uα α1 is not in N il(R) since N il(R) is prime and α1 is not nilpotent. In the sums vα (α1 Xα − α0 )Xβ and wσ Xτ all α0 and wσ are nilpotent, so none of these can cancel with the nonnil uα α1 ’s. The same is true for each zγ that is nilpotent. Thus there are nonnil zγ ’s. Since nγ > 1 for each γ, there must be γ’s with maximal values of nγ and zγ not nilpotent. Such terms cannot be cancelled in the sum. Thus no nonzero element of J is contained in N il(R). Note that if Xψ is in J for some ψ, then so is Xψ−1 = ψnψ Xψ − (ψnψ Xψ − Xψ−1 ) (or ψ0 if nψ = 1). By continuing, we would eventually have ψ0 in J, a contradiction. For the remainder of the proof we let xα denote the image of

Xα

in A .

Let t be a nonnil element of A . Then there is a nonnil element r ∈ T (R)\N il(R) and a nilpotent element g ∈ N il(A ) such that t = r − g. Since g 2 = 0, t(r + g) = r2 . Thus to show that each nonnil element of A divides each nilpotent element, it suﬃces to show that each nonnil element of T (R) divides each nilpotent in A . There is nothing to prove for the units of T (R) as each remains a unit under

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the embedding into A . So we may further reduce the problem to showing that each nonnil element r ∈ C divides each nilpotent of A . For a nonzero nilpotent b ∈ N il(R), let α = (b, r, 0, . . . ). Then rxα = b since rXα − b is one of the generators of J. Similarly, for Xα with nα > 1, let β = (α0 , α1 , . . . , αn , r, 0, . . . ). Then rxβ = xα since rXβ − Xα is in J. Since N il(R) and the set {xα } generate N il(A ), N il(A ) is a divided prime. It is only slightly more complicated to take care of statement (ii) dealing with the localizations T (R)N il(R) and AN il(A ) . The easiest thing to prove is that T (R)N il(R) embeds naturally in AN il(A ) . For this consider two equivalent fractions t/s and v/q of AN il(A ) where s, t, q, v ∈ T (R). To be equivalent, there must be a nonnil element z ∈ A such that z(sv − qt) = 0. Since A = T (R) + N il(A), there are elements u ∈ T (R) and f ∈ N il(A ) such that z = u + f . The product z(u − f ) = u2 ∈ T (R)\N il(R) since f 2 = 0. It follows that u2 (sv − qt) = 0 and the fractions t/s and v/q are also equivalent in T (R)N il(R) . To complete the proof that this embedding is surjective it suﬃces to show that each member of the set {xα } is equivalent to a quotient of the form n/q for some nilpotent n ∈ T (R) and nonnil q ∈ T (R). This is quite simple. Let xσ ∈ {xα } and consider the product q = σi with i ranging from 1 to nσ (the last nonzero σj ). This is a product of nonnil elements of T (R) so q is nonnil. By the construction of J, σn xσ = xσ−1 , σj xσ−k = xσ−(k+1) for k = nσ − j and 1 < j < nσ , and σ1 xσ−k = σ0 for k = nσ − 1. Thus qxσ = σ0 . In the localization T (R)N il(R) , we have xσ = σ0 /q. Therefore we may view T (R)N il(R) and AN il(A ) as the same ring.

As mentioned in the introduction, several speciﬁc types of φ-rings have been studied. A general deﬁnition for a φ-BLANK ring is that a φ-ring R is a φ-BLANK ring if R/N il(R) is a BLANK domain. For each of the types that have been studied so far, the speciﬁc deﬁnition involves properties of the nonnil ideals of R, sometimes with regard to the image of such ideals in φ(R). In each case, a consequence of the deﬁnition is that R is a φ-BLANK ring if and only if R/N il(R) is a BLANK domain. For example, given a φ-ring R, it is (i) a φ-chained ring if the nonnil ideals are linearly ordered – equivalent to R/N il(R) being chained (i.e., a valuation domain); (ii) a φ-pseudo-valuation ring if each nonnil prime is strongly prime with respect to T (R) (meaning if rt ∈ P for some r, t ∈ T (R), then either r ∈ P or t ∈ P ) – equivalent to R/N il(R) being a pseudo-valuation domain; (iii) a φ-Pr¨ ufer ring if for each ﬁnitely generated nonnil ideal I, φ(I) is an invertible ideal of φ(R) – equivalent to R/N il(R) being a Pr¨ ufer domain; (iv) a φ-B´ezout ring if each ﬁnitely generated nonnil ideal is principal – equivalent to R/N il(R) being a B´ezout domain; and (v) a φ-Mori ring if it satisﬁes the ascending chain condition on those nonnil ideals I for which φ(I) is divisorial as an ideal of φ(R) – equivalent to R/N il(R) being a Mori domain (a domain with a.c.c. on divisorial ideals). [For references see [4], [2], [3], [1], and [8].] For each of these special types of φ-rings, we have the following corollary to Theorem 4.3.

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Corollary 4.4. Let R be a ring with nonzero nilradical N il(R). For BLANK any one of “chained”, “pseudo-valuation”, “Pr¨ ufer”, “B´ezout”, “Mori”, and “Noetherian”, if N il(R) is prime with R/N il(R) a BLANK domain and N il(R)2 = (0), then there is a φ-BLANK ring A for which statements (I)–(VI) hold for the pair R and A. Note that with regard to idealization, if D is an integral domain and B is a D-module, then the process used in the proof of Theorem 4.3 can be used to build a divisible D-module C such that the pair R = D(+)B and A = D(+)C satisfy statements (I)–(VI). It follows that if D is Noetherian, the resulting ring A is a φ-Noetherian ring. We have been unable to extend Theorem 4.3 to rings with prime nilradicals whose squares are not zero. For some such rings it is not hard to show that it is impossible to ﬁnd a φ-ring A that satisﬁes conditions (I) through (VI). In particular, one necessary condition for the existence of A is that each nonzero nilpotent with a nonnil annihilator must annihilate N il(R). To see this assume rn = 0 for some nonnil element r ∈ R and nonzero nilpotent n. If N il(A) is divided, then for each b ∈ N il(R), there is a nilpotent m ∈ N il(A) such that rm = b. Since nr = 0, we must have nb = 0 as well. Hence nN il(R) = (0). A speciﬁc example of a ring with prime nilradical which cannot be embedded in a φ-ring is the ring R = K[Y, Z]/(Y3 , YZ). The nilradical of R is the prime ideal yK[y, z] (where the lower case letters represent the images of the indeterminates in R) and y has a nonnil annihilator, namely the element z. As y 2 = 0, it is impossible to embed R in a ring whose nilradical is divided.

Bibliography [1] D. F. Anderson and A. Badawi, On φ-Pr¨ ufer rings and φ-B´ezout rings, to appear in Houston J. Math. [2] A. Badawi, On φ-pseudo-valuation rings, Advances in Commutative Ring Theory (Fez, Morocco, 1997), Lecture Notes Pure Appl. Math., Marcel Dekker 205 (1999), 101–110. [3] A. Badawi, On φ-pseudo-valuation rings II, Houston J. Math. 26 (2000), 473– 480. [4] A. Badawi, On φ-chained rings and φ-pseudo-valuation rings, Houston J. Math. 27 (2001), 725–736. [5] A. Badawi, On nonnil-Noetherian rings, Comm. Algebra 31 (2003), 1669– 1677. [6] A. Badawi, On divided rings and φ-pseudo-valuation rings, Int. J. of Commutative Rings 1 (2002), 51–60. [7] A. Badawi, On divided commutative rings, Comm. Algebra 27 (1999), 1465– 1474. [8] A. Badawi and T. Lucas, On φ-Mori rings, preprint. [9] D. E. Dobbs, Divided rings and going-down, Paciﬁc J. Math. 67 (1976), 353– 363. [10] L. Fuchs and L. Salce, Modules over Non-Noetherian Domains, Mathematical Surveys and Monographs, Vol 84, American Mathematical Society, Providence, 2001. [11] R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers Pure Appl. Math. Vol 90, Queen’s University Press, Kingston, 1992. [12] R. Gilmer, W. Heinzer, and M. Roitman, Finite generation of power of ideals, Proc. Amer. Math. Soc. 127 (1999), 3141-3151. [13] J. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York/Basel, 1988. 211

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[14] I. Kaplansky, Commutative Rings – rev. ed., The University of Chicago Press, Chicago, 1974.

Chapter 12

On the Ideal Generated by the Values of a Polynomial by

Jean-Luc Chabert and Sabine Evrard Abstract Pn k ∈ K[X] where Let D be a Dedekind domain and let f (X) = k=0 ak X K denotes the quotient ﬁeld of D. According to P´ olya, for every maximal + wq (deg f ) ideal m of D with norm q, one has: vm (f (D)) ≤ vm (f (X)) P where vm denotes the valuation on K associated to m, wq (n) = k≥1 [n/q k ], vm (f (D)) = inf{vm (f (b)) | b ∈ D}, and vm (f (X)) = inf{vm (ak ) | 0 ≤ k ≤ n}. Vˆ ajˆ aitu gives another bound for vm (f (D)) in the case where D = Z. We extend this bound to more general cases and, in particular, to Dedekind domains: as soon as deg(f ) ≤ p(q − 1) where p = char(D/m), one has vm (f (D)) < vm (f (X)) + νm (f ) with νm (f ) = Card {k | vm (ak ) = vm (f (X))}.

1

Introduction

Let D be a Dedekind domain with quotient ﬁeld K and let f=

n

ak X k ∈ K[X]

(12.1)

k=0

be a polynomial of degree n. We denote by C(f ) the content of f , that is, the ideal of D generated by the coeﬃcients of f and by D(f ) the divisor of f , that is, the ideal generated by the values of f on D. In this introduction we assume that f is primitive, that is, C(f ) = D. When f ∈ Z[X], it is well-known that the gcd of the values of f on Z divides n!. For Dedekind domains, this result was generalized by P´ olya [8, S4] in the following way (see also [5, II.3.3]): the ideal D(f ) divides the nth factorial ideal n!D where n!D is deﬁned by mwN (m) (n) (12.2) n!D = m∈max(D), N (m)≤n

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with N (m) = Card(D/m)

(12.3)

$n% . ql

(12.4)

and wq (n) =

l≥1

Writing the ideal D(f ) in the following form D(f ) = mdm (f ) ,

(12.5)

m∈max(D)

this divisibility relation may be written as inequalities. For each maximal ideal m of D, one has: (12.6) dm (f ) ≤ wN (m) (n). The aim of this chapter is to state another divisibility relation making use of the number of coeﬃcients of f not belonging to m instead of the degree of f . More precisely, let / m}; (12.7) µm (f ) = Card {ak | ak ∈ we are going to prove that dm (f ) < µm (f ).

(12.8)

But this inequality holds only for small values of n = deg(f ), namely: deg(f ) ≤ char(D/m) × (N (m) − 1).

(12.9)

Vˆajˆ aitu [9, Theorem 2] proved Inequality (12.8) when D = Z. Here, we generalize it to every Dedekind domain D (Proposition 4.1). Then, we extend it to the ideal D(f, E) generated by the values of f on a subset E of D when D = V is a discrete valuation domain (Proposition 5.4).

2

Technical Preliminaries

Notation. For every polynomial f , we denote by µ(f ) the number of nonzero coefﬁcients of f . The following technical result is implicitly contained in the proof of [9, Thm 2]. Lemma 2.1. Let k be a ﬁeld with characteristic p > 0 such that k p = k (for instance, a ﬁnite ﬁeld). Let f ∈ k[X] be a nonzero polynomial and let z ∈ k be a nonzero root of f with multiplicity m. If m < p, then m < µ(f ). In order to prove this lemma we associate to every polynomial f an integer s(f ) < µ(f ) deﬁned by means of the following algorithm: Algorithm A. s ← 0 , fs ← f

On the Ideal Generated by the Values of a Polynomial

215

while µ(fs ) > 1 do begin s ← s + 1 fs ← X vXfs(fs ) end s(f ) = s where vX (g) denotes the least degree of the monomials of g and the symbol denotes the formal derivation of polynomials. It is clear that the procedure is ﬁnite since, for every g = 0, one has: g µ < µ(g). X vX (g) Consequently, s(f ) < µ(f ). Proof. of Lemma 2.1. Assume that m ≥ µ(f ). Then z is a root of all the polynomials fs constructed in Algorithm A with multiplicity m − s ≥ m − s(f ) > m − µ(f ) ≥ 0. By deﬁnition of s(f ), one has µ(fs(f ) ) ≤ 1. If µ(fs(f ) ) = 1, then fs(f ) is of the form bX h and z cannot be a root of fs(f ) . Consequently, fs(f ) = 0, and hence, fs(f )−1 (X) = b1 X h1 + . . . + bl X hl with l ≥ 2, bj ∈ k ∗ , h1 < h2 < · · · < hl , hj − h1 = pmj with mj ∈ N∗ . By hypothesis, for j = 1, . . . , l, bj = cpj with cj ∈ k ∗ . Thus, fs(f )−1 (X) = X h1

l

cpj X pmj

⎛ ⎞p l = X h1 ⎝ cj X m j ⎠ .

j=1

j=1

Since z is a root of fs(f )−1 , z is a root of lj=1 cj X mj . Consequently, the multiplicity of z as a root of fs(f )−1 is a nonzero multiple of p. But this multiplicity is m−s(f )+1, so m ≥ p, which contradicts the hypothesis. Corollary 2.2. Let k be a ﬁeld of characteristic p > 0 such that k p = k and let f be a nonzero polynomial in k[X]. If x ∈ k is a root of f with multiplicity mx < p then, for every y ∈ k, y = x, one has : mx < µ(f (X + y)). Proof. It suﬃces to use Lemma 2.1 with the polynomial g(X) = f (X + y) and its root z = x − y.

216

3

Chabert and Evrard

Polynomials with Coeﬃcients in a Discrete Valuation Domain

Hypotheses and notation for Section 3. Let V be a discrete valuation domain with ﬁnite residue ﬁeld. Denote by K the quotient ﬁeld of V , v the corresponding valuation of K, m the maximal ideal of V , π a generator of m, k = V /m the residue ﬁeld, p the characteristic of k, and q = pf its cardinality. For every nonzero polynomial f (X) =

n

ai X i ∈ K[X],

(12.10)

i=0

we consider the following integers: v(f ) = inf

v(ai ),

(12.11)

d(f ) = inf v(f (a)),

(12.12)

ν(f ) = #{i | v(ai ) = v(f )}.

(12.13)

0≤i≤n

a∈V

Note that ν(f ) = µ(f˜) where f˜ denotes the image of

1 f π v(f )

in k[X].

Clearly, v(f ) ≤ d(f ),

(12.14)

and we also know that (see for instance [8] or [5, Corollary II.2.13]) d(f ) ≤ v(f ) + wq (deg(f )) where wq is deﬁned by: wq (n) =

$n% . ql

(12.15)

(12.16)

l≥1

In particular, if deg(f ) < q, then d(f ) = v(f ). Here we prove another inequality for d(f ): Proposition 3.1. With the previous hypotheses and notation, for every nonzero polynomial f in K[X] such that deg(f ) ≤ p(q − 1) + 1,

(12.17)

v(f ) ≤ d(f ) < v(f ) + ν(f ).

(12.18)

one has: Proof. We may replace f by π −v(f ) f and assume that f is primitive in V [X], that is, v(f ) = 0. Note that ν(f ) ≥ 1 and, if ν(f ) = 1, then necessarily d(f ) = 0. Consequently, one may also assume that d(f ) ≥ 2.

On the Ideal Generated by the Values of a Polynomial

217

First, we recall some classical results concerning the values of a polynomial. Let u0 = 0, u1 , . . . , uq−1 be a complete system of representatives of V modulo m. We extend the sequence ur in the following way: for r = r0 + r1 q + . . . + rl q l

where 0 ≤ ri < q,

we let u r = u r0 + u r1 π + . . . + u rl π l . Clearly, the following sequence of polynomials gi (X) =

i−1

(X − uj ) , i ∈ N

j=0

is a basis of the V -module V [X]. Then let f (X) =

n

bi gi (X) with bi ∈ V.

i=0

We know, and this is easy to check, that the ideal generated by the values of f on V is equal to the ideal generated by the values of f on u0 , u1 , . . . , un where n = deg(f ) [5, Corollary II.2.9], that is, the ideal generated by the bi j
Let i0 be the least integer i such that v(bi ) = 0 (f is assumed to be primitive). The hypothesis on n = deg(f ) implies that $ % i0 i0 i0 i0 ≤ ≤ 1, ≤ 2 < 2 q q p(q − 1) n and hence,

$ i0 % $ i0 % i0 d(f ) ≤ wq (i0 ) = = ≤ . ql q q l≥1

Finally, i0 ≥ q d(f ). We denote by b the canonical image in k of an element b of V and by g the canonical image in k[X] of an element g of V [X]. It follows from the choice of i0 and from the construction of the gi ’s that f (X) =

n i=0

bi gi (X) =

n

bi gi (X) = gi0 (X)h(X) where h(X) ∈ k[X].

i=i0

Since ur = us as soon as q divides r − s and i0 ≥ d(f ), the q elements of k = V /m are roots of gi0 , and hence of f , with multiplicity ≥ d(f ). On the other hand, there

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exists at least one root of f in k ∗ with a multiplicity < p since otherwise we would have: n = deg(f ) ≥ deg(f ) ≥ d(f ) + (q − 1)p ≥ 2 + (q − 1)p > n. Thus, there exists a root z ∈ k ∗ of f with a multiplicity m such that d(f ) ≤ m < p. It follows from Lemma 2.1 that m < µ(f ), and hence, d(f ) < ν(f ). Example 3.2. Let p be a prime number. For f (X) = X(X (p−1)(q−1) − 1) + π, one has d(f ) = 1 = ν(f ) − 1 while wq (n) = p − 2 ( > ν(f ) − 1 as soon as p ≥ 5 ). Let us introduce another notation: for each a ∈ V , let νa (f ) = ν(f (X + a)). In particular, ν0 (f ) = ν(f ). Let ν˜ (f ) = inf νa (f ) = inf ν(f (X + a)). a∈V

a∈V

Of course, v(f (X)) = v(f (X + a)) and d(f (X)) = d(f (X + a)). Consequently, Corollary 3.3. If deg(f ) ≤ p(q − 1) + 1, then v(f ) ≤ d(f ) < v(f ) + ν˜(f ).

(12.19)

Example 3.4. Let p be a prime number ≥ 5, let V = Z(p) , and let f (X) = (X − 1)p−1 − 1. Then, on the one hand, for every a ∈ Z, f (X + a) ≡ (X + (a − 1))p−1 − 1

(mod p).

If a ≡ 1 (mod p), then νa (f ) = p − 1. If a ≡ 1 (mod p), then νa (f ) = 2. On the other hand, f (1) = −1, and hence, d(f ) = 0. Finally, d(f ) = 0 < ν˜ (f ) − 1 = 1 < ν(f ) − 1 = p − 2. Thus, we may have strict inequalities. Nevertheless: Remark. For n ≤ p(q − 1) < q 2 , one has wq (n) = [ nq ] ≤ p − 1, and hence, it follows from Inequalities (12.15) and (12.18) that, if f ∈ V [X] is primitive of degree

then

n ≤ p(q − 1) + 1,

(12.20)

$ % n d(f ) ≤ min , ν(f ) − 1 . q

(12.21)

Moreover, both inequalities are sharp. Inequality (12.20) is sharp as shown by the following example: f (X) = X 2 (X q−1 − 1)p , deg(f ) = p(q − 1) + 2 , ν(f ) ≤ 2 , d(f ) = 2.

On the Ideal Generated by the Values of a Polynomial

219

Inequality (12.21) is sharp in the following sense: for every integer ν between 1 and p, there exists a polynomial f primitive in V [X] of degree n ≤ p(q − 1) + 1 such that d(f ) = ν(f ) − 1 = ν − 1: For 0 ≤ k ≤ p − 1, the polynomial fk (X) = (X q − X)k satisﬁes d(fk ) = k and ν(fk ) = k + 1 with deg(fk ) = kq ≤ (p − 1)q ≤ p(q − 1) + 1.

4

Polynomials with Coeﬃcients in a Dedekind Domain

Now we globalize the previous results. Hypotheses and notation for section 4. Let D be a Dedekind domain with quotient ﬁeld K. For every maximal ideal m of D, we denote by vm the corresponding valuation of K, by pm the characteristic of the residue ﬁeld D/m, and by qm its cardinality (ﬁnite or inﬁnite). For every polynomial f=

n

ai X i ∈ K[X],

(12.22)

i=0

C(f ) denotes the content of f , that is, the fractional ideal of D generated by the coeﬃcients of f and D(f ) denotes the divisor of f , that is, the fractional ideal generated by the values of f on D. For every maximal ideal m of D, we introduce the following integers: vm (f ) = inf

vm (ai ),

(12.23)

dm (f ) = inf vm (f (a)),

(12.24)

νm (f ) = #{i | vm (ai ) = vm (f )}.

(12.25)

0≤i≤n

a∈D

Obviously, C(f ) =

mvm (f ) ,

(12.26)

mdm (f ) .

(12.27)

m∈max(D)

D(f ) =

m∈max(D)

Clearly, the ideal C(f ) divides the ideal D(f ) and it is known [5, Proposition II.3.3] that D(f ) divides C(f ) × n!D where the ideal n!D is deﬁned by Formula (12.2), in other words, for every maximal ideal m of D, analogously to Formulas 12.14 and 12.15, one has the inequalities: vm (f ) ≤ dm (f ) ≤ vm (f ) + wN (m) (deg(f )) where N (m) and wN (m) are deﬁned by Formulas (12.3) and (12.4). Proposition 3.1 may be globalized in the following way:

(12.28)

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Proposition 4.1. Let D be a Dedekind domain with quotient ﬁeld K and let f ∈ K[X] be a nonzero polynomial of degree n. With the previous notation, for every maximal ideal m of D such that n ≤ pm (qm − 1) + 1,

(12.29)

dm (f ) < vm (f ) + νm (f ).

(12.30)

one has: This proposition is the extension of Theorem 2 of Vˆajˆ aitu [9] from Z to every Dedekind domain D. Theorem 3 of [9] corresponds to the ﬁrst example below. Examples 4.2. In these three examples, f denotes a polynomial of degree n primitive in D[X]. 1) Let D = Z and denote by P the set of prime integers. Then, a generator of D(f ) divides the integer n pwp (n) × pmin([ p ],νp (f )−1) . √ 3 1 √ 3 1 p∈P, p<

n− 4 + 2

p∈P,

n− 4 + 2 ≤p≤n

2) Let D = Z[i], p0 = inf{p ∈ P | p(p2 − 1) + 1 ≥ n} and p1 = inf{p ∈ P | p(p − 1) + 1 ≥ n}. For each p ∈ P, νp (f ) denotes the number of coeﬃcients of f that are not divisible by p. Then, the ideal D(f ) divides the following ideal of Z[i]

Z[i] (1 + i)w2 (n) ×

pwp (n) ×

p≡1 (4), p1 ≤p≤n

p≡1 (4), p

×

pwp2 (n) ×

p≡3 (4), p
n

pinf([ p ],νp (f )−1)

p

inf([ pn2 ],νp (f )−1)

.

p≡3 (4), p0 ≤p≤n

3) Let D = Fq [T ] and denote by Pq the set of monic irreducible polynomials of Fq [T ]. For m = (Q) where Q ∈ Pq , one has qm = q deg(Q) . Then, a generator of D(f ) divides the polynomial pn ln n+p−1 n deg(Q)< ln ln q − ln q

Qwqdeg Q (n) × ln n ln q −

ln

pn n+p−1 ln q

Q

„» min

n qdeg(Q)

–

« ,νQ (f )−1

.

n ≤deg(Q)≤ ln ln q

Remark. Recall Theorem 1 of [9]: if the characteristic of D is 0 and if f ∈ D[X] is primitive with degree n and leading coeﬃcient a, then n+1

Card(D/D(f )) ≤ Card(D/(a.n!D)n2

).

In fact, we have just recalled a stronger result: D(f ) divides n!D and n!D divides n!D because of the containment Z ⊆ D. Thus, D(f ) divides n!D, and hence, Card(D/D(f )) ≤ Card(D/n!D).

On the Ideal Generated by the Values of a Polynomial

5

221

The Ideal Generated by the Values on a Subset

In this paragraph we extend the previous results to the ideal D(f, E) generated by the values of a polynomial f ∈ K[X] on a subset E of D. We will give the statement for the main result (Proposition 5.4) in the case when D is a discrete valuation domain. Then, by using a speciﬁc example, we will show what happens in the more general case of a Dedekind domain. Hypotheses and notation for section 5 are those of Section 3. Hence, the domain D = V is a discrete valuation domain, and we denote by E a subset of V . We introduce the integer (12.31) d(f, E) = inf v(f (x)). x∈E

Obviously, D(f, E) = md(f,E) .

(12.32)

Definition 5.1 ([1], [2]). A v-ordering of E is a sequence {uk }k∈N of elements of E such that, for every s ≥ 1, one has *s−1 + *s−1 + v (us − ut ) = inf v (x − ut ) . x∈E

t=0

t=0

There always exist v-orderings and, for every s ≥ 1, the integer *s−1 + (us − ut ) wE (s) = v

(12.33)

t=0

does not depend on the choice of the v-ordering {us }s∈N of E (see for instance [1] or [6]). Definition 5.2. The sth factorial ideal of E with respect to V is the ideal s!VE = mwE (s) . It is easy to see that if E ⊆ F ⊆ V , then s!VF divides s!VE , that is, E ⊆ F ⇒ wE (s) ≥ wF (s) ∀s ∈ N.

(12.34)

wE (s) ≥ wq (s) ∀s ∈ N.

(12.35)

In particular, Let {us }s∈N be a v-ordering of E and, for i ∈ N, let gi (X) =

i−1

(X − us ).

s=0

Then, every polynomial f ∈ K[X] of degree n may be written in the following way: f (X) =

n i=0

bi gi (X) with bi ∈ K.

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It is known (and this is easy to check) that D(f, E) is also the ideal generated by the values f (u0 ), f (u1 ), . . . , f (un ) [6, Corollary 2.8]. Thus, D(f, E) is generated by the bi j
(12.36)

In particular, d(f, E) ≤ inf v(bi ) + sup wE (i), 0≤i≤n

0≤i≤n

that is, Proposition 5.3. For every f ∈ K[X] with degree n, one has: d(f, E) ≤ v(f ) + wE (n),

(12.37)

where d(f, E), v(f ), and wE (n) are deﬁned by (12.31), (12.11) and (12.33). This well-known inequality generalizes Inequality (12.15). But our next goal is to extend Inequality (12.18) under some condition on the degree n of f . Proposition 5.4. Let E be a subset of V which contains at least r ≥ 2 distinct classes modulo m and let f ∈ K[X] be a polynomial of degree n. If n ≤ p(r − 1) + 1,

(12.38)

d(f, E) < v(f ) + νm (f ).

(12.39)

then Moreover, the previous inequality also holds as soon as 1. n < pr when m ⊆ E, 2. n ≤ pr when ∅ = E ∩ m = m. Proof. Since E ⊆ F implies d(f, E) ≥ d(f, F ), in order to prove Inequality (12.39) one may assume that E is exactly the union of r classes modulo m. Analogously to the proof of Proposition 3.1, one may also assume that f is primitive, that is, v(f ) = 0, and that d(f, E) ≥ 2. We still consider a v-ordering {us }s∈N of E, the basis of the V -module V [X] formed by the polynomials gi = 0≤s
On the Ideal Generated by the Values of a Polynomial It follows from the hypothesis that $

Thus, wE (i0 ) =

, i0 r

223

% i0 n < 1. ≤ rq rq

, and hence, i0 ≥ r d(f, E).

Because of the fact that E is a union of r classes modulo m and from the previous inequality, the image gi0 of gi0 in (V /m)[X] has a root in each class of E modulo m with a mutiplicity at least equal to d(f, E). The image f of f has the same property. Moreover, if each root of f distinct from the class m had multiplicity ≥ p, we would have the following inequalities: n = deg(f ) ≥ deg(f ) ≥ deg(g i0 ) = i0 ≥ d(f, E) + (r − 1)p ≥ 2 + p(r − 1) > n. (12.40) Consequently, there is at least one root of f distinct from the class m and with multiplicity m < p. It follows from Lemma 2.1 that m < µ(f ) = νm (f ). Finally, d(f, E) ≤ m < νm (f ). When m ⊆ E, it follows from the proof and inequalities analogous to (12.40) that the condition n < pr is enough. Finally, assume that m ⊆ E and that there is t ∈ m ∩ E. We may assume that E = E0 ∪ {t} where E0 is exactly the union of r classes modulo m distinct from m. Then, choosing t as ﬁrst element of a v-ordering {us } of E, we easily see that $s − 1% wE (s) = wE0 (s) = . rq l l≥0

Consequently, with the previous notation i0 , one has i0 − 1 ≥ r d(f, E). Then, f admits each class of E modulo m as root with mutiplicity at least equal to d(f, E). Moreover, if each root of f had a multiplicity ≥ p, we would have: n = deg(f ) ≥ deg(f ) ≥ deg(g i0 ) = i0 ≥ 1 + rp > n. We conclude in the same manner. Example 5.5. Let p ∈ P, V = Z(p) , and E = {p} ∪ N \ pN (r = p − 1). For every f ∈ Q[X], we have n ≤ p(p − 1) ⇒ d(f, E) < vp (f ) + νp (f ). Globalization. We discuss the extension of Proposition 5.4 to any Dedekind domain by showing what happens for a speciﬁc example. This seems to us a more eﬃcient way in order to understand the general case rather than proving a general result.

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Let D = Z, E = P, and f be a primitive polynomial in Z[X] of degree n. Recall that, if f (P) ⊆ Z then, for every p ∈ P, f ({p} ∪ Z \ pZ) ⊆ Z(p) [7]. Consequently, on the one hand, for every p ∈ P, one has: $ k − 1 % , dp (f, P) ≤ (p − 1)pl l≥0

on the other hand, for every p ∈ P such that p(p−1) ≥ n−1, that is, one has: dp (f, P) < νp (f ), and hence,

dp (f, P) ≤ inf

. n − 34 + 21 ≤ p,

n−1 , νp (f ) − 1 . p−1

Remark. A priori we have d(f, P) ≥ d(f, Z), but the fact that the inequality dp (f, P) < νp (f ) holds as soon as n ≤ p(p − 1) is exactly the assertion given by Proposition 3.1 for dp (f, Z).

Bibliography [1] M. Bhargava, P -orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math. 490 (1997), 101–127. [2] M. Bhargava, Generalized Factorials and Fixed Divisors over Subsets of a Dedekind Domain, J. Number Theory 72 (1998), 67–75. [3] J. Boulanger and J.-L. Chabert, Asymptotic behavior of characteristic sequences of integer-valued polynomials, J. Number Theory 80 (2000), 238–259. [4] J. Boulanger, J.-L. Chabert, S. Evrard, and G. Gerboud, The Characteristic Sequence of Integer-Valued Polynomials on a Subset, in Advances in Commutative Ring Theory, Lecture Notes in Pure and Appl. Math., Marcel Dekker 205 (1999), 161–174. [5] P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials, Amer. Math. Soc. Surveys and Monographs 48, Providence, 1997. [6] J-L. Chabert, Generalized factorial ideals, The Arabian Journal for Science and Engineering, t. 26 (2001), 51–68. [7] J.-L. Chabert, S. Chapman, and W. Smith, Basis for the Ring of Polynomials Integer-Valued on Prime Numbers, in Factorization in integral domains, Lecture Notes in Pure and Appl. Math., Marcel Dekker 189 (1997), 271–284. ¨ [8] G. P´ olya, Uber ganzwertige Polynome in algebraischen Zahlk¨ orpern, J. Reine Angew. Math. 149 (1919), 97–116. [9] M. Vˆ ajˆ aitu, The ideal generated by the values of a polynomial over a Dedekind domain, Rev. Roumaine Math. Pures Appl. 42 (1997), 155–161.

225

Chapter 13

Using Factorizations to Prove a Partition Identity by

David E. Dobbs and Timothy P. Kilbourn Abstract It is known that if m is any positive integer, then X 1 = 1, e1 ! · · · · · ek ! · de11 · · · · · dekk where the sum is e 1 d1 + · · · + e k dk positive integers). as a consequence polynomial over a

indexed by the partitions of m (i.e., by the expressions = m, where each ei and 1 ≤ d1 < · · · < dk ≤ m are A new proof is given here for this combinatorial identity, of the formula for the probability that a random monic ﬁnite ﬁeld is irreducible.

2000 Mathematics Subject Classification: Primary 13M10; Secondary 11A67, 13F15, 11A25, 12E20.

1

Introduction

It is easy to check that 1 1 1 1 1 1 1 + + + + + + = 1. 120 12 6 8 4 6 5 This equation may not seem remarkable at ﬁrst glance. After all, any positive rational number can be expressed as a sum of distinct unit fractions [11, p. 261], and 1 can be so expressed more easily, as in the equation 16 + 13 + 12 = 1. However, the ﬁrst equation is special, for its summands can be written as 1 1 1 1 , , , , 5! · 15 3! · 1! · 13 · 21 2! · 1! · 12 · 31 1! · 2! · 11 · 22 1 1 1 , , . 1! · 1! · 11 · 41 1! · 1! · 21 · 31 1! · 51 226

Using Factorizations to Prove a Partition Identity

227

These seven summands correspond to the seven partitions of 5, namely, 5·1, 3·1+1·2, 2 · 1 + 1 · 3, 1 · 1 + 2 · 2, 1 · 1 + 1 · 4, 1 · 2 + 1 · 3, and 1 · 5. In fact, the ﬁrst equation is just the case m = 5 of the following result. Theorem 1.1. If m is any positive integer, then 1≤d1 <···
1 = 1. e1 ! · · · · · ek ! · de11 · · · · · dekk

It turns out that the equation 16 + 13 + 12 = 1 is also “special,” for it is the case m = 3 of Theorem 1.1. How could one prove such a result? And what exactly does it mean? We answer the second question ﬁrst. Recall that if m is a positive integer, then a partition of m is an expression of the form m = e1 d1 + · · · + ek dk , where each ei and 1 ≤ d1 < · · · < dk ≤ m are positive integers. The sum appearing in the identity stated in Theorem 1.1 is indexed by the set of partitions of m. For this reason, we termed this identity a “partition identity” in the title of this note. For other types of identities arising from the theory of partitions, see, for instance, [9]. Now, we turn to the question of how to prove Theorem 1.1. One proof appears in [8, (22.20), p. 38], as a consequence of considerations involving partitions of arbitrary positive integers and multinomial symbols. Another combinatorial proof follows easily from Cauchy’s formula [3, Theorem 72] for the number of conjugates of a given permutation in the nth symmetric group. Our purpose here is to provide a proof with a diﬀerent ﬂavor, as a corollary of the formula for the probability that a “random” monic polynomial over a ﬁnite ﬁeld is irreducible. The next section gives the motivation and background for such factorization studies.

2

The Probability that a Polynomial is Irreducible

Our studies were motivated by the following fact. If m is any positive integer, then with probability 1, a random monic integral m-th degree polynomial is irreducible in Q[X] (and in Z[X]). The underlying notion of “probability” in question is deﬁned A , where to be limn→∞ Bm,n m,n Am,n := |{f = X m + am−1 X m−1 + · · · + a0 ∈ Z[X] | |ai | ≤ n for all i and f is irreducible in Q[X]}| and Bm,n := |{f = X m + am−1 X m−1 + · · · + a0 ∈ Z[X] | |ai | ≤ n for all i}|. Probabilistic interpretations of such limits of fractions as n → ∞ occur frequently in the literature: cf. [9, p. 269], [11, pp. 180–181]. Evidently, Bm,n = (2n+1)m ; and it 1 A = 1, thus is known that Am,n = O(nm− 2 ln n) (cf. [4]). It follows that limn→∞ Bm,n m,n establishing the above-stated motivating result (provided that all the intervening

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polynomials are decreed to be “equally likely”). Our interest here is in analogous questions for polynomials over ﬁnite ﬁelds. (As usual, Fq denotes the ﬁnite ﬁeld with q elements, where q ≥ 2 is an arbitrary prime-power.) Speciﬁcally, for each positive integer m and each prime-power q, let Am,q := |{f ∈ Fq [X] | deg(f ) = m and f is irreducible in Fq [X]}| and Bm,q := |{f ∈ Fq [X] | deg(f ) = m}|. A

m,q as the probability that an arbitrary m-th degree Of course, one can interpret Bm,q polynomial over Fq is irreducible (provided that all such polynomials are decreed to be “equally likely”). Examples show that this ratio depends on q. By considering limits as q → ∞, some underlying structure became more apparent. Probabilistic interpretations for such limits of ratios as q → ∞ have occurred often recently; see, for instance, [5] and [1]. Building on a classic result on ﬁnite ﬁelds that can be found in some lecture notes of Carlitz [2, Theorem 6.2], Proposition 3.2 (a) determines Am,q 1 = m . (It is easy to see that these limits would be unchanged if that limq→∞ Bm,q the deﬁnitions of Am,q and Bm,q were revised so as to restrict attention to monic polynomials.) In view of the probabilistic comments in the preceding paragraph, Proposition 3.2 (a) thus demonstrates a signiﬁcant way in which polynomials over ﬁnite ﬁelds behave diﬀerently from integral polynomials. Then, as a consequence of Lemma 3.1, Corollary 3.3 obtains a new proof of Theorem 1.1. Throughout, we use notation with the meanings given above, in particular for m, q, Am,q , Bm,q and Fq . As usual, µ denotes the M¨obius function; “∼” means “is asymptotic to”; and deg(f ) denotes the degree of an arbitrary polynomial f .

3

The Limiting Probability of Irreducibility Leads to the Identity

The above discussion explained the sense in which a random monic integral polynomial is likely to be irreducible, and it seems natural to try to determine the corresponding probability for polynomials over a ﬁnite ﬁeld. This is done in Proposition 3.2 (a). That result depends, in turn, on a classic result on ﬁnite ﬁelds, which is restated as the ﬁrst assertion in Lemma 3.1 and which will be very helpful in answering our basic question. A complete proof of this classic result, as given in [2, Theorem 6.2], follows via degree considerations and the M¨obius inversion formula. (As a more accessible alternative to [2], see the method of proof in either [10, Exercise 22, p. 254] or [7, pp. 567–568].) In using Lemma 3.1, we will only need to know that µ(1) = 1. Lemma 3.1. Let m be a positive integer and let q ≥ 2 be a prime-power. Then the number of monic irreducible polynomials of degree m in Fq [X] is ψq (m) :=

1 m µ( )q d . m d|m d 1≤d≤m

Using Factorizations to Prove a Partition Identity In particular, ψq (m) ∼

qm m

229

(as q → ∞).

Proposition 3.2. Let m be a positive integer and let q ≥ 2 be a prime-power. Then: (a) The probability that an m-th degree polynomial in Fq [X] is irreducible is 1 m Am,q = µ( )q d−m . Bm,q m d|m d 1≤d≤m

In particular, as q → ∞, the limiting value of this probability is

1 m.

A

m,q (b) limm→∞ (limq→∞ Bm,q ) = 0. In other words, for each > 0, there exists a positive integer M such that, for each integer m > M , there exists a prime-power Am,q < for all prime-powers q > Q. Q depending on m such that Bm,q

Proof. (a) By a fundamental counting principle, we have that Am,q = (q − 1)ψq (m) and Bm,q = (q − 1)q m . Thus, the required probability is Am,q ψq (m) 1 m = = µ( )q d−m , m Bm,q q m d|m d 1≤d≤m

where the last-displayed expression is obtained via Lemma 3.1. After this expres1 , the dominant term in the resulting sion is rewritten by multiplying through by m 1 m m−m 1 = m , and so the ﬁnal assertion is immediexpression (as q → ∞) is m µ( m )q ate. 1 = 0, the ﬁrst assertion follows from (a). The second asser(b) Since limm→∞ m tion then follows from the deﬁnition of limit. Thus, although the probability is 0 that a random monic integral m-th degree polynomial is irreducible over Z, Proposition 3.2 (a) yields that the probability of irreducibility for a random monic m-th degree polynomial over a “large” ﬁnite 1 . Note also that Proposition 3.2 (b) has some of the ﬂavor of ﬁeld is essentially m [6, Remark 4.1 (c)], which made precise the assertion that Eisenstein’s Criterion “likely” fails to apply to random monic integral polynomials of “large” degree with coeﬃcients that are “large” in absolute value. Proposition 3.2 (b) can also be viewed as an analogue of the following consequence of the Prime Number Theorem: = 0, where π(n) is, as usual, the number of positive prime integers limn→∞ π(n) n which do not exceed n. We close by showing that the above considerations of limiting probabilities lead to a new proof of the motivating identity in Theorem 1.1. We restate the result for convenience. Corollary 3.3. If m is any positive integer, then 1≤d1 <···
1 = 1. e1 ! · · · · · ek ! · de11 · · · · · dekk

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1 Proof. For m = 1, the assertion reduces to the triviality that 1!·1 1 = 1, and so we may assume henceforth that m ≥ 2. Let q ≥ 2 be a prime-power. As above, let ψq (m) denote the number of monic m-th degree irreducible polynomials in Fq [X]. Our strategy for proving Corollary 3.3 involves comparing two expressions to which ψq (m) is asymptotic. Reasoning as in the proof of Proposition 3.2 (a), we see that the number of monic m-th degree polynomials in Fq [X] is q m . Thus, to count ψq (m), we need only subtract the number of monic m-th degree reducible polynomials in Fq [X] from q m . Accordingly, we proceed to count these reducible polynomials. Of course, any monic reducible polynomial in a polynomial ring over a ﬁeld has a unique (apart from order of factors) factorization p1 · · · · · ps , where each pj is monic and irreducible (and p1 , . . . , ps are listed possibly with repetition). Using Lemma 3.1, we can determine the number of such factorizations as an integral polynomial in q. To determine the dominant term in this polynomial expression (as q → ∞), we can ignore the factorizations having at least one repeated factor pj (for the number of such factorizations is an integral polynomial whose degree in q is not maximal). Now, any factorization of a reducible monic m-th degree polynomial without repeated factors uses e1 distinct monic irreducible factors of degree d1 , e2 distinct monic irreducible factors of degree d2 ,. . . , and ek distinct monic irreducible factors of degree dk , where 1 ≤ d1 < · · · < dk ≤ m − 1 and e1 d1 + · · · + ek dk = m. Given such a partition, e1 d1 + · · · + ek dk = m, the dominant term (as q → ∞) in the number of factorizations that it describes is P

k qm q di ei (q di )ei . ei = ei = (ei ! · di ) (ei ! · dei i ) e ! · di i=1 i di

(Indeed, the ﬁnal assertion in Lemma 3.1 shows that ( qdi )ei is the dominant term (as q → ∞) in the polynomial (in q) expression that counts the number of ways of listing ei monic irreducible factors each having degree di ; and the factor e1i ! is used to avoid an overcount, since permuting factors does not create a “new” factorization.) It now follows that ψq (M ) is asymptotic (as q → ∞) to its dominant term, namely, qm −

1≤d1 <···
qm = q m (1 − (ei ! · dei i )

1≤d1 <···
1 ). (ei ! · dei i )

However, Lemma 3.1 ensures that ψq (M ) is also asymptotic (as q → ∞) to Therefore, since ∼ is an equivalence relation, q m (1 −

1≤d1 <···
and so 1−

1≤d1 <···

qm 1 as q → ∞ ei ) ∼ m (ei ! · di )

1 1 ∼ as q → ∞. (ei ! · dei i ) m

qm m .

Using Factorizations to Prove a Partition Identity

231

Of course, expressions that are asymptotic as q → ∞ but do not depend on q must be equal. Hence

1−

1≤d1 <···
1=

1 + m

1≤d1 <···

1 1 = , and so m (ei ! · dei i )

1 = (ei ! · dei i )

1≤d1 <···

1 . (ei ! · dei i )

Acknowledgement: We thank Robert M. McConnel for introducing us to the lecture notes of Carlitz [2]. Both authors were supported in part by the National Science Foundation through its Research Experiences for Undergraduates site at the University of Tennessee, Knoxville.

Bibliography [1] G. D. Bullington, On the expected number of generators of a submodule of a free module over a ﬁnite special principal ideal ring, Rend. Circ. Mat. Palermo 51 (2002), 5–50. [2] L. Carlitz, Arithmetic of Polynomials, mimeographed notes, Duke University, Durham, 1961. [3] D. I. A. Cohen, Basic Techniques of Combinatorial Theory, Wiley, New York, 1978. [4] S. D. Cohen, The distribution of Galois groups and Hilbert’s irreducibility theorem, Proc. London Math. Soc. 43 (1981), 227–250. [5] D. E. Dobbs and M. J. Lancaster, The expected dimension of a sum of vector subspaces, Bull. Austral. Math. Soc. 45 (1992), 467–477. [6] D. E. Dobbs and L. E. Johnson, On the probability that Eisenstein’s Criterion applies to an arbitrary irreducible polynomial, Advances in Commutative Rings, Lecture Notes Pure Appl. Math., Marcel Dekker 205 (1999), 241–256. [7] D. S. Dummit and R. M. Foote, Abstract Algebra, Second Edition, Wiley, New York, 1999. [8] N. J. Fine, Basic Hypergeometric Series and Applications, Mathematical Surveys and Monographs, 27, Amer. Math. Soc., Providence, 1988. [9] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 1979. [10] S. Lang, Algebra, Third Edition, Addison-Wesley, Reading, MA, 1993. [11] I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, Third Edition, Wiley, New York, 1972.

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Chapter 14

On Inside Factorial Integral Domains David E. Dobbs, Gabriel Picavet, and Martine Picavet-L’Hermitte by

Abstract The inside factorial integral domains with ﬁnite Cale bases are characterized and are shown to be certain g-domains of (Krull) dimension at most 1. In the quasilocal case, their Cale bases each have at most one element. A semiquasilocal or almost weakly factorial integral domain that is locally inside factorial must be inside factorial. Transfer of the “inside factorial” property is shown for, i.a., integral domains R ⊂ T with the same prime ideals such that U(T )/U(R) is a torsion group. Characterizations of the inside factorial integral domains are given in the classes of one-dimensional Noetherian integral domains, valuation domains, pseudo-valuation domains and Pr¨ ufer domains. An inside factorial integral domain has dimension at most 1 if and only if its integral closure is a Pr¨ ufer domain. Any overring of an inside factorial Pr¨ ufer domain is an inside factorial QR-domain.

2000 Mathematics Subject Classification: Primary 13F15, 13F05, 13G05; Secondary 13C20, 13F30. Key words and phrases: Inside factorial integral domain, Cale basis, rational generalized Krull domain, t-class group, root extension, treed domain, Pr¨ ufer domain.

1

Introduction, Background and Notation

Let us ﬁrst collect some notation. If R is a (commutative) integral domain, we let R× denote the multiplicative monoid R \ {0}; U(R) the set of units of R; R the integral closure of R (in its quotient ﬁeld); Spec(R) the set of prime ideals of R; Max(R) the set of maximal ideals of R; X (1) (R) the set of height 1 prime ideals of 233

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R; dim(R) the (Krull) dimension of R; Clt (R) the t-class group of R (in the sense of [1]); and Pic(R) the Picard group of R. It is also convenient to let N (resp., N0 ) denote the set of positive (resp., nonnegative) integers. Our interest here is in the inside factorial integral domains, a class of integral domains introduced recently by Chapman, Halter-Koch, and Krause [2]. In fact, the “inside factorial” concept, together with the associated concept of “Cale basis”, was ﬁrst introduced in [2] for monoids and then interpreted for an integral domain R by means of the monoid R× . The deﬁnitions of the “inside factorial” and “Cale basis” concepts are recalled in Deﬁnition 1.1 below. To convey some intuition about some of the key concepts that impinge on this study, we oﬀer the following three comments. Grosso modo, in an inside factorial integral domain, a Cale basis plays a role analogous to that customarily played in a UFD by a set of associate class representatives of the irreducible elements. Also, just as a DVR is arguably the most natural kind of UFD, rational valuation domains are a very natural kind of rational generalized Krull domain (deﬁnition recalled in Deﬁnition 1.3). And rational generalized Krull domains play a key role in characterizing inside factorial integral domains: see the characterization from [2] that is restated in Theorem 1.4. Before stating the aforementioned material and a couple of other background results, we devote the next three paragraphs to summarizing what is new in this paper, namely, the content of Sections 2–6. Section 2 focuses on inside factorial domains having ﬁnite Cale bases. As one may expect from the example of a DVR (which has only singleton or empty Cale bases), Proposition 2.1 shows that each inside factorial integral domain with a ﬁnite Cale basis has dimension at most 1. Moreover, a quasilocal inside factorial integral domain R is of dimension at most 1 if and only if R has a singleton Cale basis (Proposition 2.2). The inside factorial integral domains with ﬁnite Cale bases are shown to be g-domains (in the sense of [3]) in Proposition 2.3 and are characterized in Theorem 5.3. The proof of the latter result depends, in part, on Theorem 3.1 (b): if a semiquasilocal integral domain R is such that RM is an inside factorial integral domain for each M ∈ Max(R), then R is an inside factorial integral domain. An example is given to show that this assertion is false if one deletes the “semiquasilocal” hypothesis. Nevertheless, Section 3, which features our most intricate arguments, is devoted to additional instances in which “locally inside factorial” integral domains must be inside factorial. In this regard, the “almost weakly factorial” and one-dimensional Noetherian contexts are treated in Propositions 3.4 and 3.3, respectively. Section 3 also includes a new characterization of inside factorial integral domains (Corollary 3.5) and the stability of “inside factorial” under rings of fractions (Theorem 3.1 (a)). It is known (see Corollary 1.5) that the integral closure of any inside factorial integral domain is itself inside factorial. Section 4 develops additional transfer results for the “inside factorial” property, with particular emphasis on the “Spec(R) = Spec(T )” context of [4]. Also, in Section 4, we give characterizations of the inside factorial valuation domains in Proposition 4.3 and the inside factorial pseudo-valuation domains (in the sense of [5]) in Proposition 4.4. Noteworthy results in Section 5 include the facts that an inside factorial integral domain R has

On Inside Factorial Integral Domains

235

dimension at most 1 if and only if R is a Pr¨ ufer domain (Corollary 5.2); and that the class of inside factorial Pr¨ ufer domains is stable under the formation of overrings, which actually turn out to be QR-domains (Theorem 5.4). Finally, to complete the summary of our results in Sections 2–6, we turn to Section 6. In his talk at the Chapel Hill conference in October, 2003, Ulrich Krause referred to the “tamely inside factorial integral domains” introduced in [2, Deﬁnition 3] as “Cale domains.” It is relatively straightforward to see that all our results on inside factorial integral domains carry over to the “Cale domain” context. Section 6 documents how to obtain the “Cale domain” analogues of the assertions in Sections 2–5. We turn next to the speciﬁcs of some useful background material. The following fundamental deﬁnition is a reworking of a characterization of the inside factorial monoids [2, Proposition 4]. Definition 1.1. (1) Let R be an integral domain. A subset of nonunit elements Q = {qi }i∈I of R× is said to be a Cale basis of R if the following two properties hold: a. For each nonunit a ∈ R× , there exist n ∈ N, ti ∈ N0 and u ∈ U(R) such that an = u i∈I qiti , where only ﬁnitely many of the ti ’s are nonzero. b. If u i∈I qiti = v i∈I qisi where u, v ∈ U(R) and si , ti ∈ N with si = ti = 0 for almost all i ∈ I, then u = v and ti = si for all i ∈ I. (2) An integral domain R is called (an) inside factorial (integral domain) if there exists a Cale basis of R. It is easy to check that if R is an integral domain, then the empty set is a Cale basis of R if and only if R is a ﬁeld. Thus, a ﬁeld is trivially seen to be an inside factorial integral domain. Note also that if Q = {qi }i∈I is a Cale basis of an (inside factorial) integral domain R and if {ui }i∈I ⊆ U(R), then {ui qi }i∈I is a Cale basis of R. More generally, [2, Corollary 2] shows how to use a Cale basis of a given inside factorial integral domain R to obtain all the Cale bases of R. This result implies, in particular, that any two Cale bases of a given (inside factorial) integral domain have the same cardinality. The following useful theorem was given in [2] in the context of monoids, but it is straightforward to carry that proof over to the context of integral domains. (To do so, note that the height 1 prime ideals of an integral domain R which is not a ﬁeld are the sets of the form P ∪ {0}, where P ranges over the minimal semigroup-theoretic prime ideals of R× .) Theorem 1.2. ([2, Theorem 2] Structure of Cale bases) Let R be an inside factorial domain and Q ⊆ R× . Then Q is a Cale basis of R if and only if the function √ iQ : Q → X (1) (R), given by iQ (q) = Rq for each q ∈ Q, is a bijection. Just as DVRs are the important kind of valuation domains arising in the study of Krull domains, it turns out that rational valuation domains (that is, valuation

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domains with value group a subgroup of the additive group of rational numbers) are important tools in studying the rational generalized Krull domains that are deﬁned next. Definition 1.3. ([6] and [2]) Let D be an integrally closed domain. Then D is called a rational generalized Krull domain if the following four properties are satisﬁed by some family F = {Vλ }λ∈Λ of valuation overrings of D: 1. D = ∩ Vλ . λ∈Λ

2. For each index λ, Vλ is a rational valuation domain. 3. The family F has ﬁnite character. 4. Each Vλ is essential for D (that is, Vλ = DPλ for some Pλ ∈ Spec(D)). Remark Each rational generalized Krull domain D is an integral domain of ﬁnite real character (in the sense of [6, page 524]). It follows from [6, Proposition 43.7] that D = ∩ {DP | P ∈ X (1) (D)} and that this representation is of ﬁnite character, since height 1 prime ideals are essential in any integral domain of ﬁnite real character. We next state an important and very useful characterization of inside factorial integral domains that was given in [2]. First, recall from [7] that an extension R ⊆ S of (commutative) rings is called a root extension if, for each s ∈ S, there exists a natural number n (depending on s) such that sn ∈ R. Theorem 1.4. ([2, Theorem 7 (a)]) Let R be an integral domain (and R its integral closure). Then R is an inside factorial integral domain if and only if R ⊆ R is a root extension, R is a rational generalized Krull domain and Clt (R ) is a torsion group. Recall from [8] that an integral domain D is called almost weakly factorial if, for each nonzero nonunit x ∈ D, some (positive) power of x is a product of primary elements. The second assertion in Corollary 1.5 seems not to have been recorded earlier. Corollary 1.5. Let R be an inside factorial integral domain. Then R is inside factorial and almost weakly factorial. Proof. The ﬁrst assertion is an immediate consequence of Theorem 1.4. It remains to show that R is almost weakly factorial. In view of the Remark following Deﬁnition 1.3, the assertion follows from [8, Theorem 3.4 (3)]. In the spirit of the comment preceding Theorem 1.2, the referee has noted that several of our (ring-theoretic) results remain valid for monoids. We thank the referee for noting this in regard to Corollary 1.5 (with what s/he describes as “a suitable deﬁnition of an almost weakly factorial monoid”), the ﬁrst assertion in Proposition 2.3, and Theorem 3.1 (a).

On Inside Factorial Integral Domains

2

237

Inside Factorial Integral Domains with a Finite Cale Basis

As a modest generalization of the class of DVRs, we begin by studying the inside factorial integral domains having ﬁnite Cale bases. Note that for each n ∈ N, there does exist an inside factorial integral domain (indeed, a UFD) R which has a Cale basis of cardinality n. Indeed, if {pi } enumerates the prime numbers in increasing order and S is the multiplicatively closed subset of Z generated by {pj | j > n}, then R := ZS has the set {pi | 1 ≤ i ≤ n} of the ﬁrst n prime numbers as a Cale basis. This observation about rings of fractions is generalized to arbitrary inside factorial integral domains in Theorem 3.1 (a). Proposition 2.1. Let R be an inside factorial integral domain with a ﬁnite Cale basis Q; let n denote the cardinality of Q. Then dim(R) ≤ 1 and R has exactly n + 1 prime ideals. √ Proof. Write Q = {q1 , . . . , qn } and put Pi := Rqi for i = 1, . . . , n. Consider any nonzero prime ideal P of R and any nonzero element x ∈ P . Then there exist n ∈ N and ti ∈ N0 such that xn = u ni=1 qiti , where u ∈ U(R). At least one ti is nonzero since x is not a unit, and so x ∈ ∪ni=1 Pi . Hence, P ⊆ ∪ni=1 Pi and, by the Prime Avoidance Lemma (cf. [6, Proposition 4.9]), P ⊆ Pi for some i. Since Pi ∈ X (1) (R) by Theorem 1.2, it follows that P = Pi . Therefore, each nonzero prime ideal of R has height 1. An appeal to Theorem 1.2 completes the proof. The next result essentially gives a converse of Proposition 2.1 in the quasilocal case. First, recall that an integral domain R is called a treed domain if Spec(R), as a poset under inclusion, is a tree, that is, if no prime (resp., maximal) ideal of R can contain incomparable prime ideals of R. Proposition 2.2. Let R be a quasilocal inside factorial integral domain. Then the following conditions are equivalent: (1 ) R is a treed domain; (2 ) Some Cale basis of R has at most one element; (3 ) Each Cale basis of R has at most one element; (4 ) dim(R) ≤ 1. Proof. (1) ⇒ (2) by Theorem 1.2, since any quasilocal treed domain has at most one height 1 prime ideal; (2) ⇔ (3) since all Cale bases of a given inside factorial integral domain have the same cardinality; given (2), Proposition 2.1 yields that R is an integral domain with at most two prime ideals and (4) follows; and (4) ⇒ (1) is obvious. In view of Proposition 2.2, we wish to raise the question of ﬁnding necessary and suﬃcient conditions for a quasilocal one-dimensional integral domain to be inside factorial. In Theorem 5.3, we will obtain a characterization of the domains that were studied in Proposition 2.1. In the ﬁnal result of the present section, we show that

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these domains are examples of g-domains. Recall (cf. [3]) that a (commutative) ring (resp., integral domain) R is called a g-ring (resp., g-domain) if, for each P ∈ Spec(R), there exists f ∈ R \ P such that Rf ∼ = RP canonically. Proposition 2.3. Let R be an inside factorial integral domain with a ﬁnite Cale basis Q = {q1 , . . . , qn }. Then each ring of fractions of R is of the form Ra for some a ∈ R× . Moreover, R is a g-domain. Proof. If n = 0, then R is a ﬁeld, in which case the assertions are clear. Suppose henceforth that n > 0. It will be convenient to use the following notation: for each I ⊆ {1, . . . , n}, put aI := i∈I qi . We will show that each ring of fractions RS of R is of the form Ra for some a ∈ R× . Without loss of generality, the multiplicatively closed subset S is such that 0 ∈ S and S ⊆ U(R). Consider any nonzero nonunit element x ∈ R. Since R is inside factorial, n there exists k ∈ N such that xk = u i=1 qiti , where u ∈ U(R) and ti ∈ N0 . Set I := {i ∈ {1, . . . , n} | ti = 0}. Straightforward calculations verify that Rx = Rxk = RaI . It follows that RS = ∪s∈S Rs ⊆ RaJ , where J := {i ∈ {1, . . . , n} | qi divides some power of an element of S}. We claim that the reverse inclusion holds as well. To see this, assume for simplicity that J = {1, . . . , k} for i = ri qi for some some 1 ≤ k ≤ n. Then for each i = 1, . . . , k, we can write sm i k si ∈ S, mi ∈ N and ri ∈ R. Observe that s := i=1 si ∈ S and 1 r1 · · · · · rk 1 = = m1 ∈ Rs1 · ··· ·sk = Rs . k aJ q1 · · · · · qk s1 · · · · · sm k It follows that RaJ ⊆ Rs , thus establishing the above claim. Finally, for the “Moreover” assertion, it remains to prove that if P ∈ Spec(R), then there exists f ∈ R \ P such that Rf = RP . In view of the preceding paragraph, it suﬃces to observe that if S := R \ P and J is as above, then aJ ∈ S. This, in k turn, follows from the fact that P is a prime ideal of R, since aJ = i=1 qi and the i = ri qi ensures that qi ∈ P . The proof is complete. equation sm i

To close the section, we note that by reexamining the proof of Proposition 2.3 (and of Theorem 3.1 (a) below), one can use a Cale basis Q of an (inside factorial) integral domain R to describe all the saturated multiplicatively closed subsets of R that do not contain 0. Indeed, deﬁne a function ϕ from the set of all the subsets of Q to the set of all the saturated multiplicatively closed subsets of R that do not contain 0, as follows. If X ⊆ Q, then ϕ(X) is deﬁned to be the saturated multiplicatively closed subset of R generated by X. Then ϕ is a bijection. Moreover, the settheoretic inverse ψ of ϕ is given by ψ(S) := {q ∈ Q | there exists x ∈ S and n ∈ N such that q divides xn }. The veriﬁcation is routine, but tedious, and is left to the interested reader.

On Inside Factorial Integral Domains

3

239

Behavior of the Inside Factorial Property with Respect to Localization and Globalization

It is well-known that any nontrivial ring of fractions of a UFD is a UFD. As a partial converse, a result of Gilmer [9] states that if a semiquasilocal integral domain is locally a UFD, then it is a UFD. We begin this section with a kindred result for inside factorial integral domains. Note also that Theorem 3.1 (a) generalizes the example mentioned at the beginning of the previous section. Theorem 3.1. Let R be an integral domain. Then: (a) If R is an inside factorial integral domain, then RS is an inside factorial integral domain for each multiplicatively closed subset S of R that does not contain 0. Under these conditions, if Q is a Cale basis for R, then QS := Q \ S is a Cale basis for RS . (b) Suppose that the integral domain R is semiquasilocal. Then R is an inside factorial integral domain if and only if RM is an inside factorial integral domain for each M ∈ Max(R). Proof. (a) Let R be an inside factorial integral domain with Cale basis Q = {qi }i∈I . Let S be a multiplicatively closed subset of R which does not contain 0. Without loss of generality, S is saturated. It follows easily that U(RS ) = {a/b | a, b ∈ S}. We proceed to show that QS satisﬁes the ﬁrst requirement (see condition (a) in Deﬁnition 1.1) for a Cale basis of RS . Consider any nonzero nonunit y of RS . By the above comment, we can write y = x/s ∈ RS , with x ∈ R\U(R), s ∈ S. Since R is inside factorial, there exist n ∈ N and ti ∈ N0 such that xn = u i∈I qiti , where u ∈ U(R) and only ﬁnitely many of the ti are nonzero. Then y n = [(u/sn ) qi ∈Q∩S qiti ] qi ∈QS qiti = v qi ∈QS qiti , for a suitable element v ∈ U(RS ) since Q ∩ S ⊆ U(RS ). This completes the veriﬁcation of condition (a). We turn next to condition (b) in Deﬁnition 1.1. t Suppose that v qi ∈QS qiti = v qi ∈QS qi i , with ti , ti ∈ N0 and v, v ∈ U(RS ). t Clearing denominators, we get z := y qi ∈QS qiti = y qi ∈QS qi i , with y, y ∈ S. Observe that each element of a Cale basis of R is a nonunit of R. Consequently, if z ∈ U(R), then ti = ti = 0 for each i, v = v and condition (b) has been veriﬁed. Thus, we may assume, without loss of generality, that z is a (nonzero) nonunit of R. As Q is a Cale basis of R, there exists k ∈ N such that z k , y k and (y )k are each expressible as, apart from a unit factor, a product of elements of Q. Next, note that no power of y (or y ) is divisible by any element of QS (for y n = qi r, n ∈ N, qi ∈ Q, r ∈ R implies that qi ∈ S since S is saturated). Since Q satisﬁes the “uniqueness” condition (b) in Deﬁnition 1.1, we get from the ensuing factorizations of z k that kti = kti for each i, whence ti = ti for each i, whence v = v . This completes the proof that RS (is an inside factorial integral domain that) has Cale basis QS . (b) Suppose that R is semiquasilocal, with Max(R) = {M1 , . . . , Mn }. Suppose also that RMi is inside factorial for each i = 1, . . . , n. Then, by Theorem 1.4,

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RMi ⊆ (RMi ) = RM is a root extension for each i. It follows that R ⊆ R is also i a root extension. (In detail, we need only show that each u ∈ R has some power n in R. Observe that there exists d ∈ N such that ud ∈ ∩ RMi = R.) Therefore, i=1

the canonical map Spec(R ) → Spec(R) is a bijection ([7], [2]). Thus, for each i = 1, . . . , n, there exists a unique maximal ideal Mi in R that lies over Mi , and = RM RM by [6, Corollary 5.2]. Next, note by Theorem 1.4 that RM = (RMi ) i i i is a rational generalized Krull domain for each i. By verifying the conditions in Deﬁnition 1.3, we can now check that R is also a rational generalized Krull domain. n by globalization and that any localization of an (To do so, note that R = ∩ RM i i=1

is a localization of R .) Using [8, Theorem 3.5], we get that Clt (R )/Pic(R ) RM i is almost weakly factorial by Corollary 1.5. But is a torsion group, since each RM i Pic(R ) = 0 since R is semiquasilocal, and so Clt (R ) is a torsion group. Therefore, by Theorem 1.4, R is inside factorial.

It is well-known that if R ⊆ T is a root extension of integral domains, then RS ⊆ TS is also a root extension for each multiplicatively closed subset S of R. The following result establishes a strong converse that is of some independent interest. The special case of Proposition 3.2 in which R is semiquasilocal was established as part of the proof of Theorem 3.1 (b). Proposition 3.2. If R ⊆ T is an extension of integral domains such that RM ⊆ TM is a root extension for each M ∈ Max(R), then R ⊆ T is a root extension. Proof. By an easy calculation, the problem reduces to showing that if u ∈ T satisﬁes n (R :R ui ) = R for some n ∈ N, then some power of u is in R. Write 1 = i=1 n i i=1 ri , where ri ∈ (R :R u ) for all i = 1, . . . , n. (The interested reader may reformulate the argument by using the quasicompactness of Spec(R) in the Zariski topology to reach the preceding description of the problem.) We shall show that un! ∈ R. It is enough to ﬁnd a positive integer N such that each term in the n “Multinomial Theorem” expansion of ( i=1 ri )N un! contains a factor of the form n! (ri ui ) i . In other words, if 0 ≤ i1 , . . . , in ≤ N satisfy i1 + · · · + in = N , we wish at least one inequality of the form ik ≥ n! k to hold for some k = 1, . . . , n. The proof concludes by noticing that it suﬃces to choose the positive integer N ≥ n n!( i=1 1i ). It is convenient to say that an integral domain R is (a) locally inside factorial (integral domain) if RP is inside factorial for each P ∈ Spec(R). By Theorem 3.1 (a) (and the fact that RP = (RM )P RM whenever P ⊆ M are prime ideals of R), this condition is equivalent to requiring that RM be inside factorial for each M ∈ Max(R). It is clear from Theorem 3.1 (a) that each inside factorial integral domain is locally inside factorial. However, the converse is false. Indeed, consider any Dedekind domain R whose t-class group is not torsion. Then R is locally inside factorial because RM is a DVR (hence, inside factorial) for each nonzero M ∈ Max(R), but Proposition 3.4 explains why R is not inside factorial.

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Corollary 3.3. Let R be a locally inside factorial integral domain (for instance, an inside factorial integral domain). Then dim(R) ≤ 1 if and only if R is a treed domain. Proof. The “only if” assertion is trivial. Conversely, assume that R is a treed domain. Then, for each M ∈ Max(R), RM is a quasilocal treed domain which, by Theorem 3.1 (a), is inside factorial. Therefore, by Proposition 2.2, dim(RM ) ≤ 1 for all M , whence dim(R) ≤ 1. Going beyond the nonquasisemilocal case, we next oﬀer three results, as companions for Theorem 3.1 (b), that give necessary and suﬃcient conditions for certain kinds of integral domains to be inside factorial. Note that Proposition 3.4 also serves to complete the above construction of a locally inside factorial integral domain that is not inside factorial. Proposition 3.4. Let R be a one-dimensional Noetherian integral domain. Then the following conditions are equivalent: (1 ) R is an inside factorial integral domain; (2 ) Clt (R ) is a torsion group and, for each M ∈ Max(R), RM is an inside factorial integral domain; is a DVR and (3 ) Clt (R ) is a torsion group and, for each M ∈ Max(R), RM )/U(RM ) is a torsion group. U(RM Proof. (1)⇒(2): Apply Theorem 1.4 and Theorem 3.1 (a). (2)⇒(3): Let M ∈ Max(R). Since RM is inside factorial, Theorem 1.4 en = (RM ) is a root extension. Hence, the canonical map sures that RM ⊆ RM ) → Spec(RM ) is a bijection [2, Proposition 5 (b)], indeed an orderSpec(RM = (RM ) is a quasilocal Dedekind domain (by the above isomorphism. Then RM order-isomorphism and a standard consequence of the Krull-Akizuki Theorem [10, Corollary 2, page 501]), and so it is a DVR. )/U(RM ) is a torsion group. This, in turn, is It remains to prove that U(RM a consequence of the following observation. If A ⊆ B is a root extension of integral domains, then, by either the Lying-over Theorem for integral extensions or [2, Proposition 5 (a)], A∩U(B) = U(A), and it follows easily from the “root extension” hypothesis that U(B)/U(A) is a torsion group. (3)⇒(1): Assume (3). Since the class group Clt (R ) is a torsion group, it is enough to show that R ⊆ R is a root extension, by the characterization of Noetherian inside factorial integral domains in [2, Corollary 6]. We shall show that if u ∈ R \ R, then um ∈ R for some m ∈ N. As u is integral over R, it follows that the overring S := R[u] is a ﬁnitely generated R-module, and so the conductor I := (R :R S) = 0. Of course, I = R since u ∈ R. Since R is Noetherian and one-dimensional, there are only ﬁnitely many maximal ideals of R that contain I; let M1 , . . . , Mn denote these maximal ideals. Now, consider any M ∈ Max(R) such that I ⊆ M . Then SM = RM , whence um ∈ RM for all m ∈ N. Therefore, by globalization, it suﬃces to ﬁnd m ∈ N such that um ∈ RMi for all i = 1, . . . , n. Indeed, it suﬃces to ﬁnd, for each ﬁxed index i,

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a positive integer mi such that umi ∈ RMi (for a common multiple of m1 , . . . , mn is then a satisfactory value of m). For the ﬁxed index i, let vi denote a (discrete rank 1) valuation on the quo and the value group of tient ﬁeld of R such that the valuation ring of vi is RM i ) = N and H := v (R ) is a nonzero submonoid of N0 . vi is Z. Then vi (RM 0 i Mi i mi ∈ U(RMi ) ⊆ RMi beIn case u ∈ U(RMi ), there exists mi ∈ N such that u )/U(R ) is a torsion group. Assume henceforth that u ∈ U(RM ). cause U(RM M i i i h Then k := vi (u) > 0. Pick any nonzero element h ∈ H. Then vi (u ) = hvi (u) = hk ∈ H = vi (RMi ). Pick z ∈ RMi such that vi (z) = hk. Therefore, there exists ) such that uh = wz. Now, since U(RM )/U(RMi ) is a torsion group, we w ∈ U(RM i i p can ﬁnd p ∈ N such that w ∈ U(RMi ) ⊆ RMi . It follows that uhp = wp z p ∈ RMi . Thus, hp can be chosen as a satisfactory value for mi , and the proof is complete. Proposition 3.5. Let R be an almost weakly factorial integral domain. Then R is an inside factorial integral domain if and only if RM is an inside factorial integral domain for each M ∈ Max(R). Proof. The “only if” assertion is immediate from Theorem 3.1 (a) (even without the “weakly factorial” hypothesis). Conversely, assume, for each M ∈ Max(R), that RM is an inside factorial integral domain, with Cale basis QM . Consider √ any element q ∈ QM . By Theorem 1.2, q RM = P RM , for some P ∈ X (1) (R) such that P ⊆ M . We claim that we can replace q with an element q such that √ qRM = P RM and q ∈ P . Since R is almost weakly factorial, [8, Theorems 3.1 and 3.4] yields k ∈ N such that (q )k = q1 · · · · · qn , where for each index i, the element qi is a Pi -primary element for some Pi ∈ X (1) (R), and Pi = Pj if i = j. Since P RM is a prime ideal of RM , some qi ∈ P RM ∩ R = P . Thus, some P -primary element √ q (= qi ) divides (q )k . Indeed, Pi = qi R ⊆ P , which implies Pi = P since these prime ideals each have height 1. Now, consider any index j = i. We know that Pj = P . Moreover, Pj ⊆ M . (Indeed, if Pj ⊆ M , then since q ∈ Pj , we would have Pi RM = P RM = q RM ⊆ Pj RM , whence the (height 1) primes Pi RM and Pj RM would coincide and Pi = Pj , a contradiction.) Now, if W ∈ Spec(R) is such that qi ∈ W ⊆ M , then P RM = √ qi RM ⊆ W RM and P ⊆ W . It follows that P is the unique minimal element √ √ among the prime ideals of R that contain qi = q; that is, qRM = qi RM = P RM = Pi RM . One upshot of the above reasoning is that for each height 1 prime ideal P ⊆ M of R, we have associated a P -primary element q ∈ R. By Theorem 1.2, the set of such elements q constitutes a new Cale basis QM of RM . We shall QM . show that R is inside factorial with Cale basis Q := ∪ M∈Max(R)

Consider any nonzero nonunit element x ∈ R. Since R is almost weakly factorial, there exists m ∈ N such that xm = x1 · · · · ·xn , where each xi is a Pi -primary element for some height 1 prime ideal Pi of R. Fix an index i for the moment, and pick M ∈ Max(R) such that Pi ⊆ M . Then xi is a Pi RM -primary element in RM . As RM is inside factorial, there exist (shades of DVRs!) r, s ∈ N and a, b ∈ R \ M such that xri = (a/b)qis , with qi ∈ QM . Hence,

On Inside Factorial Integral Domains

243

bxri = aqis in R. Moreover, by invoking the “almost weakly factorial” condition (and [8, Theorems 3.1 and 3.4]) again, we see that there exists t ∈ N such that at and bt are each expressible as products of primary elements of R, necessarily none of which factors can be in Pi . By the unique factorization into primary elements corresponding to distinct prime ideals of R [11, Theorem 1.5], we get (upon raising the equation bxri = aqis to the t-th power) that some power of xi is associated to some power of qi . By raising the equation xm = x1 · · · · · xn to a suﬃciently high power, we n see that there exist p ∈ N and ti ∈ N0 such that xp = u i=1 qiti for some u ∈ U(R). This completes the proof that Q satisﬁes condition (a) for a Cale basis of R given in Deﬁnition 1.1. Finally, the “uniqueness” of this factorization (in the sense of condition (b) in Deﬁnition 1.1) follows from the above comments and the corresponding uniqueness exhibited by the Cale basis QM of RM . The proof is complete. Corollary 3.6. Let R be an integral domain. Then the following conditions are equivalent: (1 ) R is inside factorial; (2 ) R ⊆ R is a root extension, R is almost weakly factorial, and RM is inside factorial for each M ∈ Max(R). Proof. (1) ⇒ (2): Combine Theorem 1.4, Corollary 1.5 and Theorem 3.1 (a). (2) ⇒ (1): Assume (2). Since R ⊆ R is a root extension, the canonical map Spec(R ) → Spec(R) is a bijection [2, Proposition 5 (b)], in fact an orderisomorphism. Consider any M ∈ Max(R). Let M denote the (unique) maximal ideal of R that lies over M . By [6, Corollary 5.2] (and the fact that integral exten sions satisfy the going-up property), RM = RM . As RM ⊆ RM = RM is a root extension and root extensions ascend the “inside factorial” property [2, Proposition Then, since R is 5 (f)], RM inherits the “inside factorial” property from RM . assumed to be almost weakly factorial, Proposition 3.5 ensures that R is inside factorial. As root extensions descend the “inside factorial” property [2, Proposition 5 (f)], R is inside factorial, as desired.

4

Transfer of the Inside Factorial Property

By [2, Proposition 5 (e)], if R ⊆ T is a root extension of integral domains, then R is an inside factorial integral domain if and only if T is an inside factorial integral domain. Conversely, we next show that this ascent and descent behavior for the inside factorial property actually serves to characterize root extensions of integral domains. Proposition 4.1. Let R ⊆ T be an extension of integral domains such that R (resp., T ) is an inside factorial integral domain. Then the following conditions are equivalent: (1 ) R ⊆ T is a root extension;

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(2 ) U(T )/U(R) is a torsion group and T (resp., R) is an inside factorial integral domain (with a common Cale basis for R and T ). Proof. (1) ⇒ (2): Let R ⊆ T be a root extension. As in the proof of Proposition 3.4, it follows from the “root extension” hypothesis that U(T )/U(R) is a torsion group. Moreover, by [2, Proposition 5 (f), (e)], T (resp., R) is an inside factorial integral domain when R (resp., T ) is (in which case, there is a common Cale basis for R and T ). (2) ⇒ (1): Now, assume that both R and T are inside factorial, with a common Cale basis Q for R and T , and that U(T )/U(R) is a torsion group. We show that each x ∈ T has some integral power in R. Without loss of generality, x = 0 and x ∈ U(T ). Then there exist n ∈ N and ti ∈ N0 such that xn = u qi ∈Q qiti , where u ∈ U(T ) and only ﬁnitely many of the ti are nonzero. For each index i, we have qi ∈ Q ⊆ R. Since U(T )/U(R) is torsion, there exists k ∈ N such that uk ∈ U(R). It follows that xnk = uk qi ∈Q qikti ∈ R, as desired. We next investigate extensions R ⊂ T of inside factorial integral domains in case R and T have the same set of prime ideals. Corollary 4.2. Let R ⊂ T be an extension of integral domains such that Spec(R) = Spec(T ) as sets (so that R and T are each quasilocal). Assume also that U(T )/U(R) is a torsion group. Then: (a) R is inside factorial if and only if T is inside factorial. (b) If R and T are inside factorial, then each Cale basis of R (resp., T ) is a Cale basis of T (resp., R). (c) If R is a treed domain that is inside factorial, then so is T , dim(R) =dim(T ) ≤ 1 and, if R is not a ﬁeld, then each Cale basis of R (resp., T ) has exactly one element. Proof. The parenthetical assertion was established in [4, Proposition 3.3]. (a) Since Spec(R) = Spec(T ), we have T \ U(T ) = R \ U(R), and so x ∈ R for each nonunit x ∈ T . On the other hand, if x ∈ U(T ), then there exists k ∈ N such that xk ∈ U(R), since U(T )/U(R) is a torsion group. Therefore, R ⊆ T is a root extension. Hence, by [2, Proposition 5 (f)] (or Proposition 4.1), R is inside factorial if and only if T is inside factorial. (b) Each element of a Cale basis of an (inside factorial) integral domain is a nonunit of that integral domain. Accordingly, the assumption that R and T have the same spectra ensures that any Cale basis of either R or T is a subset of R. The assertion in (b) now follows at once from [2, Proposition 5 (e)]. (c) Apply (a) and Proposition 2.2. Recall (cf. [5]) that an integral domain R is a called a pseudo-valuation domain (PVD) if R has a (canonically) associated valuation overring V such that Spec(R) = Spec(V ). In light of Corollary 4.2, it seems reasonable to seek applications to inside factorial PVDs. Before doing so in Proposition 4.4 and Corollary 4.5, we turn ﬁrst to the case of the simplest type of PVD, namely, a valuation domain.

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Proposition 4.3. Let V be a valuation domain which is not a ﬁeld. Then V is an inside factorial integral domain if and only if V is a rational valuation domain. If these conditions hold, then V is one-dimensional, with a Cale basis consisting of only one element. Proof. Suppose ﬁrst that V is inside factorial. Being a valuation domain, V is integrally closed, and so Theorem 1.4 yields that V is a rational generalized Krull domain. Then, by [6, Proposition 43.7] (cf. the Remark following Deﬁnition 1.3), the family F in the canonical description of V in Deﬁnition 1.3 has V as its only element. In particular, V is a rational valuation domain. Conversely, suppose that V is a rational valuation domain (but not a ﬁeld). Then dim(V ) = 1 since the value group of V is contained in the additive group of the real numbers. As the singleton set F := {V } trivially satisﬁes the conditions in Deﬁnition 1.3, V is a rational generalized Krull domain. Moreover, since V is a valuation domain, Clt (V ) is a torsion group; indeed, Clt (V ) = Pic(V ) = 0. Therefore, an application of Theorem 1.4 shows that V is inside factorial. Furthermore, since V is a quasilocal treed domain but not a ﬁeld, the ﬁnal assertion follows from Proposition 2.2, to complete the proof. The next result documents how to generalize Proposition 4.3 to arbitrary PVDs. Proposition 4.4. Let R be a PVD which is not a valuation domain. Then: (a) R is an inside factorial integral domain if and only if U(R )/U(R) is a torsion group such that R is a rational valuation domain. (b) If R is an inside factorial integral domain, then dim(R) = 1, each Cale basis of R has only one element, and R is the canonically associated valuation domain of (the pseudo-valuation domain) R. Proof. Let V denote the valuation domain that is canonically associated to (the pseudo-valuation domain) R. (a) Assume ﬁrst that U(R )/U(R) is a torsion group and R is a rational valuation domain. Then R is a one-dimensional valuation domain such that R ⊆ R ⊆ V (cf. [5, Remark 1.4]). It follows that R = V since R , being a one-dimensional valuation domain, must be a maximal proper subring of its quotient ﬁeld [10, Proposition 6, page 397]. In particular, Spec(R) = Spec(V ) = Spec(R ). Since U(R )/U(R) is a torsion group, it follows easily that R ⊆ R is a root extension. Therefore, since Proposition 4.3 yields that R is an inside factorial integral domain, [2, Proposition 5 (f)] ensures that R is, too. Conversely, assume that R is an inside factorial PVD. Then Theorem 1.4 gives that R is a rational generalized Krull domain and R ⊆ R is a root extension. Therefore, U(R )/U(R) is a torsion group, thanks to Proposition 4.1. Moreover, R is quasilocal because the “root extension” condition ensures that the canonical map Spec(R ) → Spec(R) is a bijection [2, Proposition 5 (b)], in fact, an orderisomorphism with respect to inclusion. Now, since any PVD is quasilocal and treed, R has (at most) one height 1 prime ideal. By the above order-isomorphism, the same must be true for R . So, by Deﬁnition 1.3, the family F describing R as

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a rational generalized Krull domain is the singleton set {R }. In particular, R is a rational valuation domain. (b) Observe that R is a quasilocal treed domain; and, of course, dim(R ) = dim(R). Accordingly, (b) follows from Proposition 2.2 and the proof of (a). To motivate the context of the next result, recall, as has been shown by several authors, that each atomic PVD has (Krull) dimension at most 1. Also, recall from [12] that a Cohen-Kaplansky domain (CK-domain) is, by deﬁnition, an atomic domain with only ﬁnitely many pairwise nonassociated irreducible elements. Corollary 4.5. Let R be an atomic PVD which is not a ﬁeld, with canonically associated valuation domain V . Then: (a) R is an inside factorial integral domain if and only if U(V )/U(R) is a torsion group. (b) If R is an inside factorial integral domain, then dim(R) ≤ 1, each Cale basis of R has at most one element and R = V . (c) If R is a CK-domain (that is, if R has only ﬁnitely many pairwise nonassociated irreducible elements), then R is an inside factorial integral domain. Proof. (a) Since R is an atomic PVD, its associated valuation domain V is a DVR [12, Corollary 5.2] and, hence, inside factorial. By Corollary 4.2, U(V )/U(R) being a torsion group implies that R is inside factorial. Conversely, if R is inside factorial, then Proposition 4.4 gives ﬁrst that R = V and next that U(V )/U(R) = U(R )/U(R) is a torsion group. (b) Apply Proposition 4.4 (b). (c) Suppose that R is a (PVD and a) CK-domain. Then U(R )/U(R) is a torsion group and R is a DVR [12, Theorem 4.3] and, hence, inside factorial. As in the proof of Proposition 4.4, R ⊆ R is a root extension. Therefore, an application of Proposition 4.1 yields that R is inside factorial.

5

Inside Factorial Pr¨ ufer Domains

In studying inside factorial integral domains R, we have seen that it is natural to consider the case in which dim(R) ≤ 1 (cf. Propositions 2.1, 2.2, 4.3, 4.4 and Corollary 4.5). The ﬁrst result in this section shows that this case leads naturally to the subcase involving Pr¨ ufer domains. Theorem 5.1. Let R be an integrally closed inside factorial integral domain. Then dim(R) ≤ 1 if and only if R is a Pr¨ ufer domain. Proof. By Theorem 1.4, R = R is a rational generalized Krull domain and, hence, of ﬁnite character. Therefore, if dim(R) ≤ 1, then [6, Exercise 7, page 538] ensures that R is a Pr¨ ufer domain. Conversely, if R is a Pr¨ ufer domain, then R is a treed domain, as is RM for each M ∈ Max(R). Combining Theorem 3.1 and Corollary 3.3, we see that dim(RM ) ≤ 1 for each M ∈ Max(R), whence dim(R) ≤ 1.

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Corollary 5.2. Let R be an inside factorial integral domain. Then dim(R) ≤ 1 if ufer domain. and only if R is a Pr¨ Proof. By Corollary 1.5, R is an inside factorial integral domain. Therefore, by ufer domain. The assertion now Theorem 5.1, dim(R ) ≤ 1 if and only if R is a Pr¨ follows since dim(R ) = dim(R). In the next result, the “Pr¨ ufer domain” condition plays a role in a characterization of the class of inside factorial integral domains with ﬁnite Cale bases. Theorem 5.3. Let R be an integral domain. Then the following conditions are equivalent: (1 ) R is an inside factorial integral domain with a ﬁnite Cale basis; ufer domain, and RP is an inside factorial (2 ) X (1) (R) is ﬁnite, R is a Pr¨ integral domain for each P ∈ Spec(R). Proof. (1) ⇒ (2): Assume (1). Hence, X (1) (R) is ﬁnite by Theorem 1.2; and ufer dim(R) ≤ 1 by Proposition 2.1. Then Corollary 5.2 yields that R is a Pr¨ domain, and Theorem 3.1 (a) ensures that RP is inside factorial for each P ∈ Spec(R). ufer domain for each (2) ⇒ (1): Assume (2). Then RP is an inside factorial Pr¨ P ∈ Spec(R), since the class of inside factorial (resp., Pr¨ ufer) domains is stable under integral closure (resp., formation of factor rings). Then Corollary 5.2 yields that dim(RP ) ≤ 1 for each P ∈ Spec(R), whence dim(R) ≤ 1. As X (1) (R) is ﬁnite, it follows that R is semiquasilocal. Then Theorems 3.1 (b) and 1.2 combine to show that R is an inside factorial integral domain with a ﬁnite Cale basis. The next result shows that the “inside factorial Pr¨ ufer domain” property transfers to overrings. Theorem 5.4. Let R be an inside factorial Pr¨ ufer domain. Then each overring of R is also an inside factorial Pr¨ ufer domain. Moreover, R and each of its overrings are QR-domains of dimension at most 1. Proof. By Theorem 5.1, dim(R) ≤ 1. As Pic(R) ⊆ Clt (R) and R is inside factorial, Theorem 1.4 yields that Pic(R) is a torsion group. Therefore, as each nonzero twogenerated ideal I of (the Pr¨ ufer domain) R is invertible, some power of I is principal, and so [6, Proposition 27.3] gives that R has the QR-property. Now, consider any overring D of R. Of course, D is a Pr¨ ufer domain [6, Theorem 26.1 (1)]. Moreover, D = RS for some multiplicatively closed subset S of R, since R is a QR-domain. Hence, by Theorem 3.1 (a), D inherits the “inside factorial” property from R. By applying the above argument to D instead of R, we see that D is also a QR-domain of dimension at most 1, to complete the proof. In the next result, we combine the Pr¨ uferian theme of the present section with the theme of Section 2. First, recall from [13] that an integral domain R is called a strong G-domain if each of its overrings is of the form Rf for some f ∈ R. Any strong G-domain is trivially a g-domain. In view of Proposition 2.3, it is natural

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to ask when an inside factorial integral domain with a ﬁnite Cale basis is a strong G-domain. The answer to this question is given next. Corollary 5.5. Let R be an inside factorial integral domain with a ﬁnite Cale basis. Then R is a strong G-domain if and only if R is a Pr¨ ufer domain. Moreover, if these conditions hold, then R is a B´ezout domain. Proof. Suppose that R is a Pr¨ ufer domain. Then dim(R) ≤ 1 by Theorem 5.1. As X (1) (R) is ﬁnite by virtue of Theorem 1.2, it follows that Spec(R) is a ﬁnite set. In particular, R is a semiquasilocal Pr¨ ufer domain satisfying the DCC on prime ideals. Therefore, by [13, Proposition 2.4], R is a strong G-domain. Since any B´ezout domain is a Pr¨ ufer domain, the remaining assertions are consequences of the fact (cf. [13, Proposition 2.4]) that each strong G-domain is a B´ezout domain. Recall that any ﬁeld is an inside factorial integral domain. Next, we answer a basic question by characterizing the inside factorial Pr¨ ufer domains that are not ﬁelds. Proposition 5.6. Let R be a Pr¨ ufer domain which is not a ﬁeld. Then the following conditions are equivalent: (1 ) R is an inside factorial integral domain; (2 ) RM is an inside factorial integral domain for each M ∈ Max(R), Clt (R) is a torsion group, and R = ∩ {RM | M ∈ Max(R)} is of ﬁnite character; (3 ) RM is a rational valuation domain for each M ∈ Max(R), Clt (R) is a torsion group, and R = ∩ {RM | M ∈ Max(R)} is of ﬁnite character. Proof. (1) ⇒(2): Combine Theorem 3.1 (a), Theorem 1.4, Theorem 5.1 and the Remark following Deﬁnition 1.3. (2) ⇒(3): By Proposition 4.3, any inside factorial valuation domain is a rational valuation domain. (3) ⇒(1): Assume (3). Observe that dim(RM ) = 1 for each M ∈ Max(R) (cf. [10, Proposition 8, page 398]), whence dim(R) = 1. Then R is a rational generalized Krull domain because F := {RM | M ∈ Max(R)} satisﬁes all the conditions of Deﬁnition 1.3. As R is integrally closed, an application of Theorem 1.4 shows that R is an inside factorial integral domain. We close by considering the “inside factorial” property in the context of two Pr¨ ufer-like conditions, whose deﬁnitions are recalled next from [7]. Let R be an integral domain. Then R is called an almost B´ezout (AB)-domain if, for all a, b ∈ R× , there exists n ∈ N such that (an , bn ) is a principal ideal of R. Also, R is called an almost Pr¨ ufer (AP)-domain if, for all nonunits a, b ∈ R× , there exists n n n ∈ N such that (a , b ) is an invertible ideal of R. Of course, each B´ezout (resp., Pr¨ ufer) domain is an AB-domain (resp., AP-domain); and each AB-domain is an AP-domain. Our ﬁnal result gives a case in which the converse of the preceding assertion holds.

On Inside Factorial Integral Domains

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Proposition 5.7. Let R be an AP-domain such that R is an inside factorial integral domain. Then dim(R) ≤ 1 and R is both an inside factorial integral domain and an AB-domain. ufer domain such Proof. Appealing to [7, Corollary 4.8 (2)], we get that R is a Pr¨ that R ⊆ R is a root extension. Since R is an inside factorial integral domain, Corollary 5.2 yields that dim(R ) ≤ 1. Then dim(R) = dim(R ) ≤ 1 and, by [2, Proposition 5 (f)], R is an inside factorial integral domain. Moreover, Clt (R ) is a torsion group by Theorem 1.4, and so R is an AB-domain by [7, Corollary 4.8 (1)].

6

Analogous Results for Cale Domains

To obtain the “Cale domain” analogues of our results in the preceding sections, four things should be kept in mind. The ﬁrst is the following reformulation of [2, Theorem 7 (b)]: an integral domain R is a Cale domain if and only if R is an inside factorial integral domain such that R is a Krull domain. Second, when adapting some of our earlier proofs by passing from considerations of rational generalized Krull domains to Krull domains per se, it is often appropriate to let DVRs take on the roles that had been played by rational valuation domains. Third, in seven of the above results (namely, Proposition 2.1, Proposition 2.2, Proposition 2.3, Corollary 3.3, Theorem 5.1, Corollary 5.2 and Corollary 5.5), “inside factorial” (resp., “locally inside factorial”) appears only in the hypotheses but not in the conclusion. The “Cale domain” (resp., “locally Cale domain”) analogues of those results are then evidently valid (when one replaces the “inside factorial” hypothesis with the stronger “Cale domain” condition). Fourth, “inside factorial” does not ﬁgure in the hypotheses or the conclusion of Proposition 3.2. We next indicate the changes needed to produce the statements of the “Cale domain” analogues of all of our other results. In each case, we supplement the ﬁrst two guidelines that were given above with some additional details about the associated proofs. Theorem 3.1: Replace each occurrence of “an inside factorial integral domain” (in the statement of the result) with “a Cale domain”. In the proof of (a), note also that R a Krull domain implies that (RS ) = (R )S a Krull domain, since the class of Krull domains is stable under formation of rings of fractions [10, Proposition 6, | M ∈ Max(R)} and page 483]. In the proof of (b), note also that R = ∩{RM that the intersection of ﬁnitely many Krull domains is itself a Krull domain [10, Examples (3), page 480]. Proposition 3.4: Replace each occurrence of “an inside factorial integral domain” (in the statement of the result) with “a Cale domain”. For the proof of the “Cale domain” variant, use Proposition 3.4 itself, together with the “Cale domain” variant of Theorem 3.1 and the result [2, Corollary 6] that a Noetherian integral domain is a Cale domain if and only if it is inside factorial. Proposition 3.5: For the “only if” assertion, use the “Cale domain” variant of Theorem 3.1 (a). For the “if” assertion, establish that R is a Krull domain as follows. The proof of Proposition 3.5 shows that R is an inside factorial integral

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domain, and so R is a rational generalized Krull domain. By the Remark following Deﬁnition 1.3, R = ∩{RP | P ∈ X (1) (R )}, and this representation is of ﬁnite character, with RP a rational valuation domain for each P ∈ X (1) (R ). Therefore, by the very deﬁnition of a Krull domain [10, Deﬁnition 3, page 480], R is a Krull domain if (and only if) RP is a DVR for each P ∈ X (1) (R ) (since the other conditions for R to be a Krull domain are satisﬁed). To see that each such RP is indeed a DVR, choose N ∈ Max(R ) such that P ⊆ N and put M := N ∩ R ∈ Max(R). Since (RM ) is a Krull domain, it follows that RP = ((RM )R \N )P RN is a quasilocal one-dimensional Krull domain, i.e., a DVR. (It may be useful to record that the foregoing discussion has established the following criterion. An integral domain R is a Cale domain if and only if R is an inside factorial integral domain such that RP is a DVR for each P ∈ X (1) (R ).) Corollary 3.6: Replace each occurrence of “inside factorial” (in the statement of the result) with “a Cale domain”. For the proof of (1) ⇒ (2), use (Corollary 3.6 and) the “Cale domain” variant of Theorem 3.1 (a). For the proof of (2) ⇒ (1), show that R is a Krull domain by arguing as in the above discussion of the “Cale domain” variant of Proposition 3.5. Proposition 4.1: In the statement of the result, replace each occurrence of “an inside factorial integral domain” with “a Cale domain” and add the hypothesis that T is an overring of R (i.e., that T is contained in the quotient ﬁeld of R). This extra hypothesis is needed in applying the part of [2, Proposition 5 (f)] that asserts that if R ⊆ T is a root extension of integral domains and T is an overring of R, then R is a Cale domain if and only if T is a Cale domain. Corollary 4.2: In the statement of the result, replace each occurrence of “inside factorial” with “a Cale domain” or “Cale domains” as grammar warrants. Note that there is no need to add an “overring” hypothesis in the manner of the preceding result, because the “Spec(R) = Spec(T )” condition forces T to be either a ﬁeld (hence trivially a Cale domain) or an overring of R [4, Proposition 3.3]. Proposition 4.3: Change the second sentence of the statement to the following sentence. “Then V is a Cale domain if and only if V is a DVR.” The proof follows from Proposition 4.3 and the observation that a (rational) valuation domain that is a Krull domain but not a ﬁeld is the same thing as a DVR (cf. [6, Theorem 43.16]). Proposition 4.4: In the statement of the result, replace each occurrence of “an inside factorial integral domain” with “a Cale domain” and in the statement of (a), replace “rational valuation domain” with “DVR”. The proof follows as in the above discussion of the “Cale domain” variant of Proposition 4.3. Corollary 4.5: Replace each occurrence of “an inside factorial integral domain” (in the statement of the result) with “a Cale domain”. For the proofs of (a) and (c), note that all DVRs are Cale domains. Theorem 5.3: Replace each occurrence of “an inside factorial integral domain” (in the statement of the result) with “a Cale domain”. For the proof of (2) ⇒ (1), note that R = ∩{RP | P ∈ X (1) (R)} is a Krull domain because it is the intersection of ﬁnitely many Krull domains. Theorem 5.4: Replace each occurrence of “an inside factorial” (in the statement

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of the result) with “a Cale”. For the proof, note ﬁrst that a Pr¨ ufer Cale domain is the same thing as a Dedekind domain R such that Clt (R ) is a torsion group (cf. [2, Corollary 6]) and recall that each overring of a Dedekind domain is a Dedekind (hence, a Krull) domain. Proposition 5.6: In the statement of the result, replace each occurrence of “an inside factorial integral domain” with “a Cale domain” and replace “rational valuation domain” with “DVR”. For the proof of (2) ⇒ (3), recall that any Cale valuation domain which is not a ﬁeld must be a DVR. For the proof of (3) ⇒ (1), observe that the given F satisﬁes all the conditions in the deﬁnition [10, Deﬁnition 3, page 480] of a Krull domain. Proposition 5.7: Replace each occurrence of “an inside factorial integral domain” (in the statement of the result) with “a Cale domain”. The proof follows from (Proposition 5.7 and) the fact that (R ) = R .

Bibliography [1] D. F. Anderson, A general theory of class groups, Comm. Algebra 16 (1988), 805–847. [2] S. T. Chapman, F. Halter-Koch and U. Krause, Inside factorial monoids and integral domains, J. Algebra 252 (2002), 350–375. [3] G. Picavet, Autour des id´eaux premiers de Goldman d’un anneau commutatif, Ann. Univ. Clermont-Ferrand (1975), 73–90. [4] D. F. Anderson and D. E. Dobbs, Pairs of rings with the same prime ideals, Canad. J. Math. 32 (1980), 362–384. [5] J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains (II), Houston J. Math. 4 (1978), 199–207. [6] R. Gilmer, Multiplicative Ideal Theory, Pure and Appl. Math. 12, Dekker, New York, 1972. [7] D. D. Anderson and M. Zafrullah, Almost B´ezout domains, J. Algebra 142 (1991), 285–309. [8] D. D. Anderson, J. L. Mott and M. Zafrullah, Finite character representations for integral domains, Boll. UMI (7) 6-B (1992), 613–630. [9] R. Gilmer, A note on unique factorization, Delta (Waukesha) 3 (1972), 7–8. [10] N. Bourbaki, Commutative Algebra, Addison-Wesley, Reading, 1972. [11] F. Halter-Koch, Divisor theories with primary elements and weakly Krull domains, Boll. UMI (7) 9-B (1995), 417–441. [12] D. D. Anderson and J. L. Mott, Cohen-Kaplansky domains: Integral domains with a ﬁnite number of irreducible elements, J. Algebra (148) (1992), 17–41. [13] P. Maroscia, Topological properties of some classes of G-domains, Boll. UMI (5) 15-A (1978), 688–698.

252

Chapter 15

Polynomial Separation of Points in Algebras by

Sophie Frisch Abstract We show that for a wide variety of domains, including all Dedekind rings with ﬁnite residue ﬁelds, it is possible to separate any two algebraic elements a, b of an algebra over the quotient ﬁeld by integer-valued polynomials (i.e. to map a and b to 0 and 1, respectively, with a polynomial in K[x] that maps every element of D to an element of D), provided only that the minimal polynomials of a and b in K[x] are co-prime (which is obviously necessary). In contrast to this, it is impossible to separate a, b ∈ D by a n × n-integermatrix-valued polynomial (a polynomial in K[x] that maps every n×n matrix over D to a matrix with entries in D), except in the trivial case where a−b is a unit of D. (This is despite the fact that the ring of n×n-integer-matrix-valued polynomials for any ﬁxed n is nontrivial whenever the ring of integer-valued polynomials is nontrivial.)

2000 Mathematics Subject Classification: Primary 13F20; Secondary 13B25, 11C08, 15A36, 16B99.

1

Introduction

Definition 1.1. If D is a domain with quotient ﬁeld K, the ring of integer-valued polynomials of D is Int(D) = {f ∈ K[x] | ∀d ∈ D : f (d) ∈ D}. Rings of integer-valued polynomials have proved remarkably well suited for interpolation purposes. Definition 1.2. A domain D is called interpolation domain, if for all n ∈ N, for all a1 , . . . , an ∈ D (distinct) and all b1 , . . . , bn ∈ D there exists f ∈ Int(D) with f (ai ) = bi for 1 ≤ i ≤ n. 253

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Interpolation domains have been classiﬁed both among Noetherian domains and among Pr¨ ufer domains in [3] (see also Corollary 2.4). In particular, every Dedekind domain all of whose residue ﬁelds are ﬁnite is an interpolation domain. Equivalent to being an interpolation domain is the following point separation property: for any a, b ∈ D with a = b there exists a polynomial f ∈ Int(D) with f (a) = 0 and f (b) = 1. In this note we explore two possible avenues for extending this point separation property to D-algebras and K-algebras. If D is a domain for which separation of points in D by polynomials in Int(D) is possible, we ask ourselves (1) Can elements in an arbitrary K-algebra (for instance the ring of n×n matrices with entries in K) be separated by polynomials in Int(D)? (2) Can elements of D be separated by polynomials in K[x] that map a given D-algebra (for instance the ring of n × n matrices with entries in D) to itself? It turns out that the answer to (1) is yes, within reason (the minimal polynomials of the elements to be separated have to be co-prime in K[x]); but (2) may fail spectacularly for D-algebras with zero-divisors. We will show that it is impossible to separate points in D by polynomials in K[x] that map Mn (D) (the D-algebra of n × n matrices with entries in D) to itself, except in trivial cases.

2

Separation of Points in an Algebra

The basic connections between separation of points, co-maximality of null ideals, and interpolation hold in great generality, so we state them for functions - even though we are only concerned with polynomials in this paper. Throughout this paper, every ring or algebra (commutative or not) is assumed to have an identity element (and every ring homomorphism to satisfy ϕ(1) = 1). D and K stand for a domain and its quotient ﬁeld; R denotes a ring (possibly noncommutative, possibly with zero-divisors). Notation and Conventions. Let R be a ring and S a set. We say that a ring F is a ring of functions from S to R if F comes equipped with a ring homomorphism ψ : F → RS (= s∈S R) assigning to each element f ∈ F a function (which we also call f , by abuse of notation). If (F, ψ) is a ring of functions from S to R and s ∈ S, we call NF (s) = {f ∈ F | f (s) = 0} the null ideal of s in F. We say that we can separate points in S by functions from F if for all s, t ∈ S with s = t there exists an f ∈ F with f (s) = 0 and f (t) = 1. We say that we can interpolate arbitrary functions from S to R by functions in F if for all n ∈ N, for all s1 , . . . , sn ∈ S (distinct) and all r1 , . . . , rn ∈ R there exists some f ∈ F with f (si ) = ri for 1 ≤ i ≤ n.

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Lemma 2.1. Let (F, ψ) be a ring of functions from S to R. (i) For s, t ∈ S, there exists an f ∈ F with f (s) = 0 and f (t) = 1 if and only if NF (s) + NF (t) = F. (ii) To be able to interpolate arbitrary functions from S to R by functions in F it is necessary and, if F is an R-algebra and ψ an R-algebra homomorphism, also suﬃcient, to be able to separate points in S by functions in F. Proof. (i) is easy; and, (ii), separation of points is clearly necessary for interpolation. If F is an R-algebra then, given s1 , . . . , sn ∈ S which we can mutually separate, we have functions fij ∈ F with fij (si ) = 1 and fij (sj ) = 0, which we multiply to get a Lagrange interpolation function fi ∈ F with fi (si ) = 1 and fi (sj ) = 0 for j = i. Then we can use R-linear combinations of the fi to interpolate. Proposition 2.2. Let D be a domain with quotient ﬁeld K. D is an interpolation domain if and only if it has the following property: whenever a1 , a2 are algebraic elements of a K-algebra A, whose minimal polynomials in K[x] are co-prime, there exists a polynomial p ∈ Int(D) with p(a1 ) = 0 and p(a2 ) = 1. Proof. Assume D is an interpolation domain. Given a1 , a2 with co-prime minimal polynomials f1 , f2 ∈ K[x], let c1 , c2 be nonzero elements of D such that ci fi = gi ∈ D[x]. There exist polynomials pi ∈ D[x] and a nonzero constant d ∈ D such that g1 (x)p1 (x) + g2 (x)p2 (x) = d. Let h(x) ∈ Int(D) with h(0) = 1 and h(d) = 0, and set p(x) = h(g2 (x)p2 (x)) = h(d − g1 (x)p1 (x)), then p(a1 ) = 0 and p(a2 ) = 1. Conversely, the point separation property for K-algebras specialized to A = K implies separation of points in D by polynomials in Int(D) which by Lemma 2.1 (ii) implies interpolation of arbitrary functions from D to D by polynomials in Int(D). If D is a Dedekind domain with ﬁnite residue ﬁelds, the construction of the point separating integer-valued polynomial h in the preceding proof can be made fairly explicit, cf. [5]. Corollary 2.3. A domain D with quotient ﬁeld K is an interpolation domain if and only if for any f1 , f2 ∈ K[x] with K[x]f1 (x) + K[x]f2 (x) = K[x] it follows that (K[x]f1 (x) ∩ Int(D)) + (K[x]f2 (x) ∩ Int(D)) = Int(D).

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Proof. In view of Lemma 2.1 and Proposition 2.2, all that remains to show is that every two nonconstant monic polynomials in K[x] occur as minimal polynomials of two elements of the same K-algebra. This can be seen by looking at the companion matrices of the polynomials in question, bringing the smaller matrix up to the size of the larger matrix by padding with zeros at the right and bottom. Recalling the characterization of interpolation domains among Noetherian domains from [3] (Theorem 2.4.), we obtain: Corollary 2.4. Let D be a Noetherian domain with quotient ﬁeld K. Then the following are equivalent: (1 ) D is an interpolation domain (2 ) whenever a1 , a2 are algebraic elements of a K-algebra A whose minimal polynomials in K[x] are co-prime, there exists a polynomial p ∈ Int(D) with p(a1 ) = 0 and p(a2 ) = 1 (3 ) whenever I, J are ideals of K[x] with I + J = K[x] then (I ∩ Int(D)) + (J ∩ Int(D)) = Int(D) (4 ) D is one-dimensional with ﬁnite residue ﬁelds and locally unibranched (the last condition meaning that the integral closure of every localization at a prime ideal is again local).

3

Integer–Matrix-Valued Polynomials

Definition 3.1. We denote by M intn (D) the ring of polynomials in K[x] that map n × n matrices with entries in D to matrices with entries in D: M intn (D) = {f ∈ K[x] | ∀A ∈ Mn (D) : f (A) ∈ Mn (D)}. Clearly, K[x] ⊇ Int(D) ⊇ M int2 (D) ⊇ M int3 (D) ⊇ . . . ⊇ D[x], and, as Lemma 3.4 below shows,

∞

M intn (D) = D[x].

n=1

Example 3.2. One may ask if M intn (D) is not often trivial, in the sense of M intn (D) = D[x]. The answer is, for Noetherian D at least: not more often than Int(D) is trivial. (This can be shown by combining the proof of Theorem I.3.14 of [2] with Lemma 3.4 below.) It follows immediately from Lemma 3.4 that M intn (D) = D[x] (for any n ≥ 1) whenever D has a proper principal ideal of ﬁnite index.

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For example, x6 + x5 + x3 + x2 is in M int2 (Z) \ M int3 (Z). 2 (The residue class of x6 + x5 + x3 + x2 in Z2 [x] is the least common multiple of all monic polynomials in Z2 [x] of degree 2.) When investigating M intn (D), it is useful to know in what cases the null ideal of a matrix A ∈ Mn (R) in the polynomial ring R[x] is principal, not just for R = D, but also for residue class rings R = D/I. If D is a domain then ND[x] (A) = {f ∈ D[x] | f (A) = 0} is principal for every matrix A ∈ Mn (D) if and only if D is integrally closed [4]. But even for the most mundane commutative ring R we know that the null ideals in R[x] of some matrices are principal. (Thanks to Gabriel Picavet for pointing this out): Lemma 3.3. Let R be a commutative ring, f ∈ R[x] a monic polynomial and A ∈ Mn (R) the companion matrix of f . Then NR[x] (A) = f (x)R[x]. Proof. By a theorem of McCoy ([1], Theorem 7.31) the null ideal in R[x] of a matrix A ∈ Mn (R) is (Fn :R[x] Fn−1 ), where Fk denotes the ideal generated in R[x] by the k × k minors of xIn − A (In the n × n identity matrix). For any A, Fn = cA (x)R[x] (cA the characteristic polynomial of A); and if A is the companion matrix of a polynomial, then one of the (n − 1) × (n − 1) minors of xIn − A is 1, such that in this case the null ideal of A is cA (x)R[x] = f (x)R[x]. Lemma 3.4. Let D be a domain and f (x) = g(x)/c, g ∈ D[x], c ∈ D \ {0}. Then f ∈ M intn (D) if and only if g is divisible modulo cD[x] by all monic polynomials in D[x] of degree n. Proof. Suppose f ∈ M intn (D). Let g¯ denote the residue class of g in (D/cD)[x]. Then g¯ maps every matrix in Mn (D/cD) to the zero-matrix. In particular, this holds for all companion matrices of monic polynomials of degree n. Therefore g¯ is divisible in (D/cD)[x] by all monic polynomials of degree n. Conversely, this condition suﬃces for f ∈ M intn (D), because it means that g is divisible modulo cD[x] by the characteristic polynomial of every matrix in Mn (D), and therefore maps every matrix in Mn (D) to a matrix in Mn (cD). We are now ready to show that the rings M intn (D) are as unsuitable as can be for separation of points. For n > 1, M intn (D) cannot even separate elements a, b ∈ D, except in the trivial case when a − b is a unit and they can be separated by the polynomial (x − b)/(a − b) ∈ D[x]. Theorem 3.5. Let D be a domain and a, b ∈ D. If there exists a polynomial f ∈ M int2 (D) with f (a) = 1 and f (b) = 0 then a − b is a unit of D. Proof. By linear substitution we can reduce to the case a = 0. Let f ∈ M int2 (D) with f (0) = 1 and f (b) = 0. We write f as f (x) = g(x)/c with g ∈ D[x] and c = 0 in D. It follows that c = g(0) and g(x) = (x − b)h(x) with h ∈ D[x].

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By Lemma 3.4, g is divisible modulo cD[x] by all monic polynomials in D[x] of degree 2, in particular by (x − b)x. In (D/cD)[x] we have, for some k ∈ (D/cD)[x], ¯ (x − b)xk(x) = g¯(x) = (x − b)h(x) and we can cancel (x − b) which, being monic, is certainly not a zero-divisor in (D/cD)[x]. This shows that ¯h, the residue class of h in (D/cD)[x], is divisible by x. Therefore, in D[x], the constant coeﬃcient of h is divisible by c, say h(0) = dc. Looking at the constant coeﬃcient of g we have c = g(0) = −bh(0) = −bdc. Cancelling c = 0 proves −b to be a unit of D.

Bibliography [1] W. C. Brown, Matrices over Commutative Rings, Pure and Applied Mathematics 169, Dekker 1993. [2] P.-J. Cahen and J.-L. Chabert, Integer-valued Polynomials, Mathematical Surveys and Monographs 48, Amer. Math. Soc. 1997. [3] P.-J. Cahen, J.-L. Chabert and S. Frisch, Interpolation Domains, J. Algebra 225 (2000), 794–803. [4] S. Frisch, Integrally Closed Domains, Minimal Polynomials, and Null Ideals of Matrices, Comm. Algebra 32 (2004), 2015–2017. [5] S. Frisch, Interpolation by integer-valued polynomials, J. Algebra 211 (1999), 562–577.

259

Chapter 16

k-Factorized Elements in Telescopic Numerical Semigroups J. C. Rosales, Pedro A. Garc´ıa-S´ anchez, and Juan I. Garc´ıa-Garc´ıa by

Abstract The set of elements in a numerical semigroup that admit a factorization in k elements is itself a numerical semigroup. We study the multiplicity, degree of singularity, Frobenius number and embedding dimension of these semigroups when the ambient semigroup is telescopic.

1

Introduction

A numerical semigroup is a subset S of N (the set of nonnegative integers) closed under addition, 0 ∈ S and H(S) = N \ S is ﬁnite. An element x in a numerical semigroup S is a k-factorized element if there exist s1 , . . . , sk ∈ S \ {0} such that x = s1 + . . . + sk . The idea of factorization is historically related to Ring Theory. More recently this concept is studied in the more general scope of Semigroup Theory (see for instance [5, 6, 7, 8, 9, 11, 12]). Given A ⊆ N, we denote by / t λi ai | a1 , . . . , at ∈ A and λ1 , . . . , λt ∈ N , A = i=1

which is a numerical semigroup if and only if gcd(A) = 1 (see for instance [17]; gcd stands for greatest common divisor). If S = A, then we say that A is a system of generators of S. If in addition no proper subset of A generates S, then A is a minimal system of generators of S. It is well-known (see for instance [17]) that 260

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261

every numerical semigroup admits a unique minimal system of generators {n1 < · · · < np }. The integer p is known as the embedding dimension of S and n1 is the multiplicity of S, and we denote them by e(S) and m(S), respectively. In the literature one can ﬁnd a large amount of papers devoted to the study of analytically unramiﬁed one dimensional local domains via their value semigroups, which are numerical semigroups (see for instance [2, 3, 4, 10, 14, 20]). The terminology used in numerical semigroups is thus sometimes exported from Ring Theory, as occurs with the embedding dimension and multiplicity. Other invariants of interest of a numerical semigroup are #H(S) = #(N\S) (#A stands for cardinality of the set A), known as the degree of singularity of S, and the greatest integer not belonging to S, called the Frobenius number of S and denoted here by g(S). These two invariants have been widely studied in the literature from the point of view of nonnegative integer solutions to a linear Diophantine equation with positive coeﬃcients. A good overview of this can be found in [15]. If S is a numerical semigroup and k is a positive integer, denote by S(k) the set of k-factorized elements of S. We show in Section 3 that S(k) ∪ {0} is a numerical semigroup. In this chapter we study the above mentioned invariants for the semigroups S(k) , when S is a telescopic numerical semigroup. A numerical semigroup S minimally generated by {n1 < · · · < np } is telescopic if for every i ∈ {2, . . . , p}, ci ni ∈ n1 , . . . , ni−1 with ci = gcd(n1 , . . . , ni−1 )/ gcd(n1 , . . . , ni ) (see [13]). Example 1.1. Let S = 4, 6, 9. Then c2 = c3 = 2. As 2×6 ∈ 4 and 2×9 ∈ 4, 6, we have that S is telescopic. Now, let S = 3, 5, 7. In this setting, c2 = 3 and c3 = 1. Thus S is not telescopic, since 1 × 7 ∈ 3, 5. Telescopic numerical semigroups appear related to algebraic codes and they are a subclass of the class of symmetric numerical semigroups, which are important on their own since the semigroup ring associated to them is Gorenstein (see [14]); a numerical semigroup is symmetric if for every element x ∈ H(S), g(S) − x ∈ S (see . Every [1]). One can also prove that S is symmetric if and only if #H(S) = g(S)+1 2 numerical semigroup generated by two elements is telescopic and thus symmetric. It is easy to prove that g(n1 , n2 ) = n1 n2 − n1 − n2 (see [19]). In Section 4 we recall how this formula can be generalized for telescopic numerical semigroups, and study the Frobenius number and degree of singularity of the semigroups of k-factorized elements of a telescopic numerical semigroup. Section 5 shows how a system of minimal generators of these semigroups can be constructed. Finally in the last section we describe how the multiplicity, degree of singularity, Frobenius number and embedding dimension change in S(k) ∪ {0} as we increase k, with S a telescopic numerical semigroup.

2

Normal Forms and Lengths

In this section we give a tool for studying maximal lengths of factorizations of elements in commutative semigroups with some special properties. We will use

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later this tool for telescopic numerical semigroups. Let a, b ∈ Nn and let r = (r1 , r2 ) ∈ Nn × Nn . We say that a rewrites to b by r R r, denoted a→b, if a − r1 ∈ Nn and b = a − r1 + r2 . If R ⊆ Nn × Nn , then a→b r R whenever there exists r ∈ R such that a→b. The symbol →∗ denotes the reﬂexive R and transitive closure of →. The element a ∈ Nn is in normal form with respect to r R if there is no b ∈ Nn and r ∈ R such that a→b. We say that b is a normal form R of a (with respect to R) if a→∗ b and b is in normal form (with respect to R). Given R ⊆ Nn × Nn , let R be the smallest congruence containing R, known as the congruence generated by R. Let σ be a congruence on Nn . In view of R´edei’s theorem (see [16]), σ is ﬁnitely generated, that is to say, there exists a ﬁnite subset R of Nn × Nn such that σ = R. Let be a linear admissible order on Nn . Given σ and R a system of generators of σ such that r2 ≺ r1 whenever (r1 , r2 ) ∈ R, we say that R is a canonical system of generators of σ (with respect to ) if for every a ∈ Nn there exists a unique normal form of a with respect to R. Deﬁne µ : Nn → Nn , µ(a) = min {b ∈ Nn | bσa}. It turns out (see for instance [17]) that if R is a canonical system of generators of σ, then µ(a) is the normal form of a with respect to R. Assume that for every a ∈ Nn , the set [a]σ = {b ∈ Nn | bσa} is ﬁnite. Deﬁne n L(a) = max{|b| | bσa}, where |(b1 , . . . , bn )| = i=1 bi . Somehow, the map L(a) is measuring which is the largest decomposition of [a]σ in the monoid Nn /σ. If n b = (b1 , . . . , bn ) ∈ Nn is such that L(a) = |b|, then [a]σ = i=1 bi [ei ]σ , with ei the element in Nn all of whose coordinates are zero except the ith which is equal to one. Hence [a]σ is written as a sum of |b| elements of Nn /σ. Lemma 2.1. Let σ be a congruence on Nn such that [a]σ has ﬁnitely many elements for all a ∈ Nn , and let R be a canonical system of generators of σ with respect to a linear admissible order . If |r1 | ≤ |r2 | for every (r1 , r2 ) ∈ R, then for all a ∈ Nn , L(a) = |µ(a)|. Proof. Let b ∈ Nn be such that a σ b and L(a) = |b|. There exists a sequence r1

r2

r s−2

r s−1

b1 , b2 , . . . , bs ∈ Nn and r1 , . . . , rs−1 ∈ R such that b = b1 → b2 → · · · → bs−1 → bs = µ(b). From the hypothesis, we deduce that |bi | ≤ |bi+1 | for all i ∈ {1, . . . , s − 1}. Moreover, bi σbi+1 for all i, which from the choice of y, implies that |b| = |b1 | = |b2 | = · · · = |bs | = |µ(b)|. We deduce that L(a) = |b| = |µ(b)|. Since aσb, we know that µ(a) = µ(b). This leads to L(a) = |µ(a)|.

k-Factorized Elements in Telescopic Numerical Semigroups

3

263

The Semigroup of k-Factorized Elements of a Numerical Semigroup

Let S be a numerical semigroup minimally generated by {n1 , . . . , np }. We denote by S(k) the set of k-factorized elements of S. That is to say, S(k) = {s ∈ S | s = s1 + · · · + sk , for some s1 , . . . , sk ∈ S \ {0}}. p Deﬁne ϕ : Np → S by ϕ(a1 , . . . , ap ) = i=1 ai ni . Let σ be the kernel congruence of ϕ. Then S is isomorphic to Np /σ. By making an abuse of notation, for s ∈ S, we set L(s) = L(x) with x ∈ Np such that ϕ(x) = s (observe that this deﬁnition does not depend on the choice of x ∈ ϕ−1 (s)). In other words, L(s) is the maximum k ∈ N such that s = s1 + · · · + sk with s1 , . . . , sk ∈ S \ {0}, the maximum length of the factorizations of s in S. The following two results are straightforward to prove. Lemma 3.1. Let k be a positive integer and let S be a numerical semigroup. Then S(k) = {s ∈ S | L(s) ≥ k}. Lemma 3.2. Let S be a numerical semigroup and let s, t ∈ S. Then L(s + t) ≥ L(s) + L(t). Proposition 3.3. Let S be a numerical semigroup. Then S(k) ∪ {0} is a numerical semigroup. Proof. Let n1 < . . . < np be the minimal system of generators of S. By using Lemmas 3.2 and 3.3, we deduce that S(k) ∪ {0} is a monoid. We prove that every integer greater than or equal to kn1 +g(S)+1 belongs to S(k) (and thus N\(S(k) ∪{0}) has ﬁnitely many elements). Let x be an integer greater or equal to kn1 + g(S) + 1. Then x − kn1 ≥ g(S) + 1 ∈ S. Hence there exist a1 , . . . , ap ∈ N such that x − kn1 = a1 n1 + · · · + ap np and thus L(x) ≥ k + a1 + · · · + ap ≥ k, which in view of Lemma 3.1 implies that x ∈ S(k) .

4

Telescopic Numerical Semigroups

Let S be a numerical semigroup minimally generated by {n1 < n2 < · · · < np } and let σ be the kernel congruence of the map ϕ deﬁned in the preceding section. In [18] the ﬁrst two authors prove that the semigroup S is telescopic if there exists positive integers c2 , . . . , cp and dij ∈ N, with i ∈ {2, . . . , p}, j ∈ {1, . . . , i − 1}, such that σ is generated by {(ci ei , di1 e1 + · · · + dii−1 ei−1 ) | i ∈ {2, . . . , p}}. It turns out that these ci are precisely gcd{n1 , . . . , ni−1 }/ gcd{n1 , . . . , np }. Every telescopic numerical semigroup is a free numerical semigroup (see [18]; the concept of free semigroup was introduced in [3]). The main diﬀerence is that when we talk of a free numerical semigroup we must specify an arrangement of its minimal

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system of generators (the one that yields a presentation as the one given above; if one swaps the positions of two generators, then a presentation might have no longer a stair shape), and we say that a numerical semigroup S minimally generated by {n1 , . . . , np }, with n1 < · · · < np , is telescopic when it is free with respect to the arrangement n1 , n2 , . . . , np . Free semigroups have been used by several authors to construct families of symmetric numerical semigroups (and thus one dimensional local Gorenstein rings; see for instance [20]). Given a numerical semigroup S and a ∈ S, the Apry set of a in S is the set Ap(S, a) = {x ∈ S | x − a ∈ S}. It is easy to prove that #Ap(S, a) = a and that g(S) = x − m(S), where x is the greatest element in Ap(S, m(S)). Theorem 4.1. Let S be a telescopic numerical semigroup minimally generated by {n1 < · · · < np }. Let R = {(ci ei , di1 e1 + · · · + dii−1 ei−1 ) | i ∈ {2, . . . , p}} be a system of generators of σ, the kernel congruence of ϕ : Np → S, ϕ(a1 , . . . , ap ) = pi=1 ai ni . Then p 1. Ap(S, n1 ) = { i=2 λi ni | 0 ≤ λ2 < c2 , . . . , 0 ≤ λp < cp }, 2. #Ap(S, n1 ) = n1 = c2 · · · cp , 3. S is symmetric, 4. #H(S) =

g(S)+1 , 2

5. g(S) = (c2 − 1)n2 + · · · + (cp − 1)np − n1 , 6. for every i ∈ {2, . . . , p}, ci = min{k ∈ N \ {0} | kni ∈ n1 , . . . , ni−1 }, 7. R is a canonical system of generators of σ with respect to the reverse lexicographical order rlex in which n1 n2 · · · np , 8. ϕ restricted to the set {(a1 , . . . , ap ) | 0 ≤ a2 < c2 , . . . , 0 ≤ ap < cp } is a bijection, 9. for every positive integer k, the map ϕ gives a one to one correspondence p between the set {(a1 , . . . , ap ) | i=1 ai ≥ k, 0 ≤ a2 < c2 , . . . , 0 ≤ ap < cp } and S(k) . Proof. (1), (2), (3), (5) and (6) can be found in [18]. (4) is a consequence of (3). (7) From the deﬁnition of telescopic numerical semigroup and using that (di1 , . . . , dii−1 , 0, . . . , 0)

rlex

(0, . . . , 0, ci , 0, . . . , 0)

for all i ∈ {2, . . . , p}, we can consider that R is reduced with respect to rlex (in this setting this translates to dij < cj for all 2 ≤ j < i and 2 ≤ i ≤ p; see [17] and [18] for further details). By [17], in order to prove that R is a canonical system of generators of σ we only have to prove that the elements (di1 , . . . , dii−1 , 0, . . . , 0) +

k-Factorized Elements in Telescopic Numerical Semigroups

265 R

sij and (dj1 , . . . , djj−1 , 0, . . . , 0) + sji reduce to the same element via →, for every i, j ∈ {2, . . . , p}, where in this case sij = (0, . . . , 0, cj , 0, . . . , 0). Let ri be ((0, . . . , 0, ci , 0, . . . , 0), (di1 , . . . , dii−1 , 0, . . . , 0)). Assume that i = j, then rj

(di1 , . . . , dii−1 , 0, . . . , 0) + sij →(di1 , . . . , dii−1 , 0, . . . , 0) + (dj1 , . . . , djj−1 , 0, . . . , 0) (note that rj is the only rule that can be applied here) and ri

(dj1 , . . . , djj−1 , 0, . . . , 0) + sji →(dj1 , . . . , djj−1 , 0, . . . , 0) + (di1 , . . . , dii−1 , 0, . . . , 0) (again rj is the only rule that can be used here), which implies that (di1 , . . . , dii−1 , 0, . . . , 0) + sij and (dj1 , . . . , djj−1 , 0, . . . , 0) + sji reduce to the same element via R for all i, j ∈ {2, . . . , p}. (8) We only have to take into account that the normal form of every element is unique and belongs to the set {(a1 , . . . , ap ) | 0 ≤ a2 < c2 , . . . , 0 ≤ ap < cp }. (9) Let s ∈ S and µ(s) = (a1 , . . . , ap ). Clearly, ai < ci for all i ∈ {2, . . . , p}. From n1 < . . . < np we deduce that |(0, . . . , 0, ci , 0, . . . , 0)| ≤ |(di1 , . . . , dii−1 , 0, . . . , 0)|. Thus by Lemma 2.1, we know that L(s) = |(a1 , . . . , ap )|. By using now Lemma 3.1 we obtain that s ∈ S(k) if and only if |(a1 , . . . , ap )| ≥ k. Corollary 4.2. Let S be a telescopic numerical semigroup minimally generated by {n1 < · · · < np }. For every i ∈ {2, . . . , p}, let ci = gcd{n1 , . . . , ni−1 }/ gcd{n1 , . . . , ni }. Then #H(S(k) ∪ {0}) = #H(S) + # (a1 , . . . , ap ) | 0 <

p

/ ai < k, 0 ≤ a2 < c2 , . . . , 0 ≤ ap < cp

.

i=1

In particular if k ≤ min{c2 , . . . , cp }, #H(S(k) ∪ {0}) = #H(S) +

(k + p − 1)! − 1. (k − 1)!p!

Proof. Follows from (9) of Theorem 4.1 and from the fact that the cardinality of #{(a1 , . . . , ap ) ∈ Np | 0 <

p

ai < k}

i=1

= #{(a1 , . . . , ap ) ∈ Np | 0 ≤ = #{(a0 , . . . , ap ) ∈ Np+1 | = =

p+1+k−1−1 k−1

(k + p − 1)! − 1. (k − 1)!p!

p

i=1 p

ai < k} − 1

ai = k − 1} − 1

i=0

−1

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Example 4.3. If S = n1 , n2 is a numerical semigroup, by using that gcd(n1 , n2 ) = 1, we have that the least positive integers c2 , d21 fulﬁlling that c2 n2 = d21 n1 are n1 and n2 , respectively. From this fact, it follows easily that every numerical semigroup generated by two elements is telescopic. Now we can use Corollary 4.2 to compute the number of elements of the set #H(n1 , n2 (k) ∪ {0}). We already know that 2 −1) and g(n1 , n2 ) = n1 n2 − n1 − n2 . Using this fact and #H(n1 , n2 ) = (n1 −1)(n 2 the fact that the cardinality of {(a1 , a2 ) | 0 < a1 + a2 < k, a2 < n1 } is equal to (k−1)(k−2) 1 −1) if k ≤ n1 and equal to (2k−1)(n + k − 1 if k > n1 , we obtain that 2 2 #H(S(n1 , n2 , k)∪{0}) =

(n1 −1)(n2 −1) 2 (n1 −1)(n2 −1) 2

+ +

(k−1)(k+2) , 2 (2k−n1 )(n1 −1) 2

+ k − 1,

if 1 ≤ k ≤ n1 , if k > n1 .

Corollary 4.4. Let S be a telescopic numerical semigroup minimally generated by {n1 < · · · < np }. For every i ∈ {2, . . . , p}, let ci = gcd{n1 , . . . , ni−1 }/ gcd{n1 , . . . , ni }. Let k be an integer greater than two. If c2 − 1 + · · · + cp − 1 < k − 1, set j (k) = 1, otherwise let j (k) = max{i ∈ {2, . . . , p} | ci − 1 + · · · + cp − 1 ≥ k − 1}. Then max(S \ (S(k) ∪ {0})) = ((k − 1) −

p

(ci − 1))nj (k)

i=j (k) +1

+ (cj (k) +1 − 1)nj (k) +1 + · · · + (cp − 1)np . Moreover, g(S(k) ∪ {0}) =

max(S \ (S(k) ∪ {0})), (c2 − 1)n2 + · · · + (cp − 1)np − n1 ,

p if i=2 (ci − 1) ≤ k − 1, p if i=2 (ci − 1) > k − 1.

Proof. From (8) and (9) of Theorem 4.1 we know that ϕ restricted to the set / p (a1 , . . . , ap ) | ai ≥ k, 0 ≤ a2 < c2 , . . . , 0 ≤ ap < cp i=2

is a bijection with the set S(k) . By using now that n1 < . . . < np , we obtain that the maximum element in (S \ (S(k) ∪ {0})) is equal to ((k − 1) −

p

(ci − 1))nj (k) + (cj (k) +1 − 1)nj (k) +1 + · · · + (cp − 1)np .

i=j (k) +1

p Assume that i=2 (ci − 1) > k − 1, then j (k) ≥ 2 and therefore max(S \ (S(k) ∪ {0})) ≤ (c2 − 2)n2 + · · · + (cp − 1)np < (c2 − 1)n2 + · · · + (cp − 1)np − n1 . Hence g(S(k) ∪ {0}) = (c2 − 1)n2 + · · · + (cp − 1)np − n1 . p If we have that i=2 (ci − 1) ≤ k − 1, then max(S \ (S(k) ∪ {0})) ≥ (c2 − 1)n2 + · · · + (cp − 1)np > (c2 − 1)n2 + · · · + (cp − 1)np − n1 . Thus g(S(k) ∪ {0}) = max(S \ (S(k) ∪ {0})).

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As a consequence of the above result we obtain that the Frobenius number of S(k) ∪ {0} is equal to the Frobenius number of S if and only if c2 − 1 + · · · + cp − 1 > k − 1. Example 4.5. Let n1 , n2 be a numerical semigroup. We mentioned already that c2 = n1 , whence in this case the above condition c2 − 1 + · · · + cp − 1 > k − 1 is equivalent to n1 > k. From Corollary 4.4, we obtain the following formula for the Frobenius number of n1 , n2 (k) ∪ {0}: (k − n1 )n1 + (n1 − 1)n2 , if n1 ≤ k, g(n1 , n2 (k) ∪ {0}) = (n1 − 1)n2 − n1 , if n1 > k.

5

Minimal System of Generators

It is well-known (see for instance [17]) that if S is a numerical semigroup with minimal system of generators {n1 , . . . , np }, then {n1 , . . . , np } = (S \ {0}) \ ((S \ {0}) + (S \ {0})). The following result provides a method for computing a minimal system of generators of S(k) ∪ {0}. Lemma 5.1. Let S be a numerical semigroup and let k be a positive integer. Then s is a minimal generator of S(k) ∪ {0} if and only if k ≤ L(s) ≤ 2k − 1. Proof. Necessity. Assume that {n1 < . . . < np } is the minimal system of generators of S. If L(x) < k, then x ∈ S(k) ∪ {0}. If L(x) ≥ 2k, then x = a1 n1 + · · · + ap np for some (a1 , . . . , ap ) ∈ Np with a1 + · · · + ap ≥ 2k. Let (b1 , . . . , bp ), (c1 , . . . , cp ) ∈ Np be such that (a1 , . . . , ap ) = (b1 , . . . , bp ) + (c1 , . . . , cp ) and b1 + · · · + bp ≥ k and c1 + · · · + cp ≥ k. In this case, both y = b1 n1 + · · · + bp np and z = c1 n1 + · · · + cp np belong to S(k) ∪ {0}. As x = y + z, we have that x is not a minimal generator of S(k) ∪ {0}. Suﬃciency. Since k ≤ L(x), x ∈ S(k) ∪ {0}. Moreover, if there exist y, z ∈ S(k) ∪ {0} such that x = y + z, then L(x) = L(y + z) ≥ L(y) + L(z) ≥ 2k, in contradiction with L(x) ≤ 2k − 1. Theorem 5.2. Let S be a telescopic numerical semigroup minimally generated by {n1 < · · · < np }. For every i ∈ {2, . . . , p}, let ci = gcd{n1 , . . . , ni−1 }/ gcd{n1 , . . . , ni }. Let k be an integer greater than two. There is a one-to-one correspondence between the set of minimal generators of S(k) ∪ {0} and the set / p ai ≤ 2k − 1, 0 ≤ a2 < c2 , . . . , 0 ≤ ap < cp (a1 , . . . , ap ) | k ≤ i=2

(ϕ gives this correspondence). Furthermore, if 2k − 1 < min{c2 , . . . , cp }, then the cardinality of the minimal system of generators of S(k) ∪ {0} is equal to (2k + p − 1)! (k + p − 1)! − . (2k − 1)!p! (k − 1)!p!

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Example 5.3. Let S = 20, 35, 37. It is easy to prove that S is telescopic and that its relations are the following: 4 · 35 = 5 · 37 =

7 · 20, 4 · 20 + 3 · 35.

Therefore, for this semigroup c2 = 4 and c3 = 5. Thus, by using Theorem 5.2 we can ﬁnd a system of generators of S(2) ∪ {0} after applying the function ϕ(x, y, z) = 20x + 35y + 37y to the elements of the set {(a1 , a2 , a3 ) | 2 ≤ a2 + a3 ≤ 3, 0 ≤ a2 < 4, 0 ≤ a3 < 5} . By Theorem 5.2, and since 2k − 1 = 3 < c2 = 4 and 2k − 1 = 3 < c3 = 5, the above 6! 4! set has 3!3! − 1!3! = 16 elements and the minimal system of generators of S(2) ∪ {0} is {40, 55, 57, 60, 70, 72, 74, 75, 77, 90, 92, 94, 105, 107, 109, 111}. We use now Corollary 4.4 for computing g(S(2) ∪ {0}). Since c2 − 1 + c3 − 1 = 7 > 1 = k − 1, we obtain g(S(2) ∪ {0}) = g(S) = (c2 − 1)n2 + (c3 − 1)n3 − n1 = 3 · 35 + 4 · 37 − 20 = 233. Let us try now to compute the minimal system of generators of S(3) ∪ {0}. The set given by Theorem 5.2 is now equal to {(a1 , a2 , a3 ) | 3 ≤ a2 + a3 ≤ 5, 0 ≤ a2 < 4, 0 ≤ a3 < 5} . In this case 2k − 1 = 5 < c2 = 4 and 2k − 1 = 5 < c3 = 5. Hence we can only assert that cardinality of the minimal system of generators of S(3) ∪ {0} is less than or 8! 5! − 2!3! = 46. The reader can check that the above set has 41 elements equal to 5!3! and that the minimal system of generators of S(3) ∪ {0} is the set {60, 75, 77, 80, 90, 92, 94, 95, 97, 100, 105, 107, 109, 110, 111, 112, 114, 115, 117, 125, 127, 129, 130, 131, 132, 134, 142, 144, 145, 146, 147, 148, 149, 151, 162, 164, 166, 168, 179, 181, 183}. By using again Corollary 4.4, we get that as c2 − 1 + c3 − 1 = 7 > 2 = k − 1, g(S) = g(S(2) ∪ {0}) = g(S(3) ∪ {0}) = 233.

6

Behavior of Some Invariants as k Increases

Once we have S a numerical semigroup, we can construct the following chain of numerical semigroups . . . ⊆ S(k) ∪ {0} ⊆ . . . S(2) ∪ {0} ⊆ S(1) ∪ {0} = S We ﬁnish this chapter with the following theorem, which shows how the invariants studied in this chapter change as we move in this chain when S is a telescopic numerical semigroup, and summarizes the results obtained in this chapter.

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269

Theorem 6.1. Let S be a telescopic numerical semigroup with {n1 , . . . , np } its minimal system of generators, k be a positive integer and ci = gcd{n1 , . . . , ni−1 }/ gcd{n1 , . . . , ni }. 1. m(S(k+1) ∪ {0}) = m(S(k) ∪ {0}) + n1 = (k + 1)n1 . (k+p−1!) − 1. 2. #H(S(k) ∪ {0}) ≤ #H(S) + (k−1)!(p)! g(S(k) ∪ {0}) if (c2 − 1) + · · · + (cp − 1) > k, 3. g(S(k+1) ∪ {0}) = n1 + g(S(k) ∪ {0}) if (c2 − 1) + · · · + (cp − 1) ≤ k,

4. e(S(k) ∪ {0}) ≤

(2k+p−1)! (2k−1)!(p)!

−

(k+p−1)! (k−1)!(p)! .

If we focus our attention on numerical semigroups generated by two elements we can improve (2) and (4) of the above result. Let S = n1 , n2 a numerical semigroup. From (9) of Theorem 4.1, we deduce that k + 1, if 1 ≤ k < n1 , #{x ∈ n1 , n2 | L(x) = k} = n1 , if k ≥ n1 . Clearly we have that #H(S(k+1) ∪ {0}) = #H(S(k) ∪ {0}) + #{x ∈ n1 , n2 | L(x) = k} and therefore #H(S(k) ∪ {0}) + k + 1, if k < n1 , #H(S(k+1) ∪ {0}) = #H(S(k) ∪ {0}) + n1 , if k ≥ n1 . From Lemma 5.1 we deduce that if Ak is a minimal system of generators of S(k) ∪ {0} and Ak+1 is a minimal system of generators of S(k+1) ∪ {0}, then Ak+1 = (Ak \ {x ∈ n1 , n2 | L(x) = k}) ∪ {x ∈ n1 , n2 | L(x) ∈ {2k, 2k + 1}}. By using again the given formula for the cardinality of {x ∈ n1 , n2 | L(x) = k}, we obtain that ⎧ e(S(k) ∪ {0}) + 3k + 2, if 2k + 1 < n1 , ⎪ ⎪ ⎨ e(S(k) ∪ {0}) + n1 + k, if 2k < n1 ≤ 2k + 1, e(S(k+1) ∪ {0}) = e(S(k) ∪ {0}) + 2n1 − k − 1, if k < n1 ≤ 2k, ⎪ ⎪ ⎩ if n1 ≤ k. e(S(k) ∪ {0}) + n1 ,

Bibliography [1] R. Ap´ery, Sur les branches superlin´eaires des courbes algebriques, C. R. Acad. Sci. Paris 222, 1198–1200. [2] V. Barucci, D. E. Dobbs and M. Fontana, Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains, Memoirs of the Amer. Math. Soc. 598 (1997). [3] J. Bertin and P. Carbone, Semi-groups d’entiers et application aux branches, J. Algebra 49 (1987), 81–95. [4] J. Castellanos, A relation between the sequence of multiplicities and the semigroups of values of an algebroid curve, J. Pure App. Algebra 43 (1986), 119– 127. [5] S. T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths, and associated combinatorial problems, Factorization in Integral Domains (D. D. Anderson, ed.), Lecture Notes in Pure and Appl. Math., Marcel Dekker 189 (1997), 73–112. [6] S. T. Chapman, V. DeLorenzo and H. Swisher, On the asymptotic behavior of irreducibles in block semigroups, Semigroup Forum 63 (2001), 34–48. [7] S.T. Chapman, J. I. Garc´ıa-Garc´ıa, P. A. Garc´ıa-S´anchez and J. C. Rosales, On the number of factorizations of an element in an atomic monoid, Adv. Appl. Math. 29 (2002), 438–453. [8] S.T. Chapman, J. I. Garc´ıa-Garc´ıa, P. A. Garc´ıa-S´anchez and J. C. Rosales, Computing the elasticity of a Krull monoid, Linear Algebra Appl. 336 (2001), 191–200. [9] S. T. Chapman and J. C. Rosales, On the asymptotic values of length functions in Krull and ﬁnitely generated commutative monoids, J. Austral. Math. Soc., to appear. ´ [10] C. Delorme, Sous-mono¨ıdes d’intersection compl`ete de N, Ann. Scient. Ecole Norm. Sup. (4) 9, 1976, 145–154. [11] A. Geroldinger and G. Lettl, Factorization problems in semigroups, Semigroup Forum 40 (1990), 23–38. 270

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[12] F. Halter-Koch, On the asymptotic behaviour of the number of distinct factorizations into irreducibles, Ark. Mat. 31 (1993), 297–305. [13] C. Kirfel and R. Pellikaan, The minimum distance of codes in an array coming from telescopic semigroups, IEEE Trans. Inform. Theory 41(6, part 1) (1995), 1720–1732. Special issue on algebraic geometry codes. [14] E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25 (1973), 748–751. [15] J. L. Ram´ırez Alfons´ın, The Diophantine Frobenius problem, Forschungsintitut f¨ ur Diskrete Mathematik, Bonn, Report N0.00893 (2000). [16] L. R´edei, The Theory of Finitely Generated Commutative Semigroups, Pergamon, 1965. [17] J. C. Rosales and P. A. Garc´ıa-S´anchez, Finitely Generated Commutative Monoids, Nova Science Publishers, New York, 1999. [18] J. C. Rosales and P. A. Garc´ıa-S´anchez, On free aﬃne semigroups, Semigroup Forum 58(3) (1999), 367–385 (MR 2000j:13020). [19] J. J. Sylvester, Mathematical questions with their solutions, Educational Times 41 (1884), 21. [20] K. Watanabe, Some examples of one dimensional Gorenstein domains, Nagoya Math. J. 49 (1973), 101–109.

Chapter 17

Pr¨ ufer Conditions in Rings with Zero-Divisors by

1

Sarah Glaz

Introduction

In his article “Untersuchungen u ¨ ber die Teilbarkeitseigenschaften in K¨ orpernn,” J. Reine Angew. Math. 168, 1 - 36, 1932 [21], Heinz Pr¨ ufer introduced a new class of integral domains, namely those domains R in which all ﬁnitely generated ideals are invertible. He also proved that to verify this condition, it suﬃces to check that it holds for all two-generated ideals of R. This was the modest beginning of the notion of a Pr¨ ufer domain, a notion which made, and continues to make, a signiﬁcant impact on research in non-Noetherian commutative ring theory. Heinz Pr¨ ufer (1896–1934), in his short life, had no opportunity to see the rings named in his honor by Krull ([17], 1936). It is not an exaggeration to say that today there is no conference on a non-Noetherian ring theory topic where the notion of a Pr¨ ufer domain does not make an appearance. Pr¨ ufer domains acquired, through the years, a great many equivalent characterizations, each of which can, and was, extended to rings with zero divisors in a number of ways. The purpose of this chapter is to explore the relations between several extensions of the notions of a Pr¨ ufer domain to rings with zero divisors. These generalizations fall into the category which may be called homological characterizations, though work on all the rings mentioned in this chapter entails a blend of ring theoretic and homological techniques. We consider the following extensions of the Pr¨ ufer domain notion: 1. R is a semihereditary ring. 2. w.dim R ≤ 1. 3. R is an arithmetical ring. 272

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4. R is a Gaussian ring. 5. R is a Pr¨ ufer ring. In Section 2, we provide the necessary background material, including deﬁnitions, a short historical survey, and the motivation for this investigation. Section 3 is devoted to implications and counterexamples. We use a blend of known and new results to show the implications: 1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 5. We provide examples that show that none of the implications are, in general, reversible. We also explore necessary and suﬃcient conditions under which some of the implications become reversible.

2

Background

A domain D is called a Pr¨ ufer domain if every ﬁnitely generated ideal of D is invertible. This is the original deﬁnition given by Pr¨ ufer in 1932 [21]. In 1936 Krull [17] turned his attention to Pr¨ ufer domains; he named them, and proved the ﬁrst equivalent deﬁnition of such a domain, namely: Theorem 2.1 (Krull [17]). D is a Pr¨ ufer domain if and only if every localization of D by a prime (respectively maximal) ideal of D is a valuation domain. For the purpose of later generalizations to rings with zero divisors we note that for a domain D to be a valuation domain it is necessary and suﬃcient that the set of all the ideals of D be totally ordered by inclusion. Let R be a ring and let f be a polynomial in R[x]. c(f ) - the content of f is the ideal of R generated by the coeﬃcients of f . For any two polynomials f and g in R[x], we have c(f g) ⊆ c(f )c(g). f is called a Gaussian polynomial if c(f g) = c(f )c(g) for every g ∈ R[x]. A ring R is called a Gaussian ring if every polynomial with coeﬃcients in R is a Gaussian polynomial. Both deﬁnitions are due to Tsang [23]. In 1965, Tsang [23] and, independently in 1967, Gilmer [9] provided the following ufer domain: elegant characterization of a Pr¨ Theorem 2.2 (Tsang [23], Gilmer [9]). D is a Pr¨ ufer domain if and only if D is a Gaussian domain. Let R be a commutative ring, and denote by Q(R) the total ring of quotients of R. A (fractionary) ideal I of R is invertible if II −1 = R, where I −1 = {r ∈ Q(R) | rI ⊆ R}. An invertible ideal is ﬁnitely generated and contains a regular element. For an ideal I of R there is a strong relation between invertibility, projectivity, and the property of being locally principal, namely:

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Theorem 2.3. Let R be a ring, and let I be an ideal of R. Then: 1. If I is invertible, then I is projective. 2. If I is projective, then I is locally principal. 3. If I is ﬁnitely generated and regular then: I is invertible if and only if I is projective if and only if I is locally principal. In particular, the three conditions are equivalent for a ﬁnitely generated ideal of a domain R. A ring R is called a semihereditary ring if every ﬁnitely generated ideal of R is projective. The origin of this notion is obscure. The earliest source I encountered, where semihereditary rings are mentioned, is [5]. In view of the above theorem, we conclude: Corollary 2.4. D is a Pr¨ ufer domain if and only if D is a semihereditary domain. It follows that both Gaussian rings and semihereditary rings extend the notion of a Pr¨ ufer domain to rings with zero divisors. Pr¨ ufer’s original deﬁnition was generalized almost verbatim by Butts and Smith [4]. R is a Pr¨ ufer ring if every ﬁnitely generated regular ideal of R is invertible. In 1970, Griﬃn [14] studies these rings and a number of related generalized notions. Among many other results his article includes the proof of fourteen equivalent definitions for a Pr¨ ufer ring, reminiscent of similar results known to hold for Pr¨ ufer domains (see [9], and [7]). It remains to consider generalizations of Krull’s equivalent deﬁnition of a Pr¨ ufer domain. There are two equally natural candidates for this purpose. The ﬁrst is an arithmetical ring. A ring R is called an arithmetical ring if the set of ideals of every localization of R by a prime (respectively maximal) ideal of R is totally ordered by inclusion. This notion is due to Fuchs [8]. The second generalization is to a ring R with weak global dimension of R less or equal than one, w.dim R ≤ 1. Those are the rings over which every module has weak dimension at most one. Equivalently, w.dim R ≤ 1 means that every ideal of R is ﬂat. Another characterization of rings R with w.dim R ≤ 1 is that they are precisely the rings whose localizations at the prime ideals are all valuation domains (see [10]). This makes them good ufer domains as characterized by Krull. Hence both candidates for extensions of Pr¨ arithmetical rings and rings of w.dim R ≤ 1 generalize Krull’s characterization of a Pr¨ ufer domain. The diﬀerence between the two notions is that the localizations by prime ideals of an arithmetical ring need not be domains. We therefore arrived at the full list of ﬁve generalizations mentioned in the introduction. The motivation for our interest in these particular ﬁve generalizations was our interest in the notion of a Gaussian ring. This interest ﬁrst took

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the form of a joint work with W. Vasconcelos regarding a conjecture of Kaplansky: The content ideal of a Gaussian polynomial is an invertible (or locally principal) ideal. The reason behind the conjecture is that the converse holds [23]. Vasconcelos and I answered the question aﬃrmatively in a large number of cases [11, 12]. The aﬃrmative answer was later extended by Heinzer & Huneke [15] to include all Noetherian domains. Recently the question was answered aﬃrmatively for all domains by Loper & Roitman [18], and ﬁnally to nondomains provided the content ideal has zero annihilator by Lucas [19]. A ﬂurry of related research ensued, particularly investigations involving Dedekind-Mertens Lemma and various extensions of the Gaussian property. [2] and [6] provide a survey of results obtained up to year 2000 and an extensive bibliography. A related, but diﬀerent, question is: How Pr¨ ufer-like is a Gaussian ring? Various aspects of the nature of Gaussian rings were investigated in Tsang’s thesis [23], by D.D. Anderson [1], D.D. Anderson & Camillo [3], and Glaz [13]. While all of those works touch indirectly on the mentioned question, it is Glaz [13] that asks and provides some direct answers. We will cite relevant results later on in this chapter.

3 3.1

Implications and Counterexamples The relation between semihereditary rings and rings R of w.dim R ≤ 1

Theorem 3.1. Semihereditary rings R have w.dim R ≤ 1. Proof. This implication is well-known. We sketch one possible proof here: Since every principal ideal of a semihereditary ring R is projective, we obtain that for every prime ideal P of R, RP is a domain. Moreover, ﬁnitely generated ideals of RP are projective and therefore free. Thus RP is a Bezout domain, and hence a valuation domain. This implies that w.dim R = sup{w.dim RP | P runs over all prime ideals of R} ≤ 1.

Example 3.2. A nonsemihereditary ring R with w.dim R ≤ 1. Let Q be the rational numbers, and let Q[x] be the ring of polynomials in one variable over Q. Let R be the subring of ΠQ[x], the inﬁnite product of Q[x], consisting of the sequence (x, 0, x2 , 0, x3 , 0, . . .), and all sequences that eventually consist of constants. 2 The following is a compilation of the known conditions [10, 20] under which a ring of w.dim R ≤ 1 is semihereditary. Theorem 3.3 (Glaz [10], Marot [20]). Let R be a ring. The following conditions are equivalent:

276

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1. R is a semihereditary ring. 2. R is a coherent ring of w.dim R ≤ 1. 3. w.dim R ≤ 1 and the total ring of quotients of R, Q(R), is Von Neumann regular. Proof. The implication 1 ⇔ 2 is well-known and due, independently, to Marot [20], and Glaz [10]. The implication 1 ⇔ 3 follows from a result of Marot [20], which states that a ring R is semihereditary if and only if Q(R) is Von Neumann regular, and RP is a valuation domain for every prime ideal P of R.

3.2

The relation between rings R of w.dim R ≤ 1 and arithmetical rings

Theorem 3.4 (Jensen [16]). A ring R with w.dim R ≤ 1 is an arithmetical ring. Proof. The proof is due to Jensen [16]. The remarks above regarding the nature of the localizations of both type of rings make the proof of this implication clear. We now cite a very useful characterization of arithmetical rings due to Jensen [16], which helped us ﬁnd a counterexample to the converse of Theorem 3.4. Theorem 3.5 (Jensen [16]). R is an arithmetical ring if and only if every ﬁnitely generated ideal of R is locally principal. Example 3.6. An arithmetical ring R with w.dim R > 1. R = Z4 , the ring of integers modulo 4. Since its only ideal is 2Z4 , Z4 is arithmetical. Note that Z4 is not a reduced ring since 22 = 0. By Theorem 3.7 below, 2 we conclude that w.dim Z4 > 1. Theorem 3.7 (Jensen [16]). Let R be a ring. The following conditions are equivalent: 1. w.dim R ≤ 1. 2. R is an arithmetical reduced ring.

3.3

The relation between arithmetical rings and Gaussian rings.

Theorem 3.8. An arithmetical ring is a Gaussian ring. Proof. Let f be a polynomial with coeﬃcients in an arithmetical ring R. Then by Theorem 3.5 c(f ) is a locally principal ideal. It is proved in [23] that such polynomials are Gaussian; therefore R is a Gaussian ring.

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Example 3.9. A nonarithmetical Gaussian ring. Let k be a ﬁeld, and let t and u be indeterminates over k. Let T and U be the images of t and u in k[t, u]/(t, u)2 , and let R = k[T, U ](T,U ) . R is a local ring with maximal ideal m = (T, U ). Because m2 = 0, one can easily check that R is Gaussian. But m is not principal so R is not arithmetical. 2 Of recent vintage is the investigation carried out in Glaz [13], into the nature of the weak global dimension of a Gaussian ring. Among other results we found necessary and suﬃcient conditions for a Gaussian ring R to be of w.dim R ≤ 1, or semihereditary. After a brief preliminary discussion we cite the results obtained in this paper in Theorems 3.11 and 3.12 below. Recall two conditions one may impose on the set of principal ideals of a ring R to gain some control over the behavior of its zero divisors. R is called a PF ring if the principal ideals of R are ﬂat. This condition is equivalent to R being locally a domain [10]. A ring R is called a PP ring, or weak Baer ring, if the principal ideals of R are projective. The PP condition is stronger then the PF condition. The exact relation between the two conditions is given in Theorem 3.10, below. Min R denotes the set of all minimal prime ideals of R in the induced Zariski topology from Spec R. Theorem 3.10 (Glaz [10]). Let R be a ring. The following conditions are equivalent: 1. R is a PF ring and Min R is compact. 2. R is a PP ring. 3. R is a PF ring and Q(R), the total ring of quotients of R, is a Von Neumann regular ring. We now note the conditions from [13] that allow reversal of implications involving the Gaussian condition. Theorem 3.11 (Glaz [13]). Let R be a ring. The following conditions are equivalent: 1. w.dim R ≤ 1. 2. R is a Gaussian PF ring. 3. R is a Gaussian reduced ring. Theorem 3.12 (Glaz [13]). Let R be a ring. The following conditions are equivalent: 1. R is a semihereditary ring.

278

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2. R is a Gaussian PP ring. 3. R is a Gaussian ring and Q(R) is a Von Neumann regular ring. The two results cited above yield an unexpected bonus, the equivalence, over Gaussian rings, of conditions which are not generally equivalent: The PF condition implies that a ring is reduced, but the converse is not generally true, as any reduced local ring which is not a domain shows. Theorem 3.11 implies that for Gaussian rings the two conditions coincide. A similar relation holds between the two conditions “R is a PP ring” and “Q(R) is Von Neumann regular”. The PP condition implies the Von Neumann regularity of the total ring of quotients. On the other hand the condition “Q(R) is Von Neumann regular” does not, in general, need to imply that principal ideals are projective. To see this let R be a Noetherian, local, reduced ring which is not a domain. Such a ring is, necessarily, not a PF ring. Because the ring is Noetherian it has ﬁnitely many minimal prime ideals. The total ring of quotients of R is the product of all the localizations of R by the minimal prime ideals of R. As each such localization is a ﬁeld, Q(R) is a Von Neumann regular ring. Theorem 3.12 implies that the two conditions are equivalent when the ring is Gaussian. By putting together some of the results and remarks regarding Gaussian and arithmetic rings, we obtain that if we impose on a Gaussian ring any of the zero divisor controlling conditions discussed in this section, be it PF, PP, “Q(R) is Von Neumann regular”, or “R is reduced”, we obtain that R is arithmetical. We do not, at the moment, know the precise conditions needed as an addition to the Gaussian property to make it equivalent to the arithmetic condition. But given the evidence accumulated so far, we dare make the following conjecture: Conjecture 3.13. A ring R is arithmetical if and only if R is a Gaussian ring and Q(R), the total ring of quotients of R, is an arithmetical ring.

3.4

The relation between Gaussian rings and Pr¨ ufer rings

Theorem 3.14. Let R be a Gaussian ring; then R is a Pr¨ ufer ring. Proof. There are several ways one can prove this fact, but it will be nice to deduce it easily from the latest article on Kaplansky’s Conjecture, Lucas [19]. Lucas [19] proved a slightly stronger version of the following: Let R be a ring, and let f be a Gaussian polynomial whose content ideal c(f ) contains a regular element. Then c(f ) is an invertible ideal. We conclude that all ﬁnitely generated regular ideals of a Gaussian ring are invertible. Example 3.15. A non-Gaussian Pr¨ ufer ring. Let k be a countable, algebraically closed ﬁeld, let J be an inﬁnite set, and denote by k J the set of all maps from J to k. Let N denote the set of natural numbers.

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Let L = J × NN . Quentel ([22], and see [10], page 120 for an error-free version) constructed an algebra R ⊆ k L , which satisﬁes the following four properties: 1. R is a reduced ring. 2. R = Q(R). 3. Min R, the set of all minimal prime ideals of R in the induced Zariski topology, is compact. 4. R is not a Von Neumann regular ring. Since R = Q(R), every element of R is either a unit or a zero divisor, and therefore R has no regular ideals. Thus R is a Pr¨ ufer ring. It is proved in [13] that R is not a Gaussian ring. 2 We are not sure which additional property will provide necessary and suﬃcient conditions for the reversal of the implication of Theorem 3.14. We just remark that Pr¨ ufer rings are very close to being Gaussian rings, namely: Theorem 3.16. Let R be a Pr¨ ufer ring, and let f and g be two polynomials in R[x]. If c(f ) is a regular ideal, then c(f g) = c(f )c(g). Proof. It is straightforward from Tsang’s [23] result which states that polynomials with invertible content ideal are Gaussian.

Bibliography [1] D.D. Anderson, Another generalization of principal ideal rings, J. Algebra 48 (1997), 409–416. [2] D.D. Anderson, GCD domains, Gauss’ Lemma, and contents of polynomials, Non-Noetherian commutative ring theory, Kluwer Academic Publishers (2000), 1–31. [3] D.D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), 2265–2272. [4] H.S. Butts and W. Smith, Pr¨ ufer rings, Math. Z 95 (1967), 196–211. [5] H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, 1956. [6] A. Corso and S. Glaz, Gaussian ideals and the Dedekind-Mertens Lemma, Lecture Notes Pure Appl., Math Marcel Dekker 217 (2001), 131–143. [7] M. Fontana, J.A. Huckaba and I.J. Papick, Pr¨ ufer domains, Marcel Dekker, 1997. ¨ [8] L. Fuchs, Uber die ideale arithmetischer ringe, Comment. Math. Helv. 23 (1949), 334–341. [9] R. Gilmer, Multiplicative ideal theory, Marcel Dekker, 1972. [10] S. Glaz, Commutative coherent rings, Springer-Verlag Lecture Notes 1371, 1989. [11] S. Glaz and W.V. Vasconcelos, Gaussian polynomials, Cummutative ring theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 186 (1997), 325–337. [12] S. Glaz and W.V. Vasconcelos, The content of Gaussian polynomials, J. Algebra 202 (1998), 1–9. [13] S. Glaz, The weak dimension of Gaussian rings, submitted. [14] M. Griﬃn, Pr¨ ufer rings with zero-divisors, J. Reine Angew Math. 239/240 (1970), 55–67. 280

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[15] W. Heinzer and C. Huneke, Gaussian polynomials and content ideals, Proc. Amer. Math. Soc. 125 (1997), 739–745. [16] C.U. Jensen, Arithmetical rings, Acta Math. Hungr. 17 (1966), 115–123. [17] W. Krull, Beitr¨age zur arithmetik kommutativer integrit¨atsbereiche, Math. Z. 41 (1936), 545–577. [18] K. A. Loper and M. Roitman, The content of a Gaussian polynomial is invertible, preprint. [19] T. Lucas, Gaussian polynomials and invertibility, preprint. [20] J. Marot, Sur les anneaux universellement Japonais, Ph.D. Thesis, Universite de Paris-Sud, 1977. [21] H. Pr¨ ufer, Untersuchungen uber teilbarkeitseigenschaften in korpern, J. Reine Angew. Math. 168 (1932), 1–36. [22] Y. Quentel, Sur la compacit´e du spectre minimal d’un anneau, Bull. Soc. Math. France 99 (1971), 265–272. [23] H. Tsang, Gauss’s Lemma, Ph.D. Thesis, University of Chicago, 1965.

Chapter 18

Unmixedness and the Generalized Principal Ideal Theorem by

Tracy Dawn Hamilton Abstract Recent work toward extending the theory of Cohen-Macaulayness to all commutative rings (both Noetherian and non-Noetherian) has led to the deﬁnition of weak Bourbaki unmixed rings (wB-unmixed) and weak Bourbaki height-unmixed rings (wB-ht-unmixed). In this work we study these unmixedness conditions on rings which satisfy the generalized principal ideal theorem (GPIT) or at least the principal ideal theorem (PIT). This is a natural extension from Noetherian rings since Noetherian rings satisfy GPIT and PIT. There are, however, many rings which satisfy GPIT and/or PIT which are not Noetherian. Among the results are the following: (1) In rings which satisfy GPIT, wB-ht-unmixed is equivalent to wB-unmixed. (2) Every unmixed domain (in either sense) satisﬁes PIT. (3) Locally Cohen-Macaulay rings (which are locally Noetherian and therefore satisfy GPIT) are unmixed. As a corollary to result (2) we also get that a Pr¨ ufer domain R is wB-ht-unmixed if and only if dim (R) ≤ 1.

1

Introduction

In two recent papers ([1] and [2]) we have studied wB-ht-unmixed rings and wBunmixed rings as candidates for an appropriate deﬁnition of non-Noetherian CohenMacaulayness. The question of whether there exists an appropriate deﬁnition of non-Noetherian Cohen-Macaulayness was ﬁrst posed by Sarah Glaz in 1992 [3] and then again in 1994 [4]. WB-ht-unmixedness and wB-unmixedness (which will be deﬁned in Section 2) have each been shown to satisfy, at least, some of the requirements for an appropriate deﬁnition of non-Noetherian Cohen-Macaulayness. 282

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283

To determine whether either of these types of rings is an appropriate deﬁnition of non-Noetherian Cohen-Macaulayness we must learn as much as possible about these types of rings. As with most questions about non-Noetherian rings, it is diﬃcult to proceed without placing further restrictions on the ring. In this chapter we will consider these unmixedness questions on rings which satisfy either the principal ideal theorem (PIT) or the generalized principal ideal theorem (GPIT) conditions. We will ﬁnd that in rings which satisfy GPIT, these two notions of unmixedness are equivalent. We will also ﬁnd that unmixed (in either sense) domains must satisfy PIT. These results will allow us to completely classify Pr¨ ufer domains with respect to unmixedness.

2

Unmixedness

All rings are commutative rings with unity. Recall that an ideal I is said to be height-unmixed if all the associated primes of I have equal height. That is ht(P ) =ht(Q) for all P, Q ∈ all Assoc(I) where Assoc(I) denotes the set of associated prime ideals of I. An ideal I is said to be unmixed if there are no embedded primes among the associated primes of I. That is, P ⊆ Q =⇒ P = Q for all P, Q ∈Assoc(I) . Lemma 2.1. If R is a locally ﬁnite dimensional (LFD) ring, then all heightunmixed ideals are unmixed. Proof. Suppose I is a height-unmixed ideal in a LFD ring. Let P, Q ∈Assoc(I) with P ⊆ Q. Then ht(P ) =ht(Q) since I is height-unmixed. Also, ht(Q) < ∞ since the ring is LFD. Therefore, since P ⊆ Q, we have P = Q. For a non-Noetherian ring, there are many types of associated primes which may be used in the above deﬁnitions of unmixedness. In 1984 Iroz and Rush [5] published a paper describing most of the commonly used sets of associated primes in non-Noetherian rings all of which are equivalent in a Noetherian ring. One of the most commonly studied sets of associated primes, and the one that will be used here, is the set of weak Bourbaki associated primes. The set of weak Bourbaki associated primes to an ideal I is denoted Assf (I) . A prime ideal P is a weak Bourbaki associated prime of the ideal I if P ∈ Min(I : a) for some a ∈ R. Since (I : a) = R if a ∈ I, we have Assf (I) = Min (I : a) = Min (I : a) . a∈R

a ∈I

One reason for studying these primes, as opposed to some other set of associated primes, is that the set of weak Bourbaki associated primes of an ideal I is the smallest set of commonly studied associated primes which is necessarily nonempty for each proper ideal I. Note: we will say that an ideal is wB-ht-unmixed if it is height-unmixed with respect to the set of weak Bourbaki associated primes and an ideal is wB-unmixed if it is unmixed with respect to the set of weak Bourbaki associated primes.

284

Hamilton (X)

(Y,Z) (XY,XZ) (Y) (O)

The following example shows that an unmixed ideal need not be height-unmixed even if the ring is LFD (in fact, even if the ring is Noetherian). Example 2.2. Let R = k [X, Y, Z] where k is a ﬁeld and let I = (XY, XZ) . Then both (X) and (Y, Z) are in Assf (I) . These ideals have diﬀerent heights, so I is not height-unmixed with respect to the weak Bourbaki associated primes. However, I has no embedded components with respect to the weak Bourbaki associated primes, so I is unmixed. We are almost ready for the deﬁnitions of wB-ht-unmixed and wB-unmixed rings. First, recall that in [1] a height-generated ideal was deﬁned to be a ﬁnitely generated ideal I which can be generated by ht(I) elements. That is, a ﬁnitely generated ideal I is height-generated if µ (I) ≤ht(I) where µ (I) denotes the minimal number of generators of I. It is straightforward to show that in a Noetherian ring wB-unmixed is equivalent to wB-ht-unmixed for height-generated ideals; so, for height-generated ideals in Noetherian rings, we may simply refer to the ideal being unmixed to refer to either of the conditions wB-unmixed or wB-ht-unmixed. To see why wB-ht-unmixed and wB-unmixed rings have been chosen as candidates for an appropriate deﬁnition of non-Noetherian Cohen-Macaulayness we ﬁrst recall the deﬁnition of a Cohen-Macaulay ring. Definition 2.3. A Noetherian ring R is Cohen-Macaulay if every height-generated ideal in R is unmixed. To extend this to an appropriate deﬁnition of non-Noetherian Cohen-Macaulay rings it seems reasonable to simply remove the word “Noetherian” from the deﬁnition above. When this is done you must, of course, specify what is meant by the word unmixed (i.e. unmixed or ht-unmixed). You must also specify which type of associated primes you will be using. In this chapter we will be restricting our study to the weak Bourbaki. We can then consider the following two deﬁnitions as possible appropriate deﬁnitions of non-Noetherian Cohen-Macaulay rings. Definition 2.4. [1] A ring R is said to be weak Bourbaki unmixed (wB-unmixed) if every height-generated ideal in R is wB-unmixed. It can easily be seen that a ring is wB-unmixed if every height-generated ideal I in the ring satisﬁes Assf (I) =Min(I) . Definition 2.5. [2] A ring R is said to be weak Bourbaki height-unmixed (wB-htunmixed) if every height-generated ideal in R is wB-ht-unmixed.

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The following lemma follows immediately from lemma 2.1. Lemma 2.6. Every LFD wB-ht-unmixed ring is wB-unmixed.

3

Rings Satisfying PIT or GPIT

One of the most important theorems in Noetherian ring theory is Krull’s Principal Ideal Theorem [6]. Krull’s theorem says that every Noetherian ring satisﬁes the following condition (referred to as PIT): If P is a prime ideal minimal over a principal (proper) ideal of R, then ht(P ) ≤ 1. Theorem 3.1. Every wB-ht-unmixed (resp. wB-unmixed) domain satisﬁes PIT. Proof. Let R be a domain which does not satisfy PIT. Then dim (R) ≥ 2. Since R does not satisfy PIT, there is a principal ideal I = (a) which has a minimal prime P with ht(P ) > 1. Let Q be a prime ideal in R with ht(Q) = 1 and Q ⊂ P and let x be a nonzero element in Q. Then the product ax is a nonzero element of Q. Let J = (ax) . Then J = (0) and J ⊆ Q, so ht(J) = 1. Thus, J is heightgenerated. Consider (J : x) . It is easily veriﬁed that (J : x) = (a) = I. Therefore, since P is minimal over I, we have P ∈Min(J : x) ⊆Assf (J) . Thus, J is not wBht-unmixed since P ∈Assf (J) with ht(P ) > 1 =ht(J) . Also, J is not wB-unmixed since Q ∈Min(J) ⊆ Assf (J) , so Q, P ∈Assf (J) , but Q P. Therefore, R is neither wB-ht-unmixed nor wB-unmixed. As a corollary to theorem 3.1 we get a complete classiﬁcation of the unmixedness of Pr¨ ufer domains. Pr¨ ufer domains have been described by Gilmer [7, Chapter IV] as playing a central role in multiplicative ideal theory; so having this characterizations is important to making this unmixedness theory useful. Corollary 3.2. A Pr¨ ufer domain R is wB-ht-unmixed/wB-unmixed if and only if dim (R) ≤ 1. Proof. One direction is an immediate consequence of the deﬁnition of unmixedness from which it follows that all zero-dimensional rings and all one-dimensional domains are unmixed (see [1]). Now, suppose R is an unmixed Pr¨ ufer domain. By theorem 3.1, R must satisfy PIT. In [8] it was shown that a Pr¨ ufer domain R satisﬁes PIT if and only if dim (R) ≤ 1. Therefore, dim (R) ≤ 1. A more general form of the PIT condition is the GPIT (generalized principal ideal theorem) condition. A ring satisﬁes GPIT if whenever P is a prime ideal which is minimal over a (proper) ideal generated by n elements, ht(P ) ≤ n. Every Noetherian ring satisﬁes GPIT. This fact is often referred to as Krull’s generalized principal ideal theorem. One good source for more information on these conditions is the paper by Anderson, Dobbs, Eakin and Heinzer [9]. Our next theorem demonstrates

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that within the class of rings which satisfy GPIT the concepts of wB-ht-unmixed and wB-unmixed are equivalent. Theorem 3.3. If R satisﬁes GPIT, then R is wB-ht-unmixed if and only if R is wB-unmixed. Proof. Suppose R is a ring which satisﬁes GPIT. ( =⇒ ) Suppose R is wB-ht-unmixed and let I be a height-generated ideal in R. In a ring which satisﬁes GPIT every ideal I satisﬁes ht(I) ≤ µ (I) where µ (I) denotes the minimal number of generators of I. Thus, in a ring with GPIT, I is height-generated if and only if ht(I) = µ (I) < ∞. To show that I is wB-unmixed, suppose P, Q ∈ Assf (I) with P ⊆ Q. Since R is wB-ht-unmixed we have that I is wB-ht-unmixed. Thus, ht(P ) =ht(Q) = ht(I) < ∞. So, P and Q are prime ideals with P ⊆ Q and ht(P ) =ht(Q) < ∞. Thus, P = Q and so I is wB-unmixed. Therefore, R is wB-unmixed. (⇐=) Suppose R is wB-unmixed and let I be a height-generated ideal in R. As in the ﬁrst part of this proof, we have ht(I) = µ (I) < ∞. Let n =ht(I) = µ (I) . Note that, since R satisﬁes GPIT, we have ht(P ) ≤ n for all P ∈ Min(I) . However, since ht(I) ≤ ht(P ) for all P ∈Min(I) and ht(I) = n, we have ht(P ) = n for all P ∈ Min(I) . Since I is wB-unmixed, we have Assf (I) = Min(I) . Therefore, ht(P ) = n for all P ∈ Assf (I) and thus, I is wB-ht-unmixed. Therefore, R is wB-ht-unmixed. The following result and corollary follow from applying this theorem, and the fact that the condition GPIT localizes, to a result from [1, Theorem 3]. Theorem 3.4. Let R be a ring with GPIT. If RM is a wB-ht-unmixed for every maximal ideal M in R, then R is wB-ht-unmixed. Corollary 3.5. If R is a ring such that RM is a wB-ht-unmixed ring satisfying GPIT (so RM is also wB-unmixed by theorem 3.3) for every maximal ideal M in R, then R is wB-ht-unmixed (and wB-unmixed). Using this corollary we can now show that every locally Cohen-Macaulay ring is both wB-ht-unmixed and wB-unmixed. A ring R is said to be locally Noetherian if RM is Noetherian for every maximal ideal M in R. Locally Noetherian rings were studied extensively by Heinzer and Ohm [10]. Since GPIT localizes and Noetherian rings satisfy GPIT, we have that every locally Noetherian ring satisﬁes GPIT. Example 2.2 from [10] gives an example of a non-Noetherian locally Noetherian domain D whose localizations DP are all Noetherian valuation domains. Noetherian valuation domains are Cohen-Macaulay; so this is an example of a non-Noetherian

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ring which is locally Cohen-Macaulay where locally Cohen-Macaulay is deﬁned analogously to locally Noetherian. Corollary 3.6. Every locally Cohen-Macaulay ring is wB-ht-unmixed. Proof. Suppose R is locally Cohen-Macaulay. Then RM is Cohen-Macaulay for every maximal ideal M in R. Cohen-Macaulay rings are Noetherian; therefore each localization RM is Noetherian and, therefore, satisﬁes GPIT. Cohen-Macaulay rings are wB-ht-unmixed (see [2]). So each localization RM is wB-ht-unmixed. Apply corollary 3.5 and we have that R is wB-ht-unmixed. Note that locally Cohen-Macaulay rings are also wB-unmixed (by theorem 3.3) since they satisfy GPIT. Applying this corollary to the example from [10] cited above we see that the example is wB-ht-unmixed. In this case, however, this corollary was unnecessary since the ring D given in the example is a one-dimensional domain and it can easily be shown that, in general, one-dimensional domains are wB-htunmixed.

4

Unmixedness of the Ring k [x1 , x2, . . .]

We would like to have the non-Noetherian ring k [X1 , X2 , . . .] on our list of nonNoetherian Cohen-Macaulay rings. One reason for this is that, in his work that led to the theory of Cohen-Macaulay rings, Macaulay was studying polynomial rings (in ﬁnitely many variables) over a ﬁeld. In [1] it was shown that this ring is wB-unmixed by showing that R [X1 , X2 , . . .] is wB-unmixed for every Cohen-Macaulay domain R. In the proof of this fact (see [1, theorem 4]), the only place where the fact that R is a domain was used was in part (2) of lemma 6. Lemma 4.1 below shows that the assumption that R is a domain was unnecessary. Therefore, we have R [X1 , X2 , . . .] is wB-unmixed for every Cohen-Macaulay ring R. Lemma 4.1. Let R be a Noetherian ring and let S = R [X1 , X2 , . . .] . For any prime ideal P in R we have ht(P ) = ht(P S) where ht(P S) refers to the height of the ideal P S in S. Note that the proof of this lemma depends only on the weaker condition that R is a strong S-ring (see [6] for more information on strong S-rings). It is not necessary for the ring to be Noetherian. Proof. First note that we have trivially that ht(P ) ≤ht(P S) since the extensions of a chain of distinct prime ideals in R is a chain of distinct prime ideals in S. For i ≥ 1, let Ri = R [X1 , X2 , . . . , Xi ] so S = lim Ri . Let Pi = P Ri . Since R −→

is Noetherian (and thus a strong S-ring) we have ht(P ) = ht(Pi ) [6, theorem 149, page 108].

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Now, suppose ht(P S) > n where n = ht(P ) . Then there is a chain of prime ideals Q0 ⊂ Q1 ⊂ · · · ⊂ Qn+1 = P S in S. For 1 ≤ i ≤ n + 1, choose xi ∈ Qi \ Qi−1 . Since S = lim Ri , there is a positive integer j such that {x1 , x2 , . . . , xn+1 } ∈ Rj . For 0 ≤ i ≤ n + 1, let Ti = Qi ∩ Rj . Then

−→

T0 ⊂ T1 ⊂ · · · ⊂ Tn+1 is a chain of prime ideals in Rj . So, ht(Tn+1 ) ≥ n+1. However, Tn+1 = Qn+1 ∩Rj = P S∩Rj = Pj and we have already noted that ht(Pj ) = n so we have a contradiction. Therefore, ht(P S) = ht(P ) . In [8, Proposition 6.4] it was shown that R [X1 , X2 , . . .] satisﬁes GPIT (if R is a Noetherian ring). The statement of this fact in [8] actually makes the assumption that R is a domain; however, the fact that R is a domain, is not necessary in the proof given in [8], so we will use the more general result. By applying theorem 3.3 to this result, we get the following theorem. Theorem 4.2. Let R be a Cohen-Macaulay ring. Then R [X1 , X2 , . . .] is wB-htunmixed. The desired result regarding the ring k [X1 , X2 , . . .], where k is a ﬁeld, follows as an easy corollary to this since every ﬁeld is a Cohen-Macaulay ring. Corollary 4.3. For any ﬁeld k, the ring k [X1 , X2 , . . .] is a wB-ht-unmixed ring.

5

Unmixedness and Direct Sums

An interesting result about wB-ht-unmixedness (resp. wB-unmixedness) is that wB-ht-unmixed (resp. wB-unmixed) rings which can be written as a direct sum will have wB-ht-unmixed (resp. wB-unmixed) components. Theorem 5.1. Let R be a wB-ht-unmixed (resp. wB-unmixed) ring and assume that R is the direct sum of two rings A and B, so R = A ⊕ B. Then, both A and B are wB-ht-unmixed (resp. wB-unmixed) rings. To prove this theorem we will need some preliminary lemmas. Lemma 5.2. Let A and B be commutative rings and let R = A ⊕ B. Then, for any ideal I in A we have ht (J) = ht (I) where J = I ⊕ B. Lemma 5.2 follows immediately from the fact that any prime ideal in R is of the form P ⊕ B for some prime ideal P in A or of the form A ⊕ Q for some prime ideal Q in B.

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Lemma 5.3. Let A and B be commutative rings and let R = A ⊕ B. Then, for any ideal I in A and any P ∈ Assf (I) , the prime ideal Q = P ⊕ B is in Assf (J) where J = I ⊕ B. Proof. Suppose I is an ideal in A and P ∈ Assf (I) . Let Q = P ⊕ B and J = I ⊕ B. Since P ∈ Assf (I) there is some element a ∈ R such that P ∈ Min(I : a) . We will show that Q ∈ Min(J : (a, 1)) , so Q ∈ Assf (J) . To show that Q ∈ Min(J : (a, 1)) we must ﬁrst verify that (J : (a, 1)) ⊆ Q. It is easily veriﬁed that (J : (a, 1)) = (I : a) ⊕ B. So, since (I : a) ⊆ P, we have (J : (a, 1)) = (I : a) ⊕ B ⊆ P ⊕ B = Q. Now, suppose there is a prime ideal T in R such that (J : (a, 1)) ⊆ T ⊆ Q. Then (I : a) ⊕ B ⊆ T ⊆ Q. Any prime ideal in R is of the form S ⊕ B for some prime ideal S in A or of the form A ⊕ S for some prime ideal S in B. Since T is a prime ideal and (I : a) ⊕ B ⊆ T, we have T = S ⊕ B for some prime ideal S in A. Note that (I : a) ⊆ S. Also, since S ⊕ B = T ⊆ Q = P ⊕ B we have S ⊆ P. Therefore, S = P since P is minimal among prime ideals containing (I : a) . Thus, T = S ⊕ B = P ⊕ B = Q, so Q ∈ Min(J : (a, 1)) and, therefore, Q ∈ Assf (I) . Now, we are ready to prove theorem 5.1. Proof. Clearly it is suﬃcient to prove that if R = A ⊕ B is wB-ht-unmixed (resp. wB-unmixed), then A is wB-ht-unmixed (resp. wB-unmixed). Let I be a height-generated ideal in A. Then I = (a1 , a2 , . . . , an ) for some a1 , a2 , . . . , an ∈ A where ht(I) ≥ n. Let J = I ⊕ B. Then J is generated in R by the elements (a1 , 1) , (a2 , 1) , . . . , (an , 1) . Also, by lemma 5.2, ht(J) = ht(I) . Therefore, J is height-generated. Let P, Q ∈ Assf (I) with P ⊆ Q. Then, by lemma 5.3, P ⊕ B, Q ⊕ B ∈ Assf (J) . (Also, of course, P ⊕ B ⊆ Q ⊕ B.) • If we are assuming that R is wB-ht-unmixed, then, since J is height-generated in R and since P ⊕ B ∈ Assf (J) , we have ht(P ⊕ B) = ht(J) . By lemma 5.2, ht(P ) = ht(P ⊕ B) . Therefore, ht(P ) = ht(P ⊕ B) = ht(J) = ht(I) . So, A is wB-ht-unmixed. • On the other hand, if we are assuming that R is wB-unmixed, then, since J is height-generated in R and P ⊕ B, Q ⊕ B ∈ Assf (J) with P ⊕ B ⊆ Q ⊕ B, we have P ⊕ B = Q ⊕ B. Therefore, P = Q. So, A is wB-unmixed.

The converse of this theorem (theorem 5.1) is true if the ring R is assumed to satisfy GPIT. It is easily seen that for R = A ⊕ B, we have R satisﬁes GPIT if and only if A and B satisfy GPIT. Thus, it is suﬃcient to assume that R satisﬁes GPIT or that A and B satisfy GPIT to get the converse. Also, since R, A and B all satisfy GPIT in this theorem, wB-ht-unmixed is equivalent to wB-unmixed, so we will not have to consider two cases as we did in the proof of theorem 5.1 above.

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Theorem 5.4. Suppose R is the direct sum of two rings A and B, so R = A ⊕ B. Assume R satisﬁes GPIT (or equivalently that both A and B satisfy GPIT). Then R is wB-ht-unmixed (resp. wB-unmixed) if and only if A and B are both wB-htunmixed (resp. wB-unmixed). Proof. One direction of this theorem follows immediately from theorem 5.1. Also, as was mentioned above, it is suﬃcient to prove the wB-ht-unmixed version of this theorem. We need to show that if A and B are wB-ht-unmixed rings, then R = A ⊕ B is wB-ht-unmixed. Suppose A and B are wB-ht-unmixed and that R = A ⊕ B satisﬁes GPIT (so A and B also satisfy GPIT). Throughout the following we will make use of the following notation. If J is an ideal in R, we deﬁne JA = {a ∈ A : (a, b) ∈ J for some b ∈ B} and JB = {b ∈ B : (a, b) ∈ J for some a ∈ A}. It is easy to see that J ⊆ JA ⊕ JB and, in fact, J ⊆ S ⊕ T (where S is an ideal in A and T is an ideal in B) if and only if JA ⊆ S and JB ⊆ T. Let I be a height-generated ideal in R of height n. Then I = ((a1 , b1 ) , (a2 , b2 ) , . . . , (an , bn )) for some a1 , a2 , . . . , an ∈ A and some b1 , b2 , . . . , bn ∈ B. It follows easily that IA = (a1 , a2 , . . . , an ) and IB = (b1 , b2 , . . . , bn ) . Recall that every prime ideal in R is of the form P ⊕ B or A ⊕ Q where P ∈ Spec(A) and Q ∈ Spec(B) , so n = =

ht (I) = min {ht (T ) : T ∈ Spec (R) and I ⊆ T } min {min {ht (P ⊕ B) : P ∈ Spec (A) and I ⊆ P ⊕ B} , min{ht (A ⊕ Q) : Q ∈ Spec (B) and I ⊆ A ⊕ Q}}.

It is easily seen (see lemma 5.2) that, for P ∈ Spec(A) and Q ∈ Spec(B), we have ht(P ⊕ B) = ht(P ) and ht(A ⊕ Q) = ht(Q) . Also, I ⊆ P ⊕ B if and only if IA ⊆ P and I ⊆ A ⊕ Q if and only if IB ⊆ Q. Therefore, n = =

min {min {ht (P ) : P ∈ Spec (A) and IA ⊆ P } , min{ht (Q) : Q ∈ Spec (B) and IB ⊆ Q}} min {ht (IA ) , ht (IB )} .

This gives us that n ≤ ht(IA ) and n ≤ ht(IB ) . However, IA and IB are both ideals which can be generated by n elements in the GPIT satisfying rings A and B respectively. So, ht(IA ) ≤ n and ht(IB ) ≤ n. Therefore, IA and IB are heightgenerated ideals in their respective rings of height n. Thus, since A and B are both assumed to be wB-ht-unmixed rings we have ht(P ) = n for all P ∈ Assf (IA ) and ht(Q) = n for all Q ∈ Assf (IB ). Suppose T ∈ Assf (I) . We need to show ht(T ) = n. Since T is a prime ideal T = P ⊕ B for some P ∈ Spec(A) or T = A ⊕ Q for some Q ∈ Spec(B) . Without loss of generality, suppose T = P ⊕ B for some P ∈

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Spec(A) . Since T ∈ Assf (I) , we have T ∈ Min(I : (a, b)) for some (a, b) ∈ R. So, P ⊕ B ∈ Min(I : (a, b)) . This will be true if and only if P ∈ Min(I : (a, b))A . Claim: (I : (a, b))A = (IA : a) • Note that (I : (a, b)) = ⊆ = =

{(x, y) : (ax, by) ∈ I} {(x, y) : ax ∈ IA and by ∈ IB } {(x, y) : x ∈ (IA : a) and y ∈ (IB : b)} (IA : a) ⊕ (IB : b) .

Therefore, (I : (a, b))A ⊆ (IA : a) . • Also note that x ∈ (IA : a)

=⇒ =⇒ =⇒ =⇒ =⇒ =⇒

ax ∈ IA (ax, y) ∈ I for some y ∈ B (ax, y) (1, b) ∈ I for some y ∈ B (ax, by) ∈ I for some y ∈ B (x, y) ∈ (I : (a, b)) for some y ∈ B x ∈ (I : (a, b))A .

Therefore (IA : a) ⊆ (I : (a, b))A . Thus, the claim has been proven. So, P ∈ Min(IA : a) and therefore, P ∈ Assf (IA ) . Thus, ht(P ) = ht(IA ) = n, which give us ht(T ) = ht(P ) = n = ht(I) . Thus, I is wB-ht-unmixed. Therefore, R is wB-ht-unmixed.

Bibliography [8] David F. Anderson, Valentina Barucci, and David E. Dobbs, Coherent Mori domains and the principal ideal theorem, Comm. Algebra 15(6) (1987), 1119– 1156. [9] David F. Anderson, David E. Dobbs, Paul M. Eakin, Jr., and William J. Heinzer, On the generalized principal ideal theorem, Paciﬁc J. Math. 146(2) (1990), 201–215. [7] Robert W. Gilmer, Multiplicative ideal theory, volume 12 of Pure and Applied Mathematics, Marcel Dekker, Inc., 1972. [3] Sarah Glaz, Coherence, regularity and homological dimensions of commutative ﬁxed rings, in Ngˆ o Viˆet Trung, Aron Simis and Giuseppe Valla, editors, Commutative Algebra, Workshop on Commutative Algebra, pages 89–106, World Scientiﬁc, September 1992. [4] Sarah Glaz, Homological dimensions of localizations of polynomial rings, David F. Anderson and David E. Dobbs, editors, Zero-dimensional Commutative Rings, Lecture Notes in Pure and Applied Mathematics, John H. Barrett Memorial Lectures and Conference on Commutative Ring Theory, Marcel Dekker 171 (1994), 209–222. [2] Tracy Dawn Hamilton, Weak Bourbaki height-unmixed rings: Another step towards non-Noetherian Cohen-Macaulayness, submitted for publication. [1] Tracy Dawn Hamilton, Weak Bourbaki unmixed rings: A step towards nonNoetherian Cohen-Macaulayness, Rocky Mountain J. Math. 34(2) (2004). [10] William Heinzer and Jack Ohm, Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158(2) (1971), 273–284. [5] Juana Iroz and David E. Rush, Associated prime ideals in non-Noetherian rings, Canad. J. of Math. XXXVI(2) (1984), 344–360. [6] Irving Kaplansky, Commutative rings, Alyn and Bacon, 1970.

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Chapter 19

A Note on Sets of Lengths of Powers of Elements of Finitely Generated Monoids by

Wolfgang Hassler1 Abstract Let H be a ﬁnitely generated commutative, cancellative monoid. Then every nonunit a ∈ H decomposes (in general in a highly nonunique way) into a product (19.1) a = u1 · . . . · un of irreducible elements (atoms) of H. The integer n is called the length of the factorization (19.1). We call L(a) = {n ∈ N | a has a factorization into n irreducible elements of H} the set of lengths of a. It is known that sets of lengths of elements of H have the following structure: they are, up to bounded initial and ﬁnal segments, a union of arithmetical progressions with bounded distance. We say that two sets of lengths are of the same type if their initial and ﬁnal segments coincide (up to a shift) and if their central parts have the same period. Clearly, the set of equivalence classes with respect to this equivalence relation is ﬁnite. In the present note we examine the structure of sets of lengths of powers of elements of H. We prove a theorem which asserts that there exist constants N, B ∈ N (which only depend on H) such that the following holds: if a ∈ H and k, l ≥ B with k ≡ l mod N , then L(ak ) and L(al ) are of the same type. This result carries over to Krull monoids with ﬁnite divisor class group, e.g. rings of integers of algebraic number ﬁelds. In a weaker form it is true for every Krull monoid.

2000 Mathematics Subject Classification. 11R27, 13A05, 20D60. Key words and phrases. factorization, irreducible element, set of lengths, ﬁnitely generated monoid. 1 This research was supported by a grant from the Fonds zur F¨ orderung der wissenschaftlichen Forschung, project number P16770–N12.

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Introduction and Preliminaries

By a monoid we always mean a (usually multiplicatively written) commutative semigroup H with identity element for which the cancellation law holds (i.e. ab = ac implies b = c for all a, b, c ∈ H). Let H be a monoid. An element u ∈ H\H × (where H × denotes the group of invertible elements of H) is called irreducible (or an atom) if u = ab implies that either a ∈ H × or b ∈ H × for all a, b ∈ H. The monoid H is called atomic if every nonunit a ∈ H possesses a factorization a = u1 · . . . · un

(19.2)

into irreducible elements ui of H. The integer n is called the length of the factorization (19.2). We call L(a) = LH (a) = {n ∈ N | n is length of some factorization of a} the set of lengths of a. During the last years, much eﬀort was made to determine the structure of sets of lengths for certain classes of domains and monoids, e.g. ﬁnitely generated monoids, orders of global ﬁelds, Krull monoids, Congruence monoids, one-dimensional local domains and higher dimensional algebras over perfect ﬁelds. The reader is referred to [1], [5], [6], [12], [14] and [13] for the most recent results in this area. It turned out that sets of lengths in all mentioned classes of rings and monoids (in some cases under additional ﬁniteness assumptions) have the following special structure: they are, up to bounded initial and ﬁnal segments, a union of arithmetical progressions with bounded distance (see Deﬁnition 1.2 and Theorem 1.3 for the detailed statement in the case of ﬁnitely generated monoids). As a consequence of this structure theorem, one can classify sets of lengths by their initial and ﬁnal segment and the period of their central part. Obviously, there are only ﬁnitely many classes (called “types”, see Deﬁnition 2.1). Very little is known about the behavior of the type of sets of lengths, yet. For example, given two elements a and b, what can be said about the type of L(ab) if the type of L(a) and L(b) is known? What kind of additional information on a and b is needed to determine L(ab)? In view of eﬃcient methods for the computation of sets of lengths (and for theoretical reasons) it would be desirable to have “simple” criteria to determine the type of the sets of lengths of (at least certain classes) of elements. In this note we provide such a criterion for powers of elements of a ﬁnitely generated monoid H. Our research was initiated by the following question raised by F. Halter-Koch: let a ∈ H\H × (assuming that H is ﬁnitely generated) and consider the sequence L(an ) of sets of lengths. Do the initial and ﬁnal segments of L(an ) repeat periodically if n grows? We answer this question positively by proving (cf. Theorem 2.2) that there exist positive integers N and B (which only depend on H) such that for every a ∈ H and for all integers k ≥ B and l ≥ B the congruence k ≡ l mod N already implies that L(ak ) and L(al ) are of the same type. This result immediately carries over to Krull monoids with ﬁnite divisor class group, e.g. rings of integers of algebraic number ﬁelds (Corollary 2.3). In a weaker form, the Theorem holds for every Krull monoid (see Lemma 2.4 and Corollary 2.5).

Sets of Lengths of Powers of Elements of Finitely Generated Monoids

295

We denote by N resp. N0 the set of positive resp. nonnegative integers. If r ∈ N we set N≥r = {x ∈ N | x ≥ r} (resp. N≤r = {x ∈ N | x ≤ r}). For a, b ∈ Z we set [a, b] = {x ∈ Z | a ≤ x ≤ b}. For sets A, B ⊂ Z we write A + B = {a + b | a ∈ A, b ∈ B} and if b ∈ Z, we set A + b = A + {b}. Let H be a monoid. Since H satisﬁes the cancellation law, we can form the quotient group of H. We denote it by Q(H). A submonoid H0 ⊂ H is called a saturated submonoid of H if a |H b implies a |H0 b for all a, b ∈ H0 . A monoid H is called reduced if H × = {1}. For a reduced atomic monoid H we denote by A(H) the set of atoms of H. The free monoid F (A(H)) generated by A(H) (which is isomorphic to the coproduct of |A(H)| copies of (N0 , +)) is called the factorization monoid of H. We denote it by Z(H). We have a canonical homomorphism πH : Z(H) → H which sends every (formal) product of atoms in Z(H) to the corresponding product in −1 (a) the set of factorizations of a. If H. For some a ∈ H we denote by Z(a) = πH nq z = q∈A(H) q ∈ Z(H) we set vq (z) = nq for some q ∈ A(H). For y ∈ Z(H) we set Supp(y) = {q ∈ A(H) | vq (y) > 0}. The length function | · | : Z(H) → N0 and the distance function d : Z(H) × Z(H) → N0 (see for instance [3, section 2]) are deﬁned as follows: If x, y, z ∈ Z(H), then |z| = q∈A(H) vq (z) and y x d(x, y) = max , . gcd(x, y) gcd(x, y) Here “gcd” denotes the greatest common divisor of x and y in the free monoid Z(H). Let H ⊂ D be monoids. We say that H is a divisor closed submonoid of D if d |D h implies d ∈ H for all h ∈ H, d ∈ D. For a ∈ D we denote by [[a]]D ⊂ D the smallest divisor closed submonoid of D containing a. It is easy to see that [[a]]D = {d ∈ D | d | ak for some k ∈ N}. Let H ⊂ D be a divisor closed submonoid of some atomic monoid D. Then every factorization of an element h ∈ H into atoms of D is a factorization into atoms of H and vice versa. In particular, H is atomic and LH (h) = LD (h) for every h ∈ H. Definition 1.1. i.) Let T ⊂ Z. Two elements k, l ∈ T are called successive elements of T if k = l and T ∩ [min{k, l}, max{k, l}] = {k, l}. ii.) We call ∆(T ) = {|k − l| | k and l are successive elements of T } ⊂ N the set of diﬀerences of T . (Observe that ∆(T ) = ∅ if and only if |T | ≤ 1.) iii.) Let H be an atomic monoid. We call ∆(H) = ∆(L(a)) ⊂ N a∈H

the set of diﬀerences of H.

296

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The set of diﬀerences measures the size of “gaps” which occur in the sets of lengths of an atomic monoid. Definition 1.2. A nonempty, ﬁnite set L ⊂ Z is called an almost arithmetical multiprogression bounded by M ∈ N, if there exists some d ∈ [1, M ], some subset D ⊂ [0, d − 1] and a decomposition L = L1 ∪ L∗ ∪ L2 such that ∅ = L∗ , L1 ⊂ [−M, −1] + min L∗ , L2 ⊂ max L∗ + [1, M ] and L∗ = [min L∗ , max L∗ ] ∩ (D + dZ). The integer d is called the period length of L. If D is a singleton, then L is called an almost arithmetical progression bounded by M . Theorem 1.3 (Structure Theorem for sets of lengths). Let H be a ﬁnitely generated monoid. Then there exists some M ∈ N such that L(a) is an almost arithmetical multiprogression with bound M for every a ∈ H. Proof. See [5, Proposition 8.1]. For a geometric approach see [9, section 3]. The following remark is in order. Remark 1.4. Let H be a ﬁnitely generated monoid and a ∈ H. Then [[a]]H ⊂ H is ﬁnitely generated, too (cf. [7, Proposition 6.2]). By [3, Proposition 3.3] and [3, Proposition 3.4], every ﬁnitely generated monoid is locally tame and has ﬁnite catenary degree (see [3, Deﬁnition 2.3] and [3, Deﬁnition 3.1] for the relevant deﬁnitions.) From [4, Theorem 5.1] and [4, Corollary 5.2] it then follows that the sets L(an ) are almost arithmetical progressions (rather than multiprogressions) for all suﬃciently large n ∈ N. Furthermore, their period length is min ∆([[a]]H ). Let H be an atomic monoid and a ∈ H\H × . Two integers k and l are called successive lengths of a if k = l and k and l are successive elements of L(a) (cf. Deﬁnition 1.1). A. Foroutan showed only recently (cf. [2, Theorem 3.9]) that for a ﬁnitely generated monoid H there exists some K ∈ N such that for all a ∈ H\H × and for all successive lengths k, l of a the following holds: if z ∈ Z(a) with |z| = k, then there exists some z ∈ Z(a) with |z | = l and d(z, z ) ≤ K. The smallest possible K with this property is called the strong successive distance of H. It is denoted by δ(H).

2

Sets of Lengths of Powers

We begin with the classiﬁcation of sets of lengths by their initial and ﬁnal part and by the period of their central part. Definition 2.1. We say that two almost arithmetical multiprogressions L and L bounded by M are of the same type (with respect to M ) if (L − min L) ∩ [0, 2M ] = (L − min L ) ∩ [0, 2M ] and (L − max L) ∩ [−2M, 0] = (L − max L ) ∩ [−2M, 0]. Note that if two almost arithmetical multiprogressions are of the same type then they diﬀer only by a shift and by the length of their central parts. Our aim is to show the following Theorem.

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297

Theorem 2.2 (Main Theorem). Let H be a monoid for which H/H × is ﬁnitely generated. Let M ∈ N be a constant such that every set of lengths of H is an almost arithmetical multiprogression bounded by M . Then there exist constants N, B ∈ N with the following property: if a ∈ H and m, n ∈ N≥B with m ≡ n mod N , then the sets of lengths L(am ) and L(an ) are of the same type (with respect to M ). This result immediately carries over to Krull monoids with ﬁnite divisor class group (see [10] for a comprehensive treatment of Krull monoids). To see this, we brieﬂy sketch the theory of transfer homomorphisms and block monoids. For details see [1, section 3.1] or [9, section 5]. Let H be a Krull monoid with divisor class group G. Let G0 ⊂ G denote the set of classes containing prime divisors and let B(G0 ) = g ng ∈ F(G0 ) ng g = 0 g∈G0

g∈G0

be the block monoid of G0 . Then there exists a (natural) monoid epimorphism β : H → B(G0 ) such that LH (h) = LB(G0 ) (β(h)) for all h ∈ H. In particular, {LH (h) | h ∈ H} = {LB(G0 ) (b) | b ∈ B(G0 )}. Corollary 2.3. Let H be a Krull monoid with ﬁnite divisor class group. Then there exist constants N, B ∈ N with the following property: if a ∈ H and m, n ∈ N≥B with m ≡ n mod N , then the sets of lengths L(am ) and L(an ) are of the same type. Proof. Let G be the divisor class group of H and let G0 ⊂ G denote the set of classes containing prime divisors. Clearly, the block monoid of G0 is ﬁnitely generated. Thus the assertion follows from Theorem 2.2 and from the theory of transfer homomorphisms described above. Corollary 2.3 in particular holds for the multiplicative monoid D\{0} of some Dedekind domain D with ﬁnite Picard group, e.g. for the ring of integers of some algebraic number ﬁeld. Let H be a monoid. Following [8] we say that H is an SFF-monoid (strong ﬁnite factorization monoid) if [[a]]H /[[a]]× H is a ﬁnitely generated monoid for every a ∈ H. It immediately follows from the deﬁnition (and from the fact that ﬁnitely generated monoids are atomic) that SFF-monoids are atomic. Important examples for this class of monoids are Krull monoids. This seems to be well-known. However, we include a short proof of this fact for sake of completeness. Lemma 2.4. Let H be a Krull monoid. Then H is an SFF-monoid. Proof. Without restriction we may assume that H is reduced. Then H is a saturated submonoid of some free monoid F (cf. for instance [10] or [1]). If a ∈ H, then [[a]]H ⊂ [[a]]F is a saturated submonoid and [[a]]F is ﬁnitely generated (and indeed free). But then [[a]]H is ﬁnitely generated, too (cf. [7, Proposition 6.2]). Let H be an SFF-monoid. Then Theorem 1.3 immediately implies that for every a ∈ H there exists some M (a) ∈ N such that for every n ∈ N the set of lengths

298

Hassler

LH (an ) is an almost arithmetical multiprogression bounded by M (a). This makes it possible to say that the sets of lengths of two powers am and an are of the same type. Theorem 2.2 now implies: Corollary 2.5. Let H be an SFF-monoid. Then for every a ∈ H there exist constants N (a), B(a) ∈ N such that for all m, n ∈ N≥B(a) the congruence m ≡ n mod N (a) implies that L(am ) and L(an ) are of the same type. In order to prove Theorem 2.2, we need two preliminary Lemmas. Lemma 2.6 follows from [11, Lemma 6.6] and [3, Proposition 3.4]. Lemma 2.6. Let H be a ﬁnitely generated monoid and c ∈ H. Then the sets {| min L(cd) − min L(d)| | d ∈ H} and {| max L(cd) − max L(d)| | d ∈ H} are bounded. Definition 2.7. Let H be a reduced atomic monoid. A set S ⊂ A(H) is called universally minimal (resp. universally maximal) if |ξ| = min L(πH (ξ)) (resp. |ξ| = max L(πH (ξ))) holds for every ξ ∈ Z(H) with Supp(ξ) ⊂ S. The following Lemma is crucial for our proof of Theorem 2.2. Lemma 2.8. Let H be a ﬁnitely generated monoid. Then there exist constants N, C ∈ N such that for all a ∈ H and y, z ∈ Z(a) with |y| = min L(a) and |z| = max L(a) there exist factorizations ymin, zmax ∈ Z(aN ) with the following properties: Supp(ymin ) resp. Supp(zmax ) are universally minimal resp. maximal, {q ∈ A(H) | vq (y) ≥ C} ⊂ Supp(ymin ) and {q ∈ A(H) | vq (z) ≥ C} ⊂ Supp(zmax ). Proof. Without restriction we may assume that H is reduced. Set M = {(w, w ) ∈ Z(H) × Z(H) | πH (w) = πH (w ) and |w| < |w |}. A(H) A(H) Since Z(H) ∼ × = N0 , M is partially ordered by the induced partial order of N0 A(H) N0 . By Dickson’s Theorem (see [15, Satz 12]), the set Min(M) of minimal points of M is ﬁnite, say Min(M) = {(w1 , w1 ), . . . , (wr , wr )}. Furthermore, for every (w, w ) ∈ M there exists some i ∈ [1, r] with (wi , wi ) ≤ (w, w ). We proceed with the following construction: we show that there exists some C0 ∈ N such that for all b ∈ H and u1 , u2 ∈ Z(b) with |u1 | = min L(b) and |u2 | = max L(b) there exist u1 , u2 ∈ Z(b2 ) with |u1 | = min L(b2 ), |u2 | = max L(b2 ), d(u21 , u1 ) ≤ C0 and d(u22 , u2 ) ≤ C0 . Put

S = max{vq (wi ), vq (wi ) | i ∈ [1, r] , q ∈ A(H)}. Let b ∈ H and u1 , u2 ∈ Z(b) with |u1 | = min L(b) and |u2 | = max L(b). Set Ji = {q ∈ A(H) | vq (ui ) > S} and deﬁne ξi = q vq (ui ) and ωi = q vq (ui ) q∈Ji

q∈A(H)\Ji

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299

for i ∈ [1, 2]. Set ci = πH (ξi ) and di = πH (ωi ). We claim that |ξ12 | = min L(c21 ) and |ξ22 | = max L(c22 ).

(19.3)

Assume on the contrary that there exists some σ ∈ Z(c21 ) whose length is strictly smaller than |ξ12 |. Then (σ, ξ12 ) ∈ M. Hence there exists some i ∈ [1, r] such that (wi , wi ) ≤ (σ, ξ12 ). Thus wi |Z(H) ξ12 . Since vq (wi ) ≤ S < vq (ξ1 ) for all q ∈ J1 , we infer that wi |Z(H) ξ1 . This shows that min L(c1 ) < |ξ1 |. But this is a contradiction since we assumed that u1 = ξ1 ω1 is the shortest factorization of b. The equation |ξ22 | = max L(c22 ) is proved by the same arguments. Now let D ∈ N by Lemma 2.6 be an upper bound for the sets {| min L(gh) − min L(h)| | h ∈ H} and {| max L(gh) − max L(h)| | h ∈ H}, where g ∈ {πH (z) | z ∈ Z(H) , vq (z) ≤ 2S for all q ∈ A(H)}. Then | min L(c21 d21 ) − |ξ12 || ≤ D and | max L(c22 d22 ) − |ξ22 || ≤ D. With b2 = c21 d21 we thus get | min L(b2 ) − |u21 || = | min L(b2 ) − |ξ12 | − |ω12 || ≤ D + |ω12 | ≤ D + 2S|A(H)|. Similarly, we get | max L(b2 ) − |u22 || ≤ D + 2S|A(H)|. Since the strong successive distance δ(H) of H is ﬁnite, there exists some C0 ∈ N such that for all h ∈ H, u ∈ Z(h) and l ∈ L(h) with ||u| − l| ≤ D + 2S|A(H)| there exists some factorization u ∈ Z(h) with |u | = l and d(u, u ) ≤ C0 . Thus there exist factorizations u1 , u2 ∈ Z(b2 ) such that |u1 | = min L(b2 ), |u2 | = max L(b2 ), d(u21 , u1 ) ≤ C0 and d(u22 , u2 ) ≤ C0 . Now we get to the core of the proof. Put C = max{C0 + 1, S} and let N = lcm {2γ − 2δ | 0 ≤ δ < γ ≤ (C + 1)|A(H)| } ∈ N. In the following we prove only the existence of a factorization ymin with the required properties. The existence of zmax is shown by completely analogous arguments. Let a ∈ H and y ∈ Z(H) with |y| = min L(a). We construct inductively a i i sequence (vi )i∈N0 of factorizations vi ∈ Z(a2 ) with |vi | = min L(a2 ) as follows: put v0 = y and assume that v0 , . . . , vi are already constructed. By the above i+1 i+1 construction, let vi+1 ∈ Z(a2 ) be a factorization with |vi+1 | = min L(a2 ) and d(vi2 , vi+1 ) ≤ C0 . The next step is to deﬁne a map ϕ : Z(H) −→ ([0, C − 1] ∪ {∞})A(H) by setting

ϕ(z)q =

vq (z) if vq (z) < C ∞ if vq (z) ≥ C

for z ∈ Z(H) and q ∈ A(H). Then there exist α, β ∈ [0, (C + 1)|A(H)| ] with α < β N and ϕ(vα ) = ϕ(vβ ). Put N0 = 2β −2α and deﬁne ymin = (vβ vα−1 ) N0 ∈ Q(Z(H)). We claim that ymin ∈ Z(aN ) and that ymin has the required properties of the Lemma. Suppose that ϕ(vi )q = ∞ for i ∈ N0 and q ∈ A(H). Since d(vi2 , vi+1 ) ≤ C0 , we get ||vi+1 | − 2|vi || ≤ C0 and vq (vi+1 ) − vq (vi ) > vq (vi+1 ) − 2vq (vi ) + C0 ≥ −C0 + C0 = 0. Thus vq (vβ vα−1 ) > 0. Therefore, ymin ∈ Z(H) and {q ∈ A(H) | vq (y) ≥ C} ⊂ β Supp(ymin ). Since vβ is a shortest factorization of a2 , the same argument as in the proof of (19.3) shows that every σ ∈ Z(H) with Supp(σ) ⊂ {q ∈ A(H) | vq (vβ ) ≥ S}

300

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is a shortest factorization of πH (σ). Hence Supp(ymin ) is universally minimal. This completes the proof. We are now prepared to prove our main result. Proof of the Main Theorem. By passing to H/H × we can assume without loss of generality that H is reduced. Let N and C be as in Lemma 2.8. Let D ∈ N by Lemma 2.6 be a constant such that the inequalities | min L(cd) − min L(d)| ≤ D and | max L(cd) − max L(d)| ≤ D hold for all d ∈ H and c ∈ {πH (z) | z ∈ Z(H), vq (z) < N C for all q ∈ A(H)}. Put E = max{D + N C|A(H)|, 2M }, where M is a bound in the Structure Theorem for sets of lengths of H. Since the strong successive distance δ(H) of H is ﬁnite, there exists some B0 ∈ N such that for all h ∈ H, u ∈ Z(h) and l ∈ L(h) with ||u| − l| ≤ E there exists some factorization u ∈ Z(h) with |u | = l and d(u, u ) ≤ B0 . We claim that N and B = N (B0 + 1) satisfy the assertion of the Theorem. Thus our aim is to show that m (19.4) L(a ) − min L(am ) ∩ [0, 2M ] = L(an ) − min L(an ) ∩ [0, 2M ] and m L(a ) − max L(am ) ∩ [−2M, 0] = L(an ) − max L(an ) ∩ [−2M, 0]

(19.5)

hold for all a ∈ H and m, n ∈ N≥B with m ≡ n mod N . Let a ∈ H\{1} and m, n ∈ N≥B with m < n and m ≡ n mod N . We ﬁrst prove equation (19.4). Let y ∈ Z(a) with |y| = min L(a) be arbitrary and let p y k ∈ Z(am ) and k ∈ [0, N − 1] with m = pN + k and n = qN + k. Put y1 = ymin q k n N y2 = ymin y ∈ Z(a ), where ymin ∈ Z(a ) satisﬁes the conditions in Lemma 2.8. Let I = {q ∈ A(H) | vq (y) ≥ C} and set q vq (y) and ω = q vq (y) . ξ = q∈I

q∈A(H)\I

s s Then Supp(ξ) ⊂ Supp(ymin ), whence |ξ r ymin | = min L(πH (ξ r ymin )) for all r, s ∈ N0 . p m k p m k |−min L(am )||+ Thus we get |y1 |−min L(a ) = |ξ ymin |−min L(a )+|ω | ≤ ||ξ k ymin k k |ω | ≤ D + |ω | ≤ D + N C|A(H)| ≤ E. Similarly, we obtain |y2 | − min L(an ) ≤ E. To show (19.4) it is hence enough to prove that

|y2 | + t ∈ L(an ) ⇐⇒ |y1 | + t ∈ L(am )

(19.6)

for every t ∈ [−E, E]. Let t ∈ [−E, E] with |y2 | + t ∈ L(an ). Then there exists some y2 | = |y2 | + t and d(y2 , y˜2 ) ≤ B0 . Set y˜1 = y˜2 y1 y2−1 ∈ Q(Z(H)). y˜2 ∈ Z(an ) with |˜ We claim that y˜1 ∈ Z(H). Then y˜1 ∈ Z(am ) and |˜ y1 | = |y1 | − |y2 | + |˜ y2 | = |y1 | − |y2 | + |y2 | + t = |y1 | + t. −u y˜2 , where u = q − p. From the inequality m ≥ B we We have y˜1 = ymin get the estimate p ≥ B0 . Since d(y2 , y˜2 ) ≤ B0 , we have |w| ≤ B0 , where w =

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301

q−B0 y2 gcd(y2 , y˜2 )−1 . Thus ymin divides y2 w−1 = gcd(y2 , y˜2 ). Now u = q − p ≤ q − B0 implies that q−B0 u | ymin | gcd(y2 , y˜2 ) | y˜2 . ymin

The other implication in (19.6) and equation (19.5) are proved by analogous arguments.

Bibliography [1] S. Chapman and A. Geroldinger, Krull Domains and Monoids, Their Sets of Lengths, and Associated Combinatorial Problems, D.D. Anderson, editor, Factorization in Integral Domains, Marcel Dekker 189 (1997), 73–112. [2] A. Foroutan, Monotone chains of factorizations, International Journal of Commutative Rings. [3] A. Geroldinger, The Catenary Degree and Tameness of Factorizations in Weakly Krull Domains, D.D. Anderson, editor, Factorization in Integral Domains, Marcel Dekker 189 (1997), 113–153. [4] A. Geroldinger, Chains of Factorizations and Sets of Lengths, J. Algebra 188 (1997), 331–362. [5] A. Geroldinger, A structure theorem for sets of lengths, Colloq. Math. 78 (1998), 225–259. [6] A. Geroldinger and F. Halter-Koch, Congruence monoids, Acta Arith., to appear. [7] A. Geroldinger and F. Halter-Koch, Arithmetical theory of monoid homomorphisms, Semigroup Forum 48 (1994), 333–362. [8] F. Halter-Koch, On the asymptotic behaviour of the number of distinct factorizations into irreducibles, Ark. Mat. 31 (1993), 297–305. [9] F. Halter-Koch, Finitely Generated Monoids, Finitely Primary Monoids, and Factorization Properties of Integral Domains, D.D. Anderson, editor, Factorization in Integral Domains, Marcel Dekker 189 (1997), 31–72. [10] F. Halter-Koch, Ideal Systems, Marcel Dekker, 1998. [11] W. Hassler, Properties of factorizations with successive lengths in onedimensional local domains, International Journal of Commutative Rings, to appear. [12] W. Hassler, Arithmetical properties of one-dimensional, analytically ramiﬁed local domains, J. Algebra 250 (2002), 517–532. 302

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[13] W. Hassler, Factorization properties of Krull monoids with inﬁnite class group, Colloq. Math. 92 (2002), 229–242. [14] W. Hassler, Factorization in ﬁnitely generated domains, J. Pure Appl. Algebra 186 (2004), 151–168. [15] L. R´edei, Theorie der endlich erzeugbaren kommutativen Halbgruppen, PhysikaVerlag, W¨ urzburg, 1963.

Chapter 20

UMV-Domains by

Evan Houston and Muhammad Zafrullah Abstract We call a domain R a UMV-domain if each nonzero prime ideal in the polynomial ring R[X] satisfying P ∩ R = 0 is a maximal divisorial ideal of R[X]. We characterize v-domains as integrally closed UMV-domains, and we provide characterizations of UMV-domains similar to those known for UMT-domains.

1

Introduction

Let R be an integral domain with quotient ﬁeld K. If f ∈ K[X] is an irreducible polynomial, then we call the prime ideal P = f K[X]∩R[X] an upper to zero. Uppers to zero have been used by many authors to characterize ring-theoretic properties. For example, it follows from [6, Theorem 19.15] that a domain R is a Pr¨ ufer domain if and only if R is integrally closed and P M R[X] for each upper to zero P in R[X] and each maximal ideal M of R. A corresponding result exists for Pr¨ ufer v-multiplication domains (PVMDs). A domain R is a PVMD if and only if R is integrally closed and each upper to zero in R[X] is a maximal t-ideal of R[X] [10, Proposition 2.6]. This led the authors of the present chapter to isolate this condition on uppers to zero by deﬁning UMT-domains to be those domains in which each upper to zero is a maximal t-ideal [12]. UMT-domains were further studied in [3] and [4]. The purpose of this chapter is to study UMV-domains, domains with the property that each upper to zero is a maximal v-ideal, that is, a maximal divisorial ideal. In Section 1 we study uppers to zero. We then apply these results in Section 2 to our study of UMV-domains. We show that a domain R is a UMV-domain if and only if each upper to zero P satisﬁes P −1 = R[X] and c(P )v = R (where c(P ) denotes the ideal of R generated by the coeﬃcients of the polynomials in P ). We characterize v-domains as integrally closed UMV-domains. We also point out where the similarities with UMT-domains fail to hold; to some extent the problem is that, 304

UMV-Domains

305

while maximal t-ideals are plentiful (recall that for any domain R we have R = {RM | M is a maximal t-ideal of R}), a domain may have no maximal divisorial ideals. In a brief third section, we pose several questions related to UMV-domains.

2

Uppers to Zero

We begin by reviewing terminology. Let R be a domain with quotient ﬁeld K. For fractional ideals I, J of R, (I : J) = {x ∈ K | xJ ⊆ I}, and (I :R J) = {x ∈ R | xJ ⊆ I}. We write I −1 for (R : I), and Iv for (I −1 )−1 . Finally, It = Av , where the union is taken over all ﬁnitely generated subideals A of I. Most of the facts about the v- and t-operations which we shall use can be found in [6, Section 32]. Definition 2.1. Let I = f K[X] ∩ R[X], where f is a nonconstant polynomial in K[X]. Then I is said to be almost principal if there is a nonzero element a ∈ R such that aI ⊆ f R[X]. Lemma 2.2. Let I = f K[X]∩R[X], where f is a nonconstant polynomial in K[X]. Then I −1 ∩ K[X] = (I : I). Proof. [7, Lemma 1.14]. Proposition 2.3. Let P = f K[X] ∩ R[X] with f irreducible in K[X]. Then the following statements are equivalent. (1 ) P is almost principal. (2 ) P −1 K[X]. (3 ) P −1 = (P : P ). (4 ) There is an element g ∈ R[X] \ P with gP ⊆ f R[X]. (5 ) P = (f :R[X] a) for some a ∈ R. (6 ) P = (R[X] :R[X] ψ) for some ψ ∈ K(X). Proof. The equivalence of the ﬁrst four conditions is established in [7, Proposition 1.15]. Assume (1). Then we have aP ⊆ f R[X] for some 0 = a ∈ R. Thus P ⊆ (f :R[X] a). On the other hand, if h ∈ R[X] satisﬁes ha ∈ f R[X], then ha ∈ P ⊆ f K[X]; hence h ∈ f K[X] ∩ R[X] = P . Hence (1) ⇒ (5). (5) ⇒ (6) follows from the fact that (f :R[X] a) = (R[X] :R[X] af −1 ). Finally, assume (6). If ψ ∈ K[X], then there is some element a ∈ R with aψ ∈ R[X], whence a ∈ P , contradicting that P ∩ R = 0. Thus ψ = g/f for some g ∈ R[X] \ f K[X]. It follows that g ∈ / P , and we have gP ⊆ f R[X], proving (4). The following result is the key to our characterization of UMV-domains in Theorem 3.2 below. Lemma 2.4. Let I be an ideal of R[X], and assume that I −1 = R[X] and that I ∩ R = 0. Then c(I)v = R.

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Proof. We ﬁrst observe that I −1 ⊆ K[X], since for any nonzero element a ∈ I ∩ R, we have aI −1 ⊆ R[X] ⊆ K[X]. Now let ψ ∈ I −1 \ R[X], and assume that ψ has minimal degree among elements of I −1 \ R[X]. We shall show that, in fact, ψ has degree zero. We shall then have I ⊆ (R[X] :R[X] ψ) = (R :R ψ)R[X]. It will then follow that c(I) ⊆ (R :R ψ); since (R :R ψ) is divisorial, this will imply that c(I)v = R. To see that ψ must have degree zero, write ψ = un X n + · · · + u0 . We claim that (R :R un ) ⊆ (R :R u0 ) ∩ · · · ∩ (R :R un−1 ). If not, pick b ∈ R with bun ∈ R but / R for some i < n. Then ψ = bψ − bun X n is an element of I −1 \ R[X] of bui ∈ degree smaller than that of ψ, a contradiction. Thus the claim is true. In particular, / R. Now let g ∈ (R[X] :R[X] ψ), g = am X m + · · · + a0 . Then am un ∈ R, un ∈ whence by the claim, am ui ∈ R for each i. That is, am ∈ (R[X] :R[X] ψ). It then follows that g − am X m ∈ (R[X] :R[X] ψ). By induction on the degree of g, we obtain ai ∈ (R[X] :R[X] ψ) for each i. In particular, ai un ∈ R for each i, i.e., g ∈ (R :R un )R[X]. Hence I ⊆ (R[X] :R[X] ψ) ⊆ (R :R un )R[X], and un ∈ I −1 \ R[X]. Therefore, ψ = un has degree zero, as desired. Proposition 2.5. Let R be a domain with quotient ﬁeld K, and let P = f K[X] ∩ R[X] with f irreducible in K[X]. Consider the following conditions on P . (1 ) P is maximal divisorial. (2 ) P is v-invertible. (3 ) (P : P ) = R[X]. (4 ) P −1 ∩ K = R. (5 ) c(P )v = R. (6 ) P is almost principal. (7 ) P is divisorial. Then (1) ⇒ (2) ⇔ (3) ⇒ (4) ⇔ (5), and (1) ⇒ (6) ⇒ (7). Proof. The implication (1) ⇒ (2) is proved in [5, Proposition 2.5]. That (2) ⇒ (3) is well-known and is true more generally. Indeed, let I be a v-invertible ideal of a domain R. Then (I : I) ⊆ (II −1 : II −1 ) ⊆ ((II −1 )v : (II −1 )v ) = R (by v-invertibility of I). To show that (3) ⇒ (4), let u ∈ P −1 ∩ K. By Lemma 2.2, u ∈ (P : P ) = R[X]. Hence u ∈ R[X] ∩ K = R, as desired. The equivalence (4) ⇔ (5) is straightforward. Note that (6) ⇒ (7) follows from Proposition 2.3. (3) ⇒ (2): We may assume that P −1 = R[X]. If P is not divisorial, then Pv ∩ R = 0 (since Pv P ), and (Pv )−1 = P −1 = R[X]. By Lemma 2.4, c(P )v ⊆ c(Pv )v R. This contradicts (5). Hence P is divisorial. Now let ψ ∈ (P P −1 )−1 , so that ψP P −1 ⊆ R[X]. Then ψP ⊆ Pv = P , and ψ ∈ (P : P ) = R[X]. Thus (P P −1 )−1 = R[X], and P is v-invertible. (1) ⇒ (6): By [5, Proposition 2.1], P = (R[X] :R[X] ψ) for some ψ ∈ K(X) \ R[X], and P is almost principal by Proposition 2.3.

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We do not know whether (4) ⇒ (3) or (7) ⇒ (6) in Proposition 2.5. However, other than these two possibilities, no implications other than those given exist. That condition (6) does not imply (5) can be illustrated with any Noetherian domain R having a divisorial prime P of height ≥ 2. This follows from the facts that P R[X] necessarily contains an upper to zero and that uppers to zero in Noetherian domains are automatically almost principal [13, Proposition 3.3]. That condition (2) does not imply (7) is illustrated by any domain R allowing an upper to zero P with P −1 = R[X]. We produce such an example at the end of this section. It is useful to observe that much of Proposition 2.5 holds for “nonprime uppers to zero”, that is without the assumption that f is irreducible. Proposition 2.6. Let R be a domain with quotient ﬁeld K, and let I = f K[X] ∩ R[X], where f is a nonconstant polynomial in K[X]. Consider the following conditions on I. (2 ) I is v-invertible. (3 ) (I : I) = R[X]. (4 ) I −1 ∩ K = R. (5 ) c(I)v = R. (6 ) I is almost principal. (7 ) I is divisorial. Then (2) ⇔ (3) ⇒ (4) ⇔ (5), and (6) ⇒ (7). Proof. The only implication requiring proof is (6) ⇒ (7). Assume (6), and choose a nonzero element a ∈ R with aI ⊆ f R[X]. Then aIv ⊆ f R[X] ⊆ f K[X], whence Iv ⊆ f K[X]. Hence Iv ⊆ f K[X] ∩ R[X] = I. Let P be an upper to zero which satisﬁes (P : P ) = R[X] and P −1 = R[X]. Observe that the proof of (3) ⇒ (2) of Proposition 2.5 shows that P is divisorial. Also, since (P : P ) = R[X], P is v-invertible. It then follows from [5, Proposition 2.5] that P is maximal divisorial. That is, we have the following result. Proposition 2.7. With the notation of Proposition 2.5, assume that P −1 = R[X]. Then conditions (1), (2), and (3) are equivalent. Proposition 2.8. Assume that R is integrally closed. Then (a) with the notation of Proposition 2.5, conditions (1)-(5) are equivalent. (b) with the notation of Proposition 2.6, conditions (2)-(5) are equivalent (and condition (6) actually holds).

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Proof. (a) It suﬃces to prove (5) ⇒ (1). Suppose that B is a divisorial ideal of R[X] which properly contains P . Then B ∩ R = 0. By [17, Lemme 2], B = (B ∩ R)R[X]. Hence c(B) = B ∩ R. Now R = c(P )v ⊆ c(B)v = B ∩ R. It follows that B = R. Therefore, P is a maximal divisorial ideal. (b) It suﬃces to prove (5) ⇒ (2). Write I = f c(f )−1 R[X] by [17, Lemme 1]. Then c(I) = c(f )c(f )−1 . Since c(I)v = R, this shows that c(f ) is v-invertible. It follows that I is v-invertible. Also, aI ⊆ f R[X] for any a ∈ c(f ), so (6) holds. We conclude this section with the example promised above. Although we could base the proof on Lemma 2.4, we instead use the following lemma, since it may be of some independent interest. Proposition 2.9. Let P = f K[X] ∩ R[X] be an upper to zero, and let I denote the ideal generated by the constant terms of the elements of P . If I −1 = R and P −1 = R[X], then P is almost principal. Proof. Suppose that P is not almost principal. Let ψ ∈ P −1 . Assuming that I −1 = R, we shall show that ψ ∈ R[X]. Since P is not almost principal, we have ψ ∈ K[X]. Let c denote the constant term of ψ. For g ∈ P , since ψg ∈ R[X], we have ca ∈ R, where a is the constant term of g. Hence c ∈ I −1 = R. In particular, if ψ ∈ K, then ψ ∈ R. Now, since c ∈ R, the polynomial ω = (ψ − c)/X) ∈ P −1 . By induction, we have that ω ∈ R[X]. But then ψ ∈ R[X], as desired. Example 2.10. An example of an upper to zero with P −1 = R[X]. Thus P is trivially v-invertible but is not divisorial. Let k be a ﬁeld, and let s, t be inn determinates over k. Set R = k[{st2 | n ≥ 0}]. This example was discussed in [7] and [11]. In particular, [11] contains a direct proof that the upper to zero P = (X − t)K[X] ∩ R[X] (where K denotes the quotient ﬁeld of R) is not almost principal. It is easy to see that a monomial si tj ∈ k[s, t] lies in R iﬀ i ≥ ϕ(j), where ϕ(j) is the number of 1’s in the binary expansion of j. To see that P −1 = R[X], it suﬃces to show that I −1 = R, where I is the ideal generated by the constant terms n n of P (Proposition 2.9). It is easy to see that sX 2 − st2 ∈ P for each n ≥ 1. Thus n n st2 ∈ I for each n ≥ 1. Set J = ({st2 | n ≥ 1}). Then J ⊆ I, and we need only show that J −1 = R. If u ∈ J −1 , then ust ∈ R, so that u = a/st for some a ∈ R. We shall show that a ∈ stR. For this we may assume that a is a monomial, say a = si tj . n Since a ∈ R, we have i ≥ ϕ(j). Note that for each n ≥ 1, we have (a/st)st2 ∈ R, n whence si t2 −1+j ∈ R. It follows that i ≥ ϕ(2n − 1 + j). For suﬃciently large n, we have ϕ(2n − 1 + j) = 1 + ϕ(j − 1), whence i − 1 ≥ ϕ(j − 1). It follows that a = (st)(si−1 tj−1 ) ∈ stR, as desired.

3

UMV-Domains

Definition 3.1. A domain R is a UMV-domain if each upper to zero in R[X] is a maximal divisorial ideal of R[X]. Theorem 3.2. The following statements are equivalent for a domain R with quotient ﬁeld K.

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(1 ) R is a UMV-domain. (2 ) Each upper to zero P in R[X] satisﬁes (P : P ) = R[X] = P −1 . (3 ) For each nonconstant polynomial f ∈ K[X], the ideal I = f K[X] ∩ R[X] satisﬁes (I : I) = R[X] = I −1 . (4 ) Each upper to zero P in R[X] satisﬁes P −1 = R[X] and c(P )v = R. (5 ) For each nonconstant polynomial f ∈ K[X], the ideal I = f K[X] ∩ R[X] satisﬁes I −1 = R[X] and c(I)v = R. Proof. (1) ⇒ (2): Let R be a UMV-domain, and let P be an upper to zero. Then P is maximal divisorial, and we have (P : P ) = R[X] by Proposition 2.5. Since P is divisorial, we also have P −1 = R[X]. (2) ⇒ (3): Let f, I be as given, and (we may) assume f ∈ R[X]. Factor f in K[X]: f = g1m1 · · · grmr , where each gi is irreducible in K[X], and set Pi = gi K[X] ∩ R[X]. Let ψ ∈ (I : I); recall from Lemma 2.2 that this implies that ψ ∈ K[X]. Then ψ(P1m1 · · · Prmr ) ⊆ ψI ⊆ I. Thus

ψ(P1m1 −1 P2m2 · · · Prmr )P1 ⊆ P1 .

Since (P1 : P1 ) = R[X], this implies that ψ(P1m1 −1 P2m2 · · · Prmr ) ⊆ R[X]. Rewrite this as

ψ(P1m1 −2 P2m2 · · · Prmr )P1 ⊆ R[X].

Then, using (2) and Lemma 2.2, we have ψ(P1m1 −2 P2m2 · · · Prmr ) ⊆ P1−1 ∩ K[X] = (P1 : P1 ) = R[X]. Repeating this as many times as necessary, we eventually obtain ψ ∈ R[X], as desired. Finally, I −1 ⊇ P1−1 R[X]. (3) ⇒ (5): Assume (3), and let I = f K[X] ∩ R[X] with f ∈ R[X]. We have I −1 = R[X] by assumption. Moreover, the condition (I : I) = R[X] implies that c(I)v = R by Proposition 2.6. (5) ⇒ (4): Trivial. (4) ⇒ (1): Proceeding contrapositively, suppose that R is not a UMV-domain, and let P be an upper to zero which is not maximal divisorial. Assume that P −1 = R[X]. Then P J for some divisorial ideal J of R[X], and we must have J ∩R = 0. By Lemma 2.4, c(J)v = R, whence c(P )v = R. We recall that a v-domain is a domain in which each ﬁnitely generated ideal is v-invertible. Examples include all completely integrally closed domains. Theorem 3.3. The following statements are equivalent for a domain R. (1 ) R is a v-domain. (2 ) R is an integrally closed UMV-domain.

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(3 ) R is integrally closed, and every upper to zero in R[X] is v-invertible. (4 ) R is integrally closed and every upper to zero P = f K[X] ∩ R[X] with f a linear polynomial is v-invertible. Proof. (2) ⇔ (3): This follows from the fact that in the integrally closed case v-invertibility and maximal divisoriality are equivalent for uppers to zero (Proposition 2.8). (1) ⇒ (3) Suppose that R is a v-domain. It is well-known that R is integrally closed. Let P = f K[X] ∩ R[X] be an upper to zero. Then P = f c(f )−1 R[X] by [17, Lemme 1]. Since R is a v-domain, we have (c(f )c(f )−1 )v = R. Hence (P P −1 )v = (c(f )−1 c(f )v )v R[X] = R[X], and P is v-invertible. (2) ⇒ (1) Assume that R is an integrally closed domain UMV-domain, and let A be a ﬁnitely generated ideal of R. Since a principal ideal is trivially v-invertible, we may as well assume that A is not principal. Then A = c(f ) for some nonconstant polynomial f ∈ R[X]. Set I = f K[X] ∩ R[X]. Then I = f c(f )−1 R[X] by [17, Lemme 1]. By Theorem 3.2, (I : I) = R[X]. Hence by Proposition 2.6, I is vinvertible in R[X]. It follows easily that c(f ) = A is v-invertible in R. Hence R is a v-domain. (3) ⇒ (4): Trivial. (4) ⇒ (1): Assume (4). To show that R is a v-domain, it suﬃces by [14, Lemma 3.6] to show that every two-generated ideal of R is v-invertible. Consider a two-generated ideal A = (a, b). Let f = aX + b, and set P = f K[X] ∩ R[X]. By hypothesis P is v-invertible. Again using [17, Lemme 1], we can write P = f c(f )−1 R[X], and it is easy to see that c(f ) = A is v-invertible in R. We now compare our results on UMV-domains to those on UMT-domains. Recall that a domain R is a UMT-domain if each upper to zero in R[X] is a maximal t-ideal, and that R is a Pr¨ ufer v-multiplication domain (PVMD) if each ﬁnitely generated ideal of R is t-invertible. According to [12, Proposition 3.2], a domain is a PVMD if and only if it is an integrally closed UMT-domain. Our Theorem 3.3 above provides an exact analogue to this PVMD characterization. On the other hand, our Theorem 3.2 is much less satisfying than the corresponding situation for UMT-domains. Recall from [12, Proposition 1.1] that if R is a domain and M is a maximal t-ideal of R[X] with M ∩ R = 0, then M = (M ∩ R)R[X]. Hence, since in general a t-ideal of a domain is contained in a maximal t-ideal, we have that a domain R fails to be a UMT-domain if and only if there is an upper to zero P in R[X] and a maximal t-ideal M of R with P ⊆ M R[X]. Unfortunately, however, maximal divisorial ideals need not exist. (For example, let R be a valuation domain whose maximal ideal is not principal.) The closest that we can come to an analogue of the UMT-result is the following reworking of the statement of Theorem 3.2 (1) ⇔ (3). Proposition 3.4. Let R be a domain. Then R fails to be a UMV-domain if and only if there is an upper to zero P in R[X] for which either P −1 = R[X] or P ⊆ IR[X] for some divisorial ideal I of R.

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Proof. Just take I = c(P )v is the statement of Theorem 3.2 (3). In [5, Example 3.1], Gabelli and Roitman give an example of a completely integrally closed (and therefore a v-) domain R admitting an upper to zero which is a maximal divisorial ideal but not a maximal t-ideal. In view of Theorem 3.3 (and the characterization of PVMDs as integrally closed UMT-domains), any v-domain which is not a PVMD must admit such an upper to zero. The ﬁrst non-PVMD v-domain was produced by Dieudonn´e in [2]; other examples were given in [15, 16], [9], and [8]. Now there is a host of such examples. For example, if R is such an example, then so is R[X] (as is mentioned in the next section below, if R is a vdomain, then R[X] is also a v-domain). Examples can also be produced using the D + XDS [X]-construction; see [18]. Thus, from our point of view, [5, Example 3.1] is another, welcome, addition to the list of v-domains which are not PVMDs. By the characterizations mentioned above, a v-domain which is not a PVMD is a UMV-domain which is not a UMT-domain. It is of interest to have nonintegrally closed examples of this phenomenon. We now give such an example. Example 3.5. An example of a nonintegrally closed UMV-domain which is not a UMT-domain. Begin with any domain D such that D is a v-domain but is not a PVMD, let k denote the quotient ﬁeld, let T = k[[t2 , t3 ]] (t an indeterminate), and let R be deﬁned by the following pullback diagram. R ⏐ ⏐

−−−−→

D ⏐ ⏐

φ

T = −−−−→ k = T /M. Note that t is integral over R, so R is not integrally closed. Also, since D is not a UMT-domain, neither is R [4, Theorem 3.7]. It remains to show that R is a UMV-domain. Accordingly, let P = f K[X] ∩ R[X] be an upper to zero in R[X], and let Q = f K[X] ∩ T [X] (K is the common quotient ﬁeld of R and T ). Since T is Noetherian, Q is almost principal, and we have uQ ⊆ f T [X] for some nonzero u ∈ T . Hence t2 uP ⊆ f R[X]. Thus P is almost principal and therefore divisorial. In particular, P −1 = R[X]. We next want to show that c(P )v = R. Again, since T is Noetherian (and one dimensional), we have Q M [X]. Therefore, since RM = T , we have P M [X]. Now suppose c(P )v = R, and pick y ∈ c(P )−1 \ R. Then P ⊆ (R :R y)R[X], and since the ideals of R are comparable to M , we must have M (R :R y). It is then easy to show that φ(P ) is an upper to zero in D[X] with φ(P ) ⊆ (D :D φ(y))D[X]. However, since D is a UMV-domain, this contradicts Theorem 3.2. Hence we have c(P )v = R, as desired. Another application of Theorem 3.2 completes the proof. According to [4, Theorem 1.5], a domain R is a UMT-domain if and only if RQ has Pr¨ ufer integral closure for each prime t-ideal Q of R. We have the following partial analogue.

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Theorem 3.6. If R is a UMV-domain, then RQ has Pr¨ ufer integral closure for each divisorial prime Q of R. Proof. Let R be a UMV-domain, and let Q be a divisorial prime of R. Since each upper to zero in R[X] is maximal divisorial, we have P Q for each upper to zero P . Hence in RQ [X] the prime ideal QRQ [X] contains no uppers to zero. It follows that if T is the integral closure of RQ and N is a prime of T lying over QRQ , then N T [X] contains no uppers to zero. We then have that T is a Pr¨ ufer domain by [6, Theorem 19.15]. Corollary 3.7. If R is a v-domain, then RM is a valuation domain for each divisorial prime M of R.

4

Questions

We close with several questions and comments related to the results of the ﬁrst two sections. ufer integral closure for each divisorial prime Q of R, Question 4.1. If RQ has Pr¨ is R necessarily a UMV-domain? In view of Theorem 3.6, a positive answer to this question would yield a characterization of UMV-domains comparable to the characterization of UMT-domains mentioned just before the statement of Theorem 3.6. It is known that if R is a UMT-domain (respectively, a v-domain), then so is R[X] [4, Theorem 2.4] (respectively, [1, Corollary 1.6]). This motivates our next question. Question 4.2. If R is a UMV-domain, is R[X] also a UMV-domain? Here is a related question: If R is a UMV-domain, does it follow that if P is a prime ideal of R[X, Y ] such that P has height one and satisﬁes P ∩ R = 0, then P is a maximal divisorial ideal of R[X, Y ]? If Question 4.1 has a negative answer, then the following question becomes interesting. ufer integral closure for each Question 4.3. If R has the property that RQ has Pr¨ divisorial prime Q of R, does R[X] have this same property? Recall that for uppers to zero t-maximality and t-invertibility are equivalent [12, Theorem 1.4]. However, in Example 2.10, we produced a domain R admitting an upper to zero P with P −1 = R[X] (so that P is trivially v-invertible but is not (maximal) divisorial). Question 4.4. Is there an example of a non-UMV-domain R with the property that every upper to zero in R[X] is v-invertible? We do not know whether Example 2.10 is such an example. At any rate no such example can be integrally closed, by Proposition 2.8. Assuming the existence of such examples, one could ask whether the property that all uppers to zero are v-invertible would extend to the polynomial ring.

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Question 4.5. If R has the property that each upper to zero P = f K[X] ∩ R[X] with f linear is maximal divisorial, is R necessarily a UMV-domain? Note that this is proved in the integrally closed case in Theorem 3.3. The corresponding question for the UMT-property has an aﬃrmative answer. Since this has not (to our knowledge) appeared in the literature, we state it formally and sketch a proof. Proposition 4.6. Suppose that R has the property that every upper to zero P = f K[X] ∩ R[X] with f linear is a maximal t-ideal. Then R is a UMT-domain. Proof. In a manner similar to that used in the proof of Theorem 3.6, one can show that for each maximal t-ideal M of R, we have that N T [X] contains no linear uppers to zero, where T is the integral closure of RM and N is a prime of T lying over M RM . Hence for u ∈ K, we have that P = (x − u)K[X] ∩ T [X] N [X] for each maximal ideal N of T . Thus u is the root of a polynomial not contained in N [X]. By the u, u−1 -lemma [6, Lemma 19.14], either u or u−1 lies in TN . It follows that T is a Pr¨ ufer domain. Hence R is a UMT-domain by [4, Theorem 1.5].

Bibliography [1] D.F. Anderson, E. Houston, and M. Zafrullah, Pseudo-integrality, Canad. Math. Bull. 34 (1991), 15–22. [2] J. Dieudonn´e, Sur la th´eorie de la divisibilit´e, Bull. Soc. Math. France 69 (1941), 133–144. [3] D. Dobbs, E. Houston, T. Lucas, M. Roitman, and M. Zafrullah, On t-linked overrings, Comm. Algebra 20 (1992), 1463–1488. [4] M. Fontana, S. Gabelli, and E. Houston, UMT-domains and domains with Pr¨ ufer integral closure, Comm. Algebra 26 (1998), 1017–1039. [5] S. Gabelli and M. Roitman, Maximal divisorial ideals and t-maximal ideals, manuscript. [6] R. Gilmer, Multiplicative ideal theory, Dekker, New York, 1972. [7] E. Hamaan, E. Houston, and J. Johnson, Properties of uppers to zero in R[x], Paciﬁc J. Math. 135 (1998), 65–79. [8] W. Heinzer, An essential integral domain with a nonessential localization, Canad. J. Math. 23 (1981), 400–403. [9] W. Heinzer and J. Ohm, An essential ring which is not a v-multiplication ring, Canad. J. Math. 21 (1972), 856–861. [10] E. Houston, S. Malik, and J. Mott, Characterizations of ∗-multiplication domains, Canad. Math. Bull. 27 (1984), 48–52. [11] E. Houston, Prime t-ideals in R[X], Commutative ring theory, P.-J. Cahen, D. Costa, M. Fontana, S. Kabbaj, eds., Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 153 (1994), 163–170. [12] E. Houston and M. Zafrullah, On t-invertibility II, Comm. Algebra 17 (1989), 1955–1969. [13] J. Johnson, Three topological properties from Noetherian rings, Canad. J. Math. 34 (1982), 525–534. 314

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[14] J. Mott, B. Nashier, and M. Zafrullah, Contents of polynomials and invertibility, Comm. Algebra 18 (1990), 1569–1583. [15] M. Nagata, On Krull’s conjecture concerning valuation rings, Nagoya Math. J. 4 (1952), 29–33. [16] M. Nagata, Corrections to my paper “On Krull’s conjecture concerning valuation rings”, Nagoya Math. J. 9 (1955), 209–212. [17] J. Querr´e, Id´eaux divisoriels d’un anneau de polynˆ omes, J. Algebra 64 (1980), 270–284. [18] M. Zafrullah, The D+XDS [X]-construction from GCD-domains, J. Pure Appl. Algebra 50 (1988), 93–107.

Chapter 21

On Local Half-Factorial Orders by

Florian Kainrath

1

Introduction

Let R be an atomic domain. Then every nonunit r of R has a factorization r = u1 · . . . · un , where the ui are atoms of R. The integer n is called the length of the factorization. In general r has many such factorizations. R is called half-factorial, if for any r ∈ R all the factorizations of r have the same length. In [4] F. Halter-Koch gave a complete description of half-factorial orders in quadratic number ﬁelds. In particular, this implies the following for such an order ¯ is half-factorial, too. 2. If O: 1. Any ring between O and its integral closure O e1 en ¯ then ¯ (O : O) = p1 · . . . · pn is the prime factorization of the conductor of O in O, all exponents ei are equal to one. In view of this result one may conjecture that something similiar holds for orders in arbitrary number ﬁelds. In this chapter we study the local situation, meaning we look at local orders, i. e. local integral domains R, which are noetherian, one-dimensional, whose integral closure is a ﬁnitely generated R-module, and whose residue ﬁeld is ﬁnite. We give a new criterion, when such domains are half-factorial (Theorem 3.3). As an immediate application of this criterion we obtain a result on the size of the conductor of a half-factorial order, and we show that any ring between a halffactorial local order and its quotient ﬁeld is half-factorial, too. Coming back to the global situation, I will state a conjecture for local orders at the end of this chapter, which (if true) implies that any ring between an half-factorial order in some number ﬁeld and its integral closure is half-factorial, too. The proof of the above mentioned criterion is mainly based on a result on the product of two subspaces in a ﬁnite extension of ﬁelds (Theorem 2.3), which is completely analogous to a result of Kneser on the sumset of two ﬁnite sets in an abelian group (see for example [6, Theorem 4.3,chap. 4.3]). 316

On Local Half-Factorial Orders

2

317

On the Product of Two Subspaces in a Finite Extension of Fields

Let R be a ring, X, Y ⊂ R and i ≥ 1. Set X ∗ Y = {xy | (x, y) ∈ X × Y } and deﬁne X ∗i inductively by X ∗1 = X and X ∗i = X ∗(i−1) ∗ X for i ≥ 2. Finally let XY be the subgroup of R generated by X ∗ Y . If X = {x} for some x ∈ R we will also write x ∗ Y , resp. xY instead of {x} ∗ Y , resp. {x}Y . Note that if R is a subring of R such that R ∗ X ⊂ X then XY is the R -submodule of R generated by X ∗ Y . In particular it is a R -submodule of R. If L | K is an extension of ﬁelds and V , W ⊂ L are K-subspaces of L, set × V = V \ {0}, T (V, W ) = {x ∈ L | xV ⊂ W } and T (V ) = T (V, V ). Note, if [L : K] < ∞, then T (V ) is the largest subﬁeld F of L such that V is a F -subspace of L. In the following two lemmas let L | K be a ﬁnite extension of inﬁnite ﬁelds, and V and W be two nonzero K-subspaces of L. We will also assume that L = K(x) for some x ∈ L. Note that this will be so, if and only if the set of intermediate ﬁelds of L | K is ﬁnite. Lemma 2.1. There exist nonzero v ∈ V , w ∈ W having the following property: whenever U is a K-subspace of V W containing vw, we have T (U ) ⊂ T (V W ). Proof. First note that for (v, w) ∈ V × × W × the condition stated in the lemma is equivalent to: F vw ⊂ V W for all intermediate ﬁelds F of L | K, such that F ⊂ T (V W ). Let F be the set of all intermediate ﬁelds F of L | K, such that F ⊂ T (V W ). By our assumption on L | K this is a ﬁnite set. For F ∈ F set XF = {(v, w) ∈ V × × W × | F vw ⊂ V W } . Since F ⊂ T (V W ) and since V × ∗ W × generates V W we have XF = V × × W × . Next let µ : V × × W × → L be the multiplication map, µ(v, w) = vw. Then clearly XF = µ−1 (T (F, V W )) . Now endow V × W and L with the K-Zariski topologies. Note that V × × W × is an open subset of V × W , and hence is irreducible (K being inﬁnite). Since µ is the restriction of a bilinear map to V × × W × it is continous. Further T (F, V W ) is a K-subspace of L, and hence is closed. The above equation implies now that XF is closed in V × × W × , too. So the family X = (XF )F ∈F consists of proper, closed subsets of V × × W × . The irreducibility of V × × W × implies now that the union of X does not equal V × × W × . Now it is clear that any (v, w) ∈ V × × W × not belonging to this union satisﬁes the condition stated at the beginning of the proof. Lemma 2.2. Suppose that dimK (V W ) − dimK (V ) > 0. Then there exist nonzero K-subspaces V and W of L, such that:

318

Kainrath

1. V W ⊂ V W and T (V W ) ⊂ T (V W ). 2. dimK (V W ) − dimK (V ) < dimK (V W ) − dimK (V ). 3. dimK (V ) + dimK (W ) = dimK (V ) + dimK (W ). Proof. Note that dimK (V W ) − dimK (V ) > 0 implies, that V w = V W for any nonzero w ∈ W . Choose (v, w) ∈ V × × W × as in Lemma 2.1. Since V w = V W and V × ∗ W × generates V W , there exist nonzero v ∈ V and w ∈ W , such that / V w. v w ∈ Deﬁne now V and W by V = V + v w−1 W

and

W = W ∩ v

−1

wV

.

Then v ∈ V and w ∈ W , so that in particular both are nonzero. We now show that V and W have the properties 1-3 of the lemma. 1. Obviously vw ∈