Ring Constructions & Applications (Series in Algebra, Volume 9)

ALGEBRA

RING CONSTRUCTIONS AND APPLICATIONS

Andrei V. Kelarev

World Scientific

RING CONSTRUCTIONS AND APPLICATIONS

SERIES IN ALGEBRA Editors: J. M. Howie, D. J. Robinson, W. D. Munn Vol. 1: Infinite Groups and Group Rings ed. J. M. Corson et al. Vol. 2: Sylow Theory, Formations and Fitting Classes in Locally Finite Groups M. Dixon Vol. 3: Finite Semigroups and Universal Algebra J. Almeida Vol. 4: Generalizations of Steinberg Groups T. A. Fournelle and K. W. Weston Vol. 5: Semirings — Algebraic Theory and Applications in Computer Science U. Hebisch and H. J. Weinert Vol. 6: Semigroups of Matrices J. Okninski Vol. 7: Partially Ordered Groups A. M. W. Glass Vol. 9: Ring Constructions and Applications A. V. Kelarev

Forthcoming Vol. 8: Groups with Prescribed Quotient Groups and Associated Module Theory L Kurdochenko, J. Otal and I. Subbotin

SERIES

IN

ALGEBRA

VOLUME 9

R|

NG CONSTRUCTIONS AND APPLICATIONS

Andrei V. Kelarev Department of Mathematics University of Tasmania Australia

V f e World Scientific « •

New Jersey • London • Singapore • Hong Kong Sir

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

RING CONSTRUCTIONS AND APPLICATIONS Series in Algebra - Vol. 9 Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4745-1

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to my children, Lena and Nadia

Preface

A number of diverse ring constructions have been applied in various areas of modern science, from coding theory, to cryptography, to logic, to number theory, to quantum physics and symbolic computation. The investigation of properties of ring constructions is a very large and rapidly changing research area. The following two criteria have been used in selecting material for this book. First, the author has tried to choose topics, preliminaries and proofs in order to prepare new researchers for work on directions where active investigation has been carried out recently and there are enough challenging open problems. Second, an attempt has been made to supply advanced specialists with complete concise information and references related to these questions and directions. This book is devoted to various ring constructions, their properties, and examples of applications. It would be really valuable to find a single viewpoint which makes it possible to achieve good understanding of many notions in a unified framework. The emphasis is on the fairly new concept of a groupoid-graded ring embracing and unifying a variety of constructions. Familiarity with this efficient tool makes it easy to understand properties of and similarities among other more special ring constructions, which at first glance may appear quite different. Groupoid-graded rings include as special cases many other ring constructions: polynomial and skew polynomial rings, Ore extensions, direct and semidirect products, matrix and structural matrix rings, Rees matrix rings, Morita contexts and generalized matrix rings, group and semigroup rings, skew group and semigroup rings, twisted group and semigroup rings, monomial rings, smash products, crossed products, group-graded rings, vii

Vlll

Preface

monoid-graded rings, path algebras, edge algebras, incidence rings, etc. Therefore groupoid-graded rings can be applied to the study of other less general constructions. This not only gives new results for several constructions simultaneously, but also serves the unification of known theorems. Groupoid-graded rings have been actively investigated for years, and many interesting results have been obtained in the literature. Several surveys on this direction have already been published, but there was no monograph collecting various results in a convenient form with concise preliminaries included for the reader. Therefore it is also desirable to fill this gap and summarize numerous contributions made during recent years. This is a new and rapidly developing area of research. Many new open problems have been recorded in various publications, which also provides an opportunity for postgraduate students to take part in the ongoing work. A book on this topic can serve as an excellent introduction for postgraduate students to many ring constructions as well as to most essential basic concepts of group, semigroup and ring theories used in proofs. Our exposition focuses on introducing sufficient background knowledge and preparing new researchers for work on several topics with abundant open problems and opportunities for new contributions. On the other hand, many more specialized, advanced and difficult technical results have been included without proofs but with references. In this way the text at the same time addresses the needs of advanced experts too. This approach has helped to provide valuable content and references for new and established researchers using ring constructions in their work. This book has grown from a series of publications by the author, which have been discussed with co-authors, colleagues, after the talks at various conferences and seminars, as well as with editors organizing anonymous refereeing of papers. The author is grateful to all mathematicians who have contributed to his research, and would like to use this opportunity to express sincere appreciation to Professors K.I. Beidar, A.D. Bell, S. Dascalescu, D. Easdown, K.R. Fuller, B.J. Gardner, S. Goberstein, T.E. Hall, K.J. Horadam, J.M. Howie, E. Jespers, J. Justin, R. Lidl, J. Meakin, A.V. Mikhalev, W.D. Munn, J. Okninski, F. Otto, D.S. Passman, F. Pastijn, G. Pirillo, C.E. Praeger, P. Schultz, L.N. Shevrin, I. Shparlinski, P. Shumyatsky, O.V. Sokratova, T. Stokes, M.L. Teply, P.G. Trotter, M.V. Volkov, A.P.J, van der Walt, R. Wiegandt, L. van Wyk, and to the Editor of World Scientific E.H. Chionh.

Contents

Preface

vii

Chapter 1 Preliminaries 1.1 Groupoids 1.2 Groups 1.3 Semigroups 1.4 Rings

1 1 2 4 11

Chapter 2 Graded Rings 2.1 Groupoid-Graded Rings 2.2 Semigroup-Graded Rings 2.3 Group-Graded Rings 2.4 Superalgebras

17 17 20 21 23

Chapter 3 Examples of Ring Constructions 3.1 Direct, Subdirect and Semidirect Products 3.2 Group and Semigroup Rings, Monomial Rings 3.3 Crossed Products 3.4 Polynomial and Skew Polynomial Rings 3.5 Skew Group and Semigroup Rings 3.6 Twisted Group and Semigroup Rings 3.7 Power and Skew Power Series Rings 3.8 Edge and Path Algebras 3.9 Matrix Rings and Generalized Matrix Rings 3.10 Triangular Matrix Representations

25 25 26 27 27 28 29 30 31 31 33

ix

x

3.11 3.12 3.13 3.14 3.15

Contents

Morita Contexts Rees Matrix Rings Smash Products Structural Matrix Rings Incidence Algebras

33 33 34 34 35

Chapter 4 The Jacobson Radical 4.1 The Jacobson Radical of Groupoid-Graded Rings 4.2 Descriptions of the Jacobson Radical 4.3 Semisimple Semigroup-Graded Rings 4.4 Homogeneous Radicals 4.5 Radicals and Homogeneous Components 4.6 Nilness and Nilpotency

37 37 42 45 51 64 67

Chapter 5

73

Groups of Units

Chapter 6 Finiteness Conditions 6.1 Groupoid-Graded Rings 6.2 Structural Approach of Jespers and Okniriski 6.3 Finiteness Conditions and Homogeneous Components 6.4 Classical Krull Dimension and Gabriel Dimension

77 77 82 89 102

Chapter 7

107

Pi-Rings and Varieties

Chapter 8 Gradings of Matrix Rings 8.1 Full and Upper Triangular Matrix Rings 8.2 Gradings by Two-Element Semigroups 8.3 Structural Matrix Superalgebras

111 Ill 120 129

Chapter 9 Examples of Applications 9.1 Codes as Ideals in Group Rings 9.2 Codes as Ideals in Matrix Rings 9.3 Color Lie Superalgebras 9.4 Combinatorial Applications 9.5 Applications in Logic

133 133 140 146 147 148

Chapter 10

149

Appendix A

Open Problems Glossary of Notation

153

Contents

xl

Bibliography

157

Index

201

Chapter 1

Preliminaries

In this chapter we give concise background information most frequently used in this direction of research for the convenience of the reader. Complete and detailed explanations of the algebraic concepts summarized below can be found, for example, in the following books: [Beachy (1999)], [Cohen and Cuypers (1999)], [Cohn (2000)], [Fan et al. (2000)], [Grillet (1999)], [Hazewinkel (1996)], [Howie (1976)], [Howie (1995)], [Hungerford (1980)], [Kargapolov and Merzljakov (1979)], [Lallement (1979)], [Lam (1999)], [Lambek (1976)], [Lidl and Niederreiter (1994)], [Lidl and Pilz (1998)], [Passman (1991)], [Robinson (1982)], [Rowen (1991)], and [Scherk (2000)]. Several more specialized and advanced monographs have also been included in the bibliography. 1.1

Groupoids

A set with a binary operation is called a groupoid. An associative groupoid is called a semigroup. A semigroup with identity is called a monoid. A groupoid G is called a quasigroup if the equation ab = c determines a unique element b G G for given a,c € G, and a unique element a € G for given b,c £ G. An element 1 is called an identity element or unity of the groupoid G if \g = gl = g, for every g £ G. A loop is a quasigroup with an identity element. Let G be a groupoid. An element 0 is called a zero of the groupoid G if Og = gO = 0, for every g e G. Denote by G 1 (and G°) the groupoid G with identity (respectively, zero) adjoined. If G has an identity element (or zero), then we assume that G1 = G (respectively, G° = G). Otherwise, l

2

Preliminaries

Gl = G U {1} (and G U {0}). If M is a subset of a groupoid G and x € G, then xM1 = xM U {a;} and M1x = Mx U {x}. UT Q G, then the subgroupoid generated by T in G is usually denoted by (T). It consists of all products of elements of T. Another notation is used in formal language theory. Namely, the subgroupoid (subgroupoid with identity) generated by T in G1 is denoted by T+ (respectively, T* ). The ideal (left ideal, right ideal) generated by T in G is the smallest subgroupoid J containing G such that GIUlG CI (respectively, GI C I, IG C / ) . If G is a semigroup, then the ideal (left ideal, right ideal) generated by T in G coincides with the set G^TG1 (respectively, GlT, TG1). An ideal generated by one element is called a principal ideal. For s € G, let s" denote the set of all x £ G such that s belongs to the ideal generated by x in G. The cardinality of the set G is called the order of the groupoid G and is denoted by \G\. The order of the element g is the order of the subgroupoid (g) it generates. An element g of a groupoid G is said to be periodic if the subgroupoid (g) is finite. A subset T of S is periodic if every element of T is periodic. An element e of G is called an idempotent if e = e 2 . The set of all idempotents of G is denoted by E(G).

1.2

Groups

This section contains a few most basic definitions and facts for the convenience of the reader. A semigroup G with identity 1 is called a group if every element g € G has an inverse g_1 such that gg~* = g~xg = 1. A commutative group is said to be abelian. Abelian groups are often considered in the additive notation where the operation of the group (G, +, 0) is denoted by + and the neutral element is 0. Let G be a group. A subgroup H of G is a subset closed with respect to the multiplication of the group and the operation of taking the inverse element. If T C G, then the subgroup generated by T is the smallest subgroup containing T. The set T is called a generating set of the subgroup. A subgroup is finitely generated if it has a finite generating set. The definitions of a finitely generated subsemigroup, or subring, or ideal of a ring are similar. If T is a nonempty subset of G and g e G, then the conjugate of T by g is the set T> = g-lTg

= {g'Hg

\teT}.

Groups

3

A subgroup H of H is said to be normal if the set HG =

{g-1hg\heH,g£G}

is equal to H. A right (left) coset of the subgroup H is a set of the form Hg (respectively, gH), where g £ G. The set of left cosets has the same cardinality as the set of right cosets, and this cardinality is called the index of H in G. If N is a normal subgroup, then the quotient group G/N is the set of all right cosets of N in G with the following operations: (Ngi)(Ng2) = iV<7i
4

Preliminaries

A series of subgroups of the group G {l} = N0CN1CN2C-.-CNk

=G

(1.1)

is said to be normal if Nk-i is a normal subgroup of Nk, for every i = l,...,k. A group G is solvable if it has a finite normal series (1.1) such that all factors Nk/Nk-\ are abelian, for all i = 1 , . . . , k. A group is nilpotent if it has a finite normal series (1.1) such that, for i = 0 , . . . , k — 1, the N, is a normal subgroup of the whole G and Ni+i is the center of G/iVj. A group is polycyclic if it has a finite normal series (1.1) such that all factors Nk/Nk-i are cyclic groups, for i = 1 , . . . , k. Theorem 1.4

Every finite p-group is nilpotent.

A group G is said to be finite-by-abelian-by-finite if it has subgroups {e} C H C K C G such that H is finite and normal in K, K/H is abelian, K is normal in G, and G/K is finite. A group G is abelian-by-finite if it has a normal abelian subgroup K such that G/K is finite.

1.3

Semigroups

A semigroup S is said to be left {right) cancellative if zx = zy implies x — y (respectively, xz = yz implies x — y), for all x,y,z £ S. A semigroup is cancellative if it is both left and right cancellative. A semigroup is left (right) simple if it has no proper left (respectively, right) ideals. Lemma 1.1 Every finite cancellative semigroup is a group. Every finite left and right simple semigroup is a group. A band is a semigroup entirely consisting of idempotents. A band is called a semilattice (rectangular band, left zero band, right zero band, left regular band, right regular band) if it satisfies the identity xy = yx (respectively, xyx ~ x, xy = x, xy = y, xyx = xy, xyx = yx). A partial order < is defined on the set of idempotents E(S) by e < f <=> ef = fe = e. A semilattice is called a chain if it is linearly ordered. By the height of a semilattice Y we mean the supremum of the cardinalities of all chains contained in Y.

5

Semigroups

Let S be a semigroup. The Green's equivalences C, ~R, J, H, and V are defined by xCy <=> S1x = Sxy, xTZy ^xS1 xJy&SlxSl H = £n1limdV

= yS1, =

S1yS\

= a i = TIC,

where CTZ and TIC denote the usual composition of relations. Denote by Hx (Rx, Lx) the W-class (respectively, 7£-class, £-class) of S containing x, i.e., the set of elements generating the same ideals (respectively, right ideals, left ideals) as x £ S. If e is an idempotent, then He is a subgroup of S with identity e. Furthermore, it contains all such subgroups of S. Put Pe = {x € eSe \ xy = e for some y € eSe}. Then He C Pe C Re and Pe = Re n eSe. For any two elements x, y in one W-class, Hx = Hy. We refer to [Howie (1995)] for complete explanations. Let J and J be ideals of a semigroup S such that J C. I. The Rees quotient semigroup I/J is the semigroup with zero obtained from / by identifying all elements of the ideal J with 0. If J has zero and J = {0}, then I/J = I. In the case where J = 0, we assume that I/J = J. The quotient I/J is called a factor of S. We identify all elements of I\J with their images in I/J, and say that all elements of I\J belong to I/J. Take any element g in S, put J = S' 1 g5 1 and denote by J the set of all elements which generate principal ideals properly contained in I. Then J is also an ideal of S, and I/J is called a principal factor of S. A semigroup S is called an epigroup if a power of each element belongs to a subgroup of S. A few longer terms have also been used to refer to this concept (see [Shevrin and Ovsyannikov (1996)]). Epigroups naturally appear in ring theory. Indeed, the multiplicative semigroup of every semisimple Artinian ring is an epigroup ([Okninski (1991)], §1). Since semisimple Artinian rings and their generalizations play key roles in many ring theorems, some facts concerning the structure of epigroups become useful in deducing properties of rings. The method of exploiting the structure of the multiplicative semigroup of a ring is well known. Properties of epigroups have

6

Preliminaries

been explicitly used, for example, in [Jespers and Okniriski (1995)], where it is proved that the multiplicative semigroup of homogeneous elements of every right perfect semigroup-graded ring is an epigroup, and this is used to obtain several strong results on related ring constructions. A semigroup S is said to be inverse if, for each s £ S, there exists a unique t £ S such that sts = s and tst = t. If S is an inverse semigroup, then every two idempotents of S commute, and so the set E(S) of idempotents of S is a semilattice. A semigroup is completely regular if it is a union of groups. A semigroup is said to be aperiodic or combinatorial if it has no subgroups of order greater than one. A semigroup S is said to be Archimedean if and only if, for every s, t £ S, a power of s belongs to the ideal generated in S by t. A semigroup S is said to be torsion-free if sn = tn implies s = t, for all s, t £ S and any positive integer n. Let S be a semigroup with zero 0. Then S is called a null semigroup, or a semigroup with zero multiplication, if S2 = 0. An element x of S is nilpotent if xn = 0, for a positive integer n. The whole S is nilpotent if Sn — 0, for some n. A semigroup is said to be nil if it entirely consists of nilpotent elements. A semigroup with zero is said to be Baer radical if every nonzero factor of S has a nonzero nilpotent ideal. It is known that every semigroup S has a largest Baer radical ideal B(S) (called the Baer radical of S), and S/B(S) has no nonzero nilpotent ideals. Let G be a group, / and A nonempty sets, and let P = \p\i] be a (A x 7)-matrix with entries p\i £ G, for all A £ A, i £ I. The Rees matrix semigroup M(G; I, A; P) over G with sandwich matrix P consists of all triples (g; i, A), where i £ I, A € A, and g £ G, with multiplication defined by the rule (gi; k, A1X02; i2, A2) = (gip\1i2g2; h, A2).

(1.2)

Now suppose that Q = [qxi] is a (A x 7)-matrix with entries q\i in the group G° with zero adjoined. Then the Rees matrix semigroup M°(G;I,A;Q) over G° with sandwich matrix Q consists of zero 0 and all triples (g; i, A), for i £ I, A £ A, and g £ G°, where all triples of the form (0;i,X) are identified with 0, and multiplication is denned by the same rule (1.2). A semigroup is said to be completely simple if it has no proper ideals and has an idempotent minimal with respect to the natural partial order defined above. A semigroup with zero is said to be completely 0-simple if

7

Semigroups

it has no proper nonzero ideals and has a minimal nonzero idempotent. Theorem 1.5 Each principal factor of an epigroup is completely simple, or completely 0-simple, or a null semigroup. Theorem 1.6 (i) Every completely 0-simple semigroup is isomorphic to a Rees matrix semigroup M(G°; I, A; Q) over a group with zero adjoined. (ii) A Rees matrix semigroup M(G°; I, A; Q) over a group G with zero is completely 0-simple if and only if every row and column of the sandwich matrix P has a nonzero element. (iii) A completely 0-simple semigroup is inverse if and only if it is isomorphic to a Rees matrix semigroup M(G°;I,I;Q) with identity matrix Q. (iv) A semigroup is completely simple if and only if it is isomorphic to a Rees matrix semigroup M(G; I, A; P) over a group G. A semigroup is said to be right (left) simple if it has no proper right (left) ideals. A semigroup is left (right) cancellative iixy = xz (respectively, yx = zx) implies y — z, for all x,y,z G S. It is cancellative if it is both left and right cancellative. A semigroup is called a right (left) group if it is right (left) simple and left (right) cancellative. Theorem 1.7

For any epigroup S, the following are equivalent:

(i) S is right (left) simple; (ii) S is a right (left) group; (iii) S is isomorphic to the direct product of a right (left) zero band and a group; (iv) S is a union of several of its left (right) ideals and each of these ideals is a group. Let G be a group, S = M(G; I, A; P), and let i G J, A G A. Then we put ^

= {(/>;«,A)|/>eG,t€/}l

5 « = {(/ l ;i,A) SiX =

\h€G,XeA}, {(h;i,X)\heG}.

8

Preliminaries

If S = M°(G; I, A; Q), then we include zero in all of these sets, i.e., put S,x = {0} l) {(h;i,\)

\ h £ G,i e I},

Si, = {0} U {(/i; i, A) | h G G, A G A}, SiX =

{0}U{(h;i,\)\h€G}.

Theorem 1.8 Let G be a group, and let S = M(G;I,A;P) be a completely simple semigroup (or let S = M°(G; I, A; Q) be a completely 0-simple semigroup). Then, for all i,j G / , A, fi G A, h G G, and x = (h;i,X) G S, the following conditions hold: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

the set S,x is a minimal (minimal nonzero) left ideal of S; the set Si, is a minimal (minimal nonzero) right ideal of S; Sx = S^x = S„x; xS = xSjt = Sit; x G Sx n xS = Six; the set Si\ is a left ideal of S^ and a right ideal of S*x; ifpxi = 0, then Sfx = 0; ifpxi 7^ 0, then Six is a maximal subgroup of S (maximal subgroup with zero adjoined in S); it is isomorphic to G (respectively, to G°); (ix) each maximal subgroup (subgroup with zero adjoined) in S coincides with Sj/j,, for some j G / , (J- G A; (x) M(G; I, A; P) is a right (left) group if and only if \I\ = 1 (respectively, |A| = 1); (xi) if S = M{G; I, A; P), then each S*x is a left group, and each Si, is a right group.

Theorem 1.9 If an epigroup S contains only a finite number of idempotents, then S° has a finite ideal chain 0 = SoCS1C...CSn

= S°

such that each factor Si+i/Sit where 0 < i < n — 1, is nil or is a completely 0-simple semigroup with a finite sandwich matrix. Theorem 1.10 If S is an epigroup without infinite descending idempotent chains, then S° has an ascending ideal chain 0 = 5 0 C 5i C . . . C UTST = S°

9

Semigroups

such that Sy = U^^S,, for every limit ordinal number 7 < r, and for each ordinal /x the factor 5' M+ i/S r /i is nil or is a completely O-simple semigroup. Theorem 1.11 alent:

For every epigroup S, the following conditions are equiv-

(i) S is Archimedean; (ii) S does not contain the two-element semilattice; (iii) has a completely simple ideal and the quotient semigroup modulo this ideal is nil. A linear semigroup is a semigroup of matrices with entries in a division ring. Denote by e%j or e,j the standard matrix unit with 1 in the (i,j) entry and all other entries equal to 0. The set {e„ I 1

forms a semigroup with respect to ordinary matrix multiplication. It is denoted by Bn and is called a Brandt semigroup. Theorem 1.12 Every nil subsemigroup of a linear semigroup over a division ring is nilpotent. Theorem 1.13 Let F be a division ring, and let Mn{F) be the set of all n x n matrices over F. For k = 0 , 1 , . . . , n, denote by 1^ the set of all matrices of rank < k. Then 0 = I0ChC---Cln

= Mn{F)

are the only ideals of the multiplicative semigroup Mn(F). For every k = 1 , . . . , n, the set Ik\Ik-i is a disjoint union of subsets Gap, indexed by the elements a, (3 of a certain set A^, and such that for all a,f3,j,S G Afc the following conditions hold: (i) (ii) (iii) (iv) (v)

either Gap is a linear group, or G\„ C Jfc_i; Ga/3Mn(F)Gy5 C Ga5 U 7 f c _ i ; G a « U Jfc_i is a right ideal of Mn(F), where G a * = Ux€\kGax; G,p\Jlk _i is a left ideal of Mn(F), where G*p = UaeAkGaxi GQygU/fc-i is a left ideal o/G a „Uifc_i and a right ideal ofG^^Ulk-i-

Let B be a band. A semigroup S is said to be a band B of subsemigroups Sb, b e B, if S = UbgsS'i, is a disjoint union of the Sb, and SaSb C S a 6 for all a,b £ B. If, in addition, B is a semilattice, then S is called a semilattice B

10

Preliminaries

of the Sfc. A semigroup S with zero 0 is a 0- direct union of subsemigroups St, i £ I, ii and only if S = U i e jS f , and Si D 5,- = S ^ - = 0 for i ^ j . Theorem 1.14 (i) Every rectangular band is isomorphic to a direct product of a left zero band and a right zero band. (ii) Every band is a semilattice of rectangular bands. (iii) Every completely regular semigroup is a semilattice of completely simple semigroups. Theorem 1.15 A commutative semigroup S is Archimedean if and only if it has an ideal I which is a group such that S/I is nil. Every commutative semigroup is a semilattice of Archimedean semigroups. Theorem 1.16

For a semigroup S, the following are equivalent:

(i) S is a union of a finite number of groups and does not contain the two-element right (left) zero band; (ii) S has a finite ideal chain such that every factor of S° is a union of its right (left) ideals which are subgroups with the same zero adjoined. Theorem 1.17

For a finite band S, the following are equivalent:

(i) S is a left (right) regular band; (ii) S does not contain the two-element right (left) zero band; (iii) S has a finite ideal chain such that every factor ofS° is a left (right) zero band with zero adjoined. A semigroup S is said to be a u.p.-semigroup (unique product semigroup), respectively a t.u.p.-semigroup (two unique products semigroup), if for any two nonempty finite subsets X, Y of S, one of which is not a singleton, there exists at least one, respectively two, elements uniquely presented in the form xy, where i e l and y GY. A semigroup S is said to be permutational if there exists n > 1 such that, for any n elements xi,...,xn of 5, their product can be rearranged as xi... xn = xa\... xan for a non-trivial permutation a. A semigroup S is said to be right Ore if every two nonzero principal left ideals have nonzero intersection. Theorem 1.18 abelian-by-finite.

(i) A group is permutational if and only if it is finite-by-

Rings

11

(ii) A finitely generated permutational group is abelian-by-finite. (iii) If G is a group with a permutational subsemigroup S such that S generates G as a group, then G is permutational. (iv) Every permutational cancellative semigroup embeds in a permutational group.

1.4

Rings

A ring (R,+,-,0) is a set R with two binary operations, addition + and multiplication •, such that (R, +,0) is an abelian group, 0 is a neutral element of (R, +) and a zero of (R, •), and two distributive laws r(si + s2) = rsi + rs2 and (sx + s2)r = Syr + s2r hold, for all r,s\,s2 £ R. A ring is associative if (R, •) is a semigroup. All rings considered in this book are associative (with the exception of Sections 2.4 and 9.3 introducing applications to Lie superalgebras). A ring with identity element such that every nonzero element is invertible is called a division ring. A commutative division ring is called a field. Let Z be the ring of integers, and let Q, R and C denote the fields of rational, real and complex numbers, respectively. A ring R is called an algebra over a field F, if it is a vector space over F with respect to the same addition, and with scalar multiplication by the elements of F related to the ring multiplication by the laws f(xy) = (fx)y = x(fy) and f(gx) = (fg)x, for all f,g £F, x,y e R. Let R be a ring. If T C R, then the subring generated by T is the set (T) consisting of all finite sums of products kt\t2 • • -tn, where k £ Z and all U £ T. The ideal [right ideal, left ideal) generated by T in R is the set id(T) (id r (T), id/(T)) consisting of all finite sums of elements of R^TR1 (respectively, TR1, i? 1 T). The ideals generated by one element are said to be principal. Let -R1 be the ring obtained by adjoining a unity 1 to R in the usual fashion. The ring R1 is a direct sum of the ideal R and subring (1) isomorphic to Z. If all ideals of a ring are principal, then it is called a principal ideal ring. Let p be a prime, A a ring, and let / G A[x\. Then / is said to be irreducible modulo p, if the image of / in A[x]/pyl[a;] is an irreducible polynomial. A

12

Galois ring GR(pm,r)

Preliminaries

is a ring of the form (Z/pmZ)[x}/(f(x)),

where p is a prime, m an integer, and f(x) G (Z/pmZ)[x] is a monic polynomial of degree r which is irreducible modulo p. This definition depends only on the degree r of the irreducible polynomial, but not on the choice of/. If R is an algebra over a field F with a subset T, then the subalgebra (T) generated by T is the linear space spanned by all products tit2 • • • tn, where tt G T. The algebra R1 is a direct sum of the ideal R and subalgebra (1) = F. The ideal [right ideal, left ideal) generated by T in R is the linear space id(T) (id r (T), id z (T)) spanned by the set R^TR1 (respectively, TR1, R}T). If R has an identity, then every one-sided ideal of R considered as a ring is also a vector space. If there exist positive integers n such that nx = 0 for all x in the ring R, then the smallest n with this property is called the characteristic of R. If there are no positive integers like this, then R is said to have characteristic zero. The characteristic of R is denoted by char(iZ). If / is an ideal of R, then the set of all classes x + I, for x £ R, with addition (x+I)+(y+I) = (x+y)+I and multiplication (x+I)(y+I) = xy+I forms a ring called the quotient ring of R modulo / and denoted by R/I. The ring R is said to be an ideal extension of / by R/I. An idempotent of a ring as an idempotent of its multiplicative semigroup. Two idempotents e and / are said to be orthogonal if ef = 0. An element x is nilpotent if xn = 0 for a positive integer n. A ring is nil (nilpotent) if its multiplicative semigroup is nil (respectively, nilpotent). Denote by o the circle composition on R defined byxoy = x + y — xy, for x,y £ R. Then (R, o) is a monoid with identity 0, called the adjoint monoid of R. An element y is called a right (left) quasi-inverse of x if x + y + xy = 0 (x + y + yx = 0). A ring is quasiregular if its adjoint monoid is a group or, equivalently, every element is quasiregular. The Jacobson radical J(R) of R is the largest quasiregular ideal of R. If R is a ring with 1, then the group of units of R is the largest subgroup of the multiplicative semigroup of R with identity 1. A ring R is said to be Baer radical if every nonzero homomorphic image of R has a nonzero nilpotent ideal. Every ring R has a largest Baer radical ideal B(R). It is called the Baer or prime radical of R. It coincides with the

13

Rings

set of all strongly nilpotent elements of R, i.e., elements x with the property that, for each sequence xi,x%, • • • defined by x\ = x, Xi+\ = xtyiXi, for some 2/j € R, i = 1,2,..., there exists a positive integer n such that xn = 0. The Baer radical contains all one-sided Baer radical ideals of R. The quotient ring R/B(R) does not have nonzero nilpotent ideals. A ring is said to be locally nilpotent if every finitely generated subring of it is nilpotent. The Levitzki radical of R is the largest locally nilpotent ideal C(R) of R. It contains all one-sided locally nilpotent ideals. The Brown-McCoy radical BM(R) of R is the intersection of all ideals I of R such that R/I is a simple ring with an identity element. The following inclusions hold for every ring R: B{R) C C(R) C J(R)

C BM(R).

(1.3)

These inequalities may be strict for some rings. However, in every finite ring and finite dimensional algebra all of the radicals mentioned above coincide with the largest nilpotent ideal of the ring or algebra. The first example of a nil ring which is not locally nilpotent was given by Golod. The longstanding Koethe problem is whether a ring without nonzero nil ideals has no nonzero one-sided nil ideals. If the Jacobson and prime radicals of every homomorphic image of R coincide, then R is called a Jacobson ring. A radical g is said to be hereditary if g(I) = I n g(R), for every ring R with an ideal I. The Jacobson, Baer and Levitzki radicals are hereditary. A ring R is said to be semisimple or g-semisimple if g(R) = 0. For every radical g, the quotient ring R/g(R) is semisimple. If g is the Jacobson or the Baer radical, then a p-semisimple ring is called a semiprimitive or semiprime ring, respectively. Note that a ring is semiprimitive (semiprime) if and only if it has no nonzero quasiregular (respectively, nilpotent) ideals. Semiprimitive rings are also often called Jacobson semisimple (or semisimple) rings. A ring is right Artinian (Noetherian) if every descending (ascending) chain of right ideals in the ring stabilizes. A semiprimitive ring is right Artinian if and only if it is left Artinian, and so it is called a semisimple Artinian ring. A ring R is semilocal if R/J(R) is Artinian. A ring R is said to be right (left) T-nilpotent if, for every sequence of elements r±, 7-2,... of R, there exists n such that rn ... r\ = 0 (respectively, r\... rn = 0). A semilocal ring is semiprimary (right perfect, left perfect) if J(R) is nilpotent (right T-nilpotent, left T-nilpotent).

14

Preliminaries

Theorem 1.19 is nilpotent.

The Jacobson radical of every right or left Artinian ring

Theorem 1.20 nilpotent.

Every nil ideal of every right or left Noetherian ring is

Theorem 1.21 A ring is semisimple Artinian if and only if it is a direct sum of a finite number of matrix rings over division rings. An abelian group M is called a (left) R-module if the product rx € M is defined for all r G R, x £ M, and the following laws hold: (rs)x = r(sx), r(x + y) = rx + ry and (r + s)x = rx + sx, for all r,s € R, x,y S M. If R is a ring with identity element 1, then we assume that each module is unitary, which means that 1 acts as identity on all elements of M (that is lx = x for all x £ M). A module M is irreducible if it is nonzero and has no nonzero proper submodules. A left i?-module M is faithful if, for every r € R, rM = 0 implies r = 0. A ring is primitive if it has a faithful irreducible module. Theorem 1.22 primitive rings.

(i) Every semiprimitive

ring is a subdirect product of

(ii) If R is a primitive ring, then there exists a division ring F such that either R is isomorphic to a matrix ring Fn or, for each positive integer m, R has a subring Sm which maps homomorphically onto Fm. A ring R is prime IJ ^ 0 for every nonzero ideals I, J of R. An ideal J of R is said to be prime (semiprime) if JK % I (respectively, J 2 ^ 0) for every nonzero ideals J, K of R. An ideal / of R is prime (semiprime) if and only if R/I is prime (semiprime). Theorem 1.23 rings.

Every semiprime ring is a subdirect product of prime

A class K. of rings is said to be closed under right ideals (left ideals, homomorphic images) if, for every ring R G /C, the class K contains all right ideals (left ideals, homomorphic images) of R. We say that K. is closed under (finite) sums of one-sided ideals if /C contains every ring which is a (finite) sum of its right ideals or is a (finite) sum of its left ideals belonging to /C. A class K. is said to be closed under ideal extensions if K. contains every ring R such that I € K. and R/I £ K, for an ideal I of R.

15

Rings

Lemma 1.2 The classes of semilocal, semiprimary, right perfect, left perfect, nilpotent, locally nilpotent, right T-nilpotent, left T-nilpotent rings, strongly nilpotent, quasiregular ring and Pi-rings are closed under ideal extensions, right and left ideals, homomorphic images and finite sums of one-sided ideals. Lemma 1.3 The class of all rings with nilpotent Jacobson radicals is closed under left and right ideals, ideal extensions and finite sums of onesided ideals. Lemma 1.4 The class of Jacobson rings is closed under ideal extensions, right and left ideals, homomorphic images and sums of one-sided ideals. The classes of right Artinian or Noetherian rings are not closed under finite sums of left ideals, as the semigroup ring Q[-B] of the two-element right zero band B = {a, 6} shows. Lemma 1.5 ([Kelarev (1993e)]) The class of right Artinian (right Noetherian) rings is closed under finite sums of right ideals. Suppose that the set S = spec(i?) of prime ideals of R satisfies the ascending chain condition. Define the sets Sa inductively. Let So be the set of all maximal elements in S; and for each ordinal a denote by Sa the set of all s S S such that t € S,t > s imply t £ Sp for some f3 < a. Then there exists the least ordinal a such that Sa = S. This ordinal is called the classical Krull dimension of R. It is denoted by cl—K—dim(i?). Uniform dimension or Goldie dimension of a ring is a maximal integer n such that the ring has a right ideal which is a direct sum of n nonzero right ideals. If a maximal integer with this property does not exist then the uniform dimension is assumed to be equal to infinity. The right annihilator of a subset X of the ring R is the set Ann r (X) = {r e R \ Xr = 0}.

(1.4)

A ring that satisfies the ascending chain condition on right annihilators and has finite right Goldie dimension is called a right Goldie ring. A Pi-ring (or Pi-algebra) is a ring or, respectively, algebra satisfying a polynomial identity. Every Pi-ring (or Pi-algebra) satisfies a multilinear identity, i.e., an identity of the form x \ . . . xn -t-

y l^o-e Sym(n)

KuX(j\...

X(jn — 0,

(1.5)

16

Preliminaries

where Sym(n) is the symmetric group, ka are integers (elements of the field in the case of algebras, see [Rowen (1980)]). A variety of rings is a class of all rings satisfying some fixed set of identities. The following theorem holds not only for rings, but for any algebraic structures. Theorem 1.24 (Birkhoff's Theorem) A class of rings is a variety if and only if it is closed for homomorphic images, subrings and direct products.

Chapter 2

Graded Rings

2.1

Groupoid-Graded Rings

Let G be a groupoid. An associative ring R is said to be G-graded, or graded by G, if R = ®geaRg is a direct sum of additive subgroups Rg and RgRh Q Rgh, for all g, h £ G. If G is a group or semigroup, then R is called a group-graded or semigroup-graded ring. An algebra R = ®geoRg ° v e r a field F is said to be G-graded if i? is a direct sum of the F-vector spaces Rg, g € G, and RgRh Q Rgh, for all g,h £ G. The concept of a groupoid-graded ring was first mentioned explicitly in [Schiffels (I960)]. Groupoid-graded rings include as special cases many other ring constructions: polynomial and skew polynomial rings, direct and semidirect products, matrix and structural matrix rings, Rees matrix rings, Morita contexts and generalized matrix rings, group and semigroup rings, skew group and semigroup rings, twisted group and semigroup rings, monomial rings, smash products, crossed products, group-graded rings, path algebras, edge algebras, incidence rings, etc. Therefore groupoid-graded rings can be applied to the study of other less general constructions. This not only gives new results for several constructions simultaneously, but also serves the unification of already known theorems. For example, in [Karpilovsky (1991)], Chapter 4, "most of what is known concerning the Jacobson radical of polynomial rings and skew polynomial rings (for example, all classical results of Amitsur on the Jacobson radical of polynomial rings) is obtained as an easy consequence of the corresponding results pertaining to the Jacobson radical of monoid-graded algebras". Other applications are given in the earlier survey papers (see, for example, [Jespers (1993a)], [Kelarev 17

18

Graded Rings

(1993c)], [Kelarev (1995d)], [Kelarev (1998d)], [Kelarev (1999b)], [Munn (1986)]). The general definition of a groupoid-graded ring is motivated by the following two circumstances. First, certain facts concerning ring constructions do not depend on the grading set being a group or a semigroup. Hence it is better to prove them without this irrelevant assumption. Surprisingly, the more general proof may actually turn out to be substantially shorter. The first example of a short proof of this sort using groupoids has been obtained in [Kelarev (1994a)] (see also [Kelarev (1995c)]). In particular, it is shown in [Kelarev (1995c)] that a ring R graded by a finite groupoid G is right perfect, semiprimary, semilocal, or is a Pi-ring if and only if all components Re satisfy the same property, for all idempotents e of G. Second, nonassociative groupoids are better suitable for grading nonassociative algebras, which are well known for various important applications. In the case of nonassociative algebras the associativity of the grading semigroup places serious restriction on the multiplicative structure of the algebra. We need the following technical concept equivalent to that of a groupoidgraded ring. If G is a partial groupoid, then we say that R = ®g^GRg is G-graded and R induces G if and only if the following two conditions hold: (i) RgRh C Rgh whenever gh is defined; (ii) RgRh =/= 0 implies that the product gh is defined. Then G is called the partial groupoid induced by R. Let R be a ring, G a finite set, and let R be a direct sum of additive subgroups Rg where g G G. If the set H(R) = \JgeaRg of all homogeneous elements of ii is closed under multiplication, then R is called a homogeneous sum of the Rg. If RgRh =/= 0 for some g,h G G, then there exists a unique element u in G such that RgRh Q Ru- Therefore we can introduce a partial operation on G by putting gh = u for all triples g,h,u G G such that 0 ^ RgRh C Ru. Then G becomes a partial groupoid and R is graded by G. Of course, we can define all the other products in G arbitrarily, and make G an ordinary groupoid and R a groupoid-graded ring. However, the products of the partial groupoid induced by R are determined uniquely and convey more essential information. Discussing properties of elements of partial groupoids we use the standard terminology assuming that the corresponding property holds whenever all the necessary products are defined. In particular, we say that a partial groupoid G is cancellative if each of the equalities xz = yz or zx = zy

Groupoid-Graded

Rings

19

implies x = y, for x,y,z £ G. The components Rg are called homogeneous components and elements r £ Rg, for some g £ G, are called homogeneous elements of the ring. Each element r £ R has a unique decomposition as a sum r = Ylg£Gr9 with r 9 s i? 9 , where only a finite number of the rg are nonzero. The element rg is called the projection of r on the component Rg. If T C S, then put i?r = ®terRt and r^ = ] £ t g T r t . For I C R and g £ G, put 7g = I H Rg and IT = I ^ RT- Further, for r ^ 0, denote the set of all nonzero homogeneous summands rg of r by H(r). The support of r is the set supp(r) = {g £ G \ rg ^ 0}. Put tf (0) = 0 and supp(0) = 0. Clearly, H(r) and supp(r) are finite sets. Let H(I) = LV e ji7(r). Then H(R) is the set of all homogeneous elements of R. Evidently, H(R) is a multiplicative subsemigroup of R. By the length of r we mean | supp(r)|. A graded ring R is said to have a finite support if only a finite number of the homogeneous components of Rg are nonzero. An element r of a graded ring R is rigid if xry = 0 •$=> rrrty = 0, for each t £ supp(r) and all x, y £ H(R). It is routine to verify that if the ring R is graded by a cancellative semigroup and J is an ideal of R, then all elements of minimal positive length in I are rigid. In particular, all homogeneous elements of J are rigid in this case. If G has a zero 0 and R is a G-graded ring with RQ = 0, then R is called a contracted G-graded ring. For any groupoid G, if I is an ideal of G and R is a G-graded ring, then the quotient semigroup G/I has a zero and R/Rj is a contracted G/J-graded ring. Similarly, if R = ®geoRg is a G-graded ring with a homogeneous ideal I, then R/I = © s 6 Gi? 9 /7 g is G-graded. A ring R = ®g^GRg is said to be strongly G-graded if RgRh = Rgh, f° r all g,h £ G. The grading of R is said to be non-degenerate if and only if the induced partial groupoid has an identity 1 and each of the equalities (T\R)I = 0 or (Rr)i

= 0 implies r = 0.

If R is a G-graded ring, then the largest homogeneous ideal contained in J(R) is denoted by J%T(R) or JQ{R) and is called the homogeneous part of the radical or the homogeneous radical. Let R = 0 s € G Rg be a G-graded ring. A left .R-module M = 0 s e G Mg is called a graded R-module if RgMh C Mgh, for all g,h £ G. Categories of modules over rings provide very relevant information on the structure of the rings (see [Anderson and Fuller (1992)]). This explains the interest in the investigations of properties of modules over graded rings. A graded left module is graded Noetherian if every ascending chain of

20

Graded Rings

graded submodules of the module terminates. Obviously, every left Noetherian module which is graded is also graded left Noetherian. The converse is not true in general (see [Dascalescu and Kelarev (1999)]). For a G-graded ring R, we denote by (G, R)-gi the category of G-graded i?-modules. If M is an object of this category, then the lattice of all subobjects of M in this category will be denoted by Lat(G ) fl)_ gr (M). Let K—dim(Gifl)_gr(M) denote the Krull dimension of this lattice (which is the same as the Krull dimension of M in the category (G, i?)-gr), whenever it exists. If M has Krull dimension as an i?-module, this dimension will be denoted by K—dimfl(M). The following properties of Krull dimension are well known. If N is a submodule of the .R-module M, then M has Krull dimension if and only if N and the quotient module M/N have Krull dimension, and in this case K-dim(M) = sup(K-dim(7V), K-dim(M/iV)). In particular, if M = 0 " = 1 Mi is a sum of a finite number of modules M» with Krull dimension, then M has Krull dimension and K—dim(M) = sup K—dimM;. i=\

2.2

Semigroup-Graded Rings

The Wedderburn's classical theorem tells us that every finite dimensional algebra over a perfect field is a semilattice-graded ring with two components: the radical and a semisimple subalgebra (see, for example, [Curtis and Reiner (1962), Theorem 72.19] or [Pierce (1982), §11.6]). The following interesting example is also due to Wedderburn. Every matrix ring Fn over a field F of characteristic zero contains idempotents £tj, where 1 < i,j < n, such that Fn = (BijFstj and SijSke € Feu for all 1 < i,j, k,£
Group-Graded

Rings

21

band-graded rings allow us to carry over the information from R[Sy] to R[S]. This method has been used by many authors; for example, [Wauters (1986)] deals with several applications of this idea (see also the survey [Kelarev (1999b)]). Let R = (BS£sRs be an 5-graded ring. Following [Cohen and Montgomery (1984)], we say that the grading is faithful if, for any s,t € 5 and r G Rs, each of the equalities rRt = 0 and Rtr = 0 implies that r = 0. Examples of faithfully graded rings are semigroup crossed products and semigroup algebras. The following substantially more general concept has been defined in [Munn (2002)]. An 5-graded ring R = @sesRs is T>-faithful if, for all x,y € 5 with x, y, and xy in the same X>-class of 5, a e f i I \ 0 = ^ aRy ^ 0 and b € Ry \ 0 =4> Rxb ^ 0. If 5 is a nontrivial semigroup with zero, and F is a field, then the contracted semigroup ring Fo[S] is V-faithful, but not faithful 5-graded ring (see [Munn (2002)]). Let 5 = B be a band. If each ring Rb has an identity 1;,, and l a l b = \ab for any a, b, then the ring R = (BbeBRb is called a special band-graded ring, or a special B -graded ring. This concept was introduced by Munn in [Munn (1992)]. Let B be a semilattice, and let R = (B^s^b be a £?-graded ring. A J5-graded ring R = © b € B Rb is called a strong semilattice sum if, for all a < b in B, there exist homomorphisms /*:./?(,—> Ra such that (1) / j is the identity map for every b G B; (2) fin = fa for all a < b < c in B; (3) xy = f%b{x)fbab{y) for all a, b € B and x e Ra, y 6 RbEvery strong semilattice sum can be embedded in a special semilatticegraded ring by adjoining identities to all homogeneous components. 2.3

Group-Graded Rings

There are several excellent books and surveys devoted to group rings, crossed products, group-graded rings, incidence rings, semigroup rings, and loop rings. We are trying to avoid duplicating them and include only complementing or unifying new topics in this monograph. This section contains a few facts required for our proofs (see, for example, [Beattie and Jespers (1991)], [Cohen and Montgomery (1984)], [Cohen and Rowen (1983)],

22

Graded Rings

[Jensen and Jondrup (1991)], [Kelarev (1993e)], [Kelarev (1995c)], [Passman (1989)] for details). Let us begin with one of the most useful theorems on radicals of group-graded rings due to Cohen and Montgomery [Cohen and Montgomery (1984)]. Theorem 2.1 ([Cohen and Montgomery (1984)]) Let G be a finite group of order n with identity 1, and let R be a G-graded ring. Then J(Ri)

= RiH J{R)

and J(R)n

C

JgI(R),

B(i?i) = Ri n B(R) and B(R)n C BgT(R), £ ( # i ) = Rx n L{R) and C{R)n C CgI{R). Theorem 2.2 ([Cohen and Rowen (1983)]) Let G be a finite group with identity 1, and let R be a G-graded ring. If I is a homogeneous ideal of R such that lDRi=0, then 7 | G | = 0. Theorems 2.1 and 2.2 yield the following Theorem 2.3 Let G be a finite group with identity 1, and let R be a Ggraded algebra. Then the radical J&t{R) (or Bgr(R), Csr(R)) is the largest homogeneous ideal of R with the property that J&{R) n R\ = J{R\) (respectively, BEY{R) n Ri = B(Ri), £Sr(R) n Rx = C{R{)). Theorem 2.4 ([Cohen and Montgomery (1984)]) Let G be a finite group with identity 1, and let R be a G-graded ring. (i) If P is a prime ideal of R, then there exist n < \G\ primes Q\, Qij • • •) Qn of R\ minimal over P D R\, and we have P C\ Ri = Q1nQ2n...nQn. (ii) If P C Q are prime ideals of R and P ^ Q, then PnRi ^Qr\R\. The following theorem was proved in [Beattie and Jespers (1991)] even in the more general case of group-graded rings with finite supports. Theorem 2.5 ([Beattie and Jespers (1991)]) Let G be a finite group with identity \, and let R be a G-graded ring. Then R is semilocal (semiprimary, right perfect, left perfect, nilpotent, right T-nilpotent, left T-nilpotent) if and only if R\ satisfies the same property. Theorem 2.6 ([P assman (1989)]) Let G be a finite group, and let R be a G-graded ring. If Ri is Jacobson, then R is also a Jacobson ring.

23

Superalgebras

Theorem 2.7 ([Kelarev (1993e)]) Let G be a finite group, and let R = ©geG &9 be « G-graded ring. If Ri is a Pi-ring, then R is also a Pi-ring. Proof. It is known that R is embeddable in a generalized n x n matrix ring A = (BijAij, where n = |G|, such that every An is isomorphic to Re. (As A we can take the smash product of R and Z(G).) Note that A,j is also a Pi-ring for i ^ j , because Af, = 0. Therefore Rj = Q^Rij is a sum i

of its right ideals i?^, and by Lemma 1.2, Rj is a Pi-ring. Since R is a sum of the left ideals Rj, R is a Pi-ring, as well. It follows that A is a. Pi-ring, and so the same can be said of R. • Lemma 2.1 Let G be a group, N a normal semigroup of G, and R a G-graded algebra. Then R = ®gNeG/NRgN ?s G/N-graded. Lemma 2.2 Let G be a group, R a G-graded ring, and let H be a subgroup ofG. Then J(RH) D RH n J(R). 2.4

Superalgebras

An algebra graded by the group Z2 = {0,1} is called a superalgebra. A Lie superalgebra is a superalgebra L = Lo + L\ such that, for all a, b € {0,1}, x € La, y € Lt,, z G L, two identities hold: [x,y] = {{x,y},z} =

-(-l)a\

[x,{y,z]}-{-lTb{y,[x,z)\.

Lie superalgebras have received considerable attention in the literature in view of important applications, in particular, related to physics. Let F be a field with characteristic ^ 2,3, F* = F\{0], and let G be an additive abelian group. Consider a bilinear alternating form s : G x G —* F*, i.e., a function satisfying e(g + h,k) =

e(g,k)e(h,k),

e(g,h + k) =

e(g,h)e(g,k),

e(g,h)

=e(h,g)~1,

24

Graded Rings

for g,h,k e G. A G-graded algebra L = ®g&GLg is called a color Lie superalgebra if, for all a S Lg, b € L^, c £ L, [a,b] =

-e(g,h)[b,a],

[[a, b],c} = [a, [6, c]] - £(5, /i)[6, [a, c]]. If G = 0, then we get an ordinary Lie algebra; if G = Z 2 , £(0,0) = £(0,1) = e(l, 0) = 1, e(l, 1) = —1 then L is a Lie superalgebra. Every associative superalgebra R = R0 + R1 gives us a Lie superalgebra with multiplication defined by [x,y]=xy-(-l)abyx,

(2.1)

for all a,b £ {0,1}, x £ Ra, y e Ry. A general way of obtaining color Lie superalgebras is to introduce a commutator on a G-graded associative algebra R = ®g€GRg by setting [x,y] = xy - e(x, y)yx. We refer to [Bahturin et al. (1992)] and [Montgomery (1993)] for earlier results on Lie superalgebras and relevant concepts. The associative enveloping algebras of Lie superalgebras or, more generally, color Lie algebras are graded by finite groups (see, for example, [Bergen and Passman (1995)]). Superalgebras have been used, for example, in the solution of the famous Specht problem by Kemer ([Kemer (1988)]), and in the proof of a semigroup-graded analogue of the well-known NagataHigman theorem ([Chanyshev (1990)]).

Chapter 3

Examples of Ring Constructions

This chapter contains the definitions of several ring constructions: direct, subdirect and semidirect products, group and semigroup rings, monomial rings, crossed products, polynomial and skew polynomial rings, skew group and semigroup rings, twisted group and semigroup rings, power and skew power series rings, Laurent series ring, edge and path algebras, matrix rings and generalized matrix rings, triangular matrix representations, Morita contexts, Rees matrix rings, smash products, structural matrix rings, and incidence algebras. It is explained how these constructions can be regarded as groupoid-graded rings.

3.1

Direct, Subdirect and Semidirect Products

If Ri, i £ / , is a collection of rings, then the direct product Yiiei Ri i s the set of all functions / : J —> U,e/.R; such that f(i) £ Ri, with operations defined by (f + g)(i) = f(i)+g(i) and (fg)(i) = f(i)g(i). A ring Q C RisI Ri is called a subdirect product of the rings Ri, i £ I, if Q(i) = Ri for all i £ I. A direct sum of the rings Ri,i £ I, is the set of all functions f £ Q such that only a finite number of images /(z) are nonzero for i £ I. Obviously, each direct sum is G-graded for any groupoid G defined on the set I. A ring R is called a semidirect product of its ideal RQ and a subring R\ if R = i?i © RQ, that is, the additive group of R is a direct sum of the additive groups of R\ and RQ. If we denote the two-element semilattice by Y2 = {0,1}, then it is clear that R = ©Sgy2-R5 is a semilattice-graded ring. 25

26

3.2

Examples of Ring

Constructions

Group and Semigroup Rings, Monomial Rings

Let R be a ring, and let 5 be a semigroup. The semigroup ring R[S] consists of all sums of the form ^2seS rss, where rs £ R for all s £ S, and only a finite number of the coefficients rs are nonzero, with addition and multiplication defined by the rules

H r»s + J2 r'«s = J2^s + r*)s> s£S

s€S

\s£S

J \tes

seS

J

s,tes

If G is a group (monoid), then R[G] is called a group ring (respectively, monoid ring). If S is a semigroup with zero 6, then the contracted semigroup ring RQ[S] is the quotient ring of R[S] modulo the ideal R6. Thus Ro[S] consists of all finite sums of the form Y^7=iriSi> w n e r e rt £ R, 8 ^ s, S S, and all elements of R6 are identified with zero. If R is a field, then we use the terms semigroup algebra, contracted semigroup algebra, monoid algebra, contracted monoid algebra, and group algebra. Theorem 3.1 (Maschke's Theorem) Let F be a field, and let G be a finite group. Then F[G] is an Artinian ring which is semisimple if and only if char(F) does not divide \G\. Theorem 3.2 Let F be a field of characteristic p > 0, and let G be a finite p-group. Then the radical of F[G] is equal to the augmentation ideal n

LO(F{G\)

n

= {J2fi9i \fi£F,gi£G,YJfi i=l

= 0}i=\

Lemma 3.1 Let S be a semigroup such that the semigroup algebra C[S] is semilocal (semiprimary, right perfect, left perfect). Then every subgroup of S is finite and every nil factor of S is locally nilpotent (respectively, nilpotent, right T-nilpotent, left T-nilpotent). Let X* be the free monoid generated by a set X, and let M C X*. Elements of X* are called monomials. Denote by I the ideal generated by M in the polynomial ring R[X\. The quotient ring R[X]/I is called a monomial ring. It is isomorphic to the contracted semigroup ring RQ[S] of

Crossed

27

Products

the Rees quotient monoid S = X*/(X*MX*). If / is an ideal generated by monomials in the commutative ring R[xi..., xn], then the quotient ring R[xi... ,xn]/I is also called a monomial ring. The monograph [Villarreal (2001)] is devoted to monomial algebras. Every semigroup ring R[S] is an S-graded ring with components Rs, where s € S. US has zero 0, then the contracted semigroup ring is also graded by S with components Ro = 0 and Rs = Rs, for 0 ^ s s S. If B is a band and S = UfcgsSb is a band B of semigroups Sb, then the semigroup ring R[S] = ®beBR[Sb] is a 5-graded ring. If, in addition, R and all the Sb have identities, then R[S] is a special band-graded ring. We refer to [Passman (1977); Passman (1986)] and [Okninski (1991)] for detailed accounts on group and semigroup algebras, respectively.

3.3

Crossed Products

Let R be a ring with 1, and let S be a semigroup. A crossed product R* S of S over R consists of all finite sums Y^ses r «* with addition and multiplication defined by the distributive law and two rules xy xr

= =

r(x,y)xy r

a{x)

x

(twisting)

(3.1)

(action)

(3.2)

such that R is associative and the set S forms a multiplicative subgroup of R isomorphic to S. Every crossed product R * S is an 5-graded ring with components Ra = i?s. We refer to [Passman (1986); Passman (1989)] for further information on crossed products.

3.4

Polynomial and Skew Polynomial Rings

if xn = K 1 • • > 0} is a free commutative monoid with n generators, then 7?[Xn] is called a commutative polynomial ring in n variables. It is also denoted by R[xi,... ,xn], if it is clear from the context that commutative rings are considered. If X* is the free (noncommutative) monoid generated by X = {xi,... ,xn}, then i?[X*] is called a polynomial ring in n noncommuting variables. This ring is also denoted by R[xi,...,xn], when it is clear from the context that noncommutative rings are considered.

28

Examples of Ring

Constructions

A (left) skew derivation on a ring R is a pair (a, 6), where a is a ring endomorphism of R and 5 is a (left) a-derivation on R, that is, an additive homomorphism from R to itself such that S(ab) = a{a)6(b) + S(a)b,

(3.3)

for all a,b G R. If (a, 6) is a skew derivation on R, then the skew polynomial ring (or Ore extension) is denoted by R[S; a, 5]. It consists of all finite sums 5Zr=o rix%i where r* G # , with multiplication defined by the distributive law and the rule xr = a(r)x + S(r),

(3.4)

for all r £ R. Many quantized algebras can be expressed in terms of iterated skew polynomial rings (see [Goodearl (1992)]). The skew Laurent polynomial ring R(x; a, 6) is the set of finite sums Y^i=m.riX'> where fi G R and m , n G Z, n > m. If a = 1, then the ring R[x; 5] = R[x; 1,6} is called a skew polynomial ring of derivative type, and R(x; 6) = R(x; 1,6) is called a skew Laurent polynomial ring of derivative type. If a = 1 and 6 = 0, then we get the usual polynomial ring R[x] = R[x; 1,0] and Laurent polynomial ring R(x; 1,0). Let S be any semigroup, and let / be a homomorphism from the infinite monogenic monoid x+ = {x}+ into S. Then every skew polynomial ring of automorphism type R[x; a] = (BsesRs is an 5-graded ring with components Rs = X3/fx')=« R-xli where the sum is assumed to be equal to zero if there are no elements xl with f(xl) = s. A similar assertion is true for every skew polynomial ring of automorphism type in n commuting variables: R[xi,x2,...,xn;a]

=

R[xi;a}[x2;o-}...[xn;a],

where the automorphism a naturally extends to an automorphism of each ring R[xi,... ,Xk\o~], for k = 1 , . . . ,n — 1. 3.5

Skew Group and Semigroup Rings

Let R * S be a semigroup crossed product with twisting r and action a. If the twisting r is trivial, that is T(X, y) = 1 for all x,y G S, then R * S is denoted by R[S;cr] and is called a skew semigroup ring. If 5 is a group (or monoid), then this ring is called a skew group ring (respectively, skew monoid ring).

Twisted Group and Semigroup

29

Rings

Suppose that a is a homomorphism of a semigroup S into the endomorphism monoid (automorphism group) of the ring R. The skew semigroup ring of endomorphism (respectively, automorphism) type R[S; a] consists of all finite sums Y^i=i r»s*> w n e r e r% G -R, Sj G S, with multiplication defined by the distributive law and risi • r2s2 = ria(s1)(r2)s1s2,

(3.5)

for all r\,r2 G R, Si,s2 G S. If 5 is a group, then R[S;a] is called a skew group ring of automorphism type. Every skew semigroup ring R[S; a] is an S-graded ring with components Rs, where s G S. If S has zero 0, then the contracted semigroup ring is also graded by S with components Ro = 0 and Rs = Rs, for 0 ^ s G S. 3.6

Twisted Group and Semigroup Rings

Let R * S be a semigroup crossed product with twisting r and action a. If the action a is trivial, i.e., a{x) = x for all x, then R * S is denoted by i?T[5] and is called a twisted semigroup ring. If S is a group (or monoid), then this ring is called a twisted group ring (respectively, twisted monoid ring). Twisted group and semigroup rings are most often considered in the case where R is a field. The general definition of twisted group and semigroup rings above leads to the following concept of a cocycle. Let S be a semigroup, C a commutative semigroup, and let r : S x S —> C be a mapping. Denote by ET or ET (C, S) the set C x S with multiplication defined by the rule (c, s)(d, t) = (cdr(s, t), st) for all c,deC,s,t£

S.

(3.6)

The mapping T is called a cocycle if the following cocycle equation is satisfied: T(s,t)r(st,u)

= T(s,tu)T(t,u),

(3.7)

for all s, t, u € S. The cocycle equation implies the associativity of ET. If C is cancellative, then the set ET is a semigroup if and only if T is a cocycle. The semigroup ET is called the extension semigroup. Thus, if R is a field, then the associativity of the twisted semigroup ring RT \S] is equivalent to r being a cocycle. It is worth noting that the cocycle equation arises in

30

Examples of Ring

Constructions

the topology of surfaces, in quantum mechanics, in group cohomology, in combinatorial designs, and constructing error-correcting codes (see [Baliga and Horadam (1995)], [de Launey et al. (2000)], [Horadam (1996)], [Horadam (2000)], [Horadam and de Launey (1993)], [Horadam and de Launey (1995)], [Horadam and Perera (1997)] and [Horadam and Udaya (2000)] for details).

3.7

Power and Skew Power Series Rings

If [a, 5) is a skew derivation on R, then the ring R[[x; a, 6]] of skew power series of R is the set

R[[x; a, 6}} = 0

'n

G R

and the ring of skew Laurent series of R is the set

R((x,a,S))

=< ^

nxl | n e R,m £ Z

I i>m

with multiplication defined by (3.4). If a = 1, then the ring i?[[x;£]] = R[[x; 1,5]] is called a skew power series ring of derivative type, and the ring R((x;5)) = R((x;l,5)) is called a skew Laurent series ring of derivative type. If 5 = 0, then the ring ^[[ZJCT]] = R[[x; a, 0]] is called a skew power series ring of endomorphism type, and R((x;5)) = R((x;l,S)) is called a skew Laurent series ring of endomorphism type. If, in addition, a is an automorphism, then these rings are called the ring of power and Laurent series of automorphism type. If a = 1 and (5 = 0, then we get the usual power series ring R[[x]] = R[[x; 1,0]] and R((x; 1,0)). Let S be any semigroup, and let / be a homomorphism from the infinite monogenic monoid x+ into S. Then every skew power series ring of automorphism type i?[[x;cr]] = (BsesRs is a n 5-graded ring with components Rs = ^2f(xi)=s Rxl, where the sum is assumed to be equal to zero if there are no elements xl with f(xl) = s. This transfers to all skew power series rings of automorphism type in commuting n variables: R[[xi,x2,

...,x„;o-]] = R[[xi;
[[xn;a]],

31

Edge and Path Algebras

where the automorphism a naturally extends to an automorphism of each ring R[[xi,..

3.8

.,Xk;a]],

for k = 1 , . . . , n - 1.

Edge and P a t h Algebras

Let F be a field. Given a graph G = (V,E) with vertices {xi,...,xn}, the edge algebra associated with G is the subalgebra F[G] of the commutative algebra F[x\,... ,xn] generated by the monomials XiXj for all edges (xi,Xj) G E. Since every edge algebra is a homogeneous subalgebra of the commutative polynomial ring, it can be graded by any commutative semigroup. Let F be a field, and let D = (V, E) be a directed graph. A path in D of length n > 0 is a sequence of vertices Vo, v\,..., vn such that (i>j_i, Vi) G E, for all i = 1 , . . . ,n. In particular, a single vertex is a path of length 0. Denote by P(D) the set of all paths of D, and introduce a multiplication on P°(D) = P(D) U {0} by putting, for any two paths u = ui,...,um and v = v1,...,vn, _

( M l , . . . ,Um,V2,..

10

.,Vn

if i um = vi otherwise.

(3.8)

The path algebra of D over F is denoted by FD and is defined as the contracted semigroup ring F0[P°(D)]. The study of path algebras, in particular, path algebras of quivers and their homomorphic images, plays essential role in the representation theory.

3.9

Matrix Rings and Generalized Matrix Rings

Let R be a ring, and let I be any set. The set of all matrices (upper triangular matrices) with entries in R and rows and columns indexed by the elements of I will be denoted by Mi(R) (respectively, Ui(R)). If |7| = n is a positive integer, then this set is denoted by Mn(R) (respectively, Un(R)), and it is known that it forms a ring with respect to the usual matrix multiplication. This ring is called the full matrix ring (upper triangular matrix ring) over R. A matrix is row (column) finite if each of its rows (columns) has a finite number of nonzero entries. The set of all row (column) finite matrices of MT(R) forms a ring denoted by MJ rf) (i?) (respectively, M$o!)(R)) and

32

Examples of Ring

Constructions

called the ring of row (column) finite matrices over R. The intersection M 7 (rcf) (i?) = Mff){R)nMff)(R) is an ideal in both M^(R) and M\C1)(R). It is called the ring of row and column finite matrices over R. A matrix is row (column) bounded if it has a finite number of nonzero columns (rows). The set of all row (column) bounded matrices of Mi(R) is a ring. It is denoted by MJ rb) (i?) (respectively, MJ cb) (i2)) and called the ring of row (column) bounded matrices over R. The intersection M\ (R) — Mf (R) D Mj (R) is the ring of matrices with finite numbers of nonzero entries. A ring R = ®™j=1Rij is called a generalized matrix ring if

RijRk,i

c

Ri,e 0

if j = k; otherwise.

It is said to be a generalized upper (lower) triangular matrix ring if Rij = 0 for all 1 < j < i < n (respectively, 1 < i < j < n). A generalized lower triangular 2 x 2 matrix ring was used to construct examples of nonsingular rings (see [Goodearl (1976)]). Generalized row finite matrix rings are defined in a similar fashion, as well as column finite, row and column finite, row bounded, column bounded matrix rings. All of these rings are graded by Brandt semigroups, finite groups, and rectangular bands. To illustrate the standard grading methods, consider the ring K = Mn(R). Recall that the Brandt semigroup Bn consists of 0 and all standard matrix units. Putting Ko = 0 and Ks = Rs for 0 ^ s £ Bn, we see that K = © s £ B Ks is graded by the Brandt semigroup Bn. Take any group G of order n. In order to index the rows and columns of K by the elements of G, we introduce a one-to-one mapping from { 1 , . . . ,n} to G. The z-th row and i-th column are indexed by 4>(i). For g e G, put Lg = E ^ - i ^ O f f ^ . r T h e n K = ®geGL9 i s a G-graded ring. If H = {(i,j) | 1 < i,j < n} is a rectangular band, then K = ®heH Ph is an i7-graded ring, where Ph = Reitj. There are also other ways of grading Mn(R) by rectangular bands, as mentioned in Section 2.2.

Triangular Matrix Representations

3.10

33

Triangular Matrix Representations

If R is a ring with 1 graded by the Brandt semigroup Bn, and if i? 0 and all components Rij = Rei . are equal to zero for all i > j , then R is also called a generalized triangular matrix representation of R, since it can be viewed in matrix notation. An internal characterization of generalized triangular matrix representations of rings in terms of sets of left triangulating idempotents has been obtained, and a number of important properties of these rings have been described in [Birkenmeier et al. (2000)]. Diverse applications of generalized triangular matrix representations occur in operator theory, quasitriangular Hopf algebras, and various Lie algebras (Kac-Moody, Lie, Virasoro, and Heisenberg). A variety of conditions have been used in order to obtain generalized triangular matrix representations of algebras (see [Birkenmeier et al. (2000)] for references). Triangular matrix representations are also graded by groups, Brandt semigroups, and rectangular bands, as illustrated in Section 3.9.

3.11

Morita Contexts

If R = ®seB2Rs is a contracted i?2-graded ring, then Ren

) -*^ei2 : *^e2\ > •*^e22j

is called a Morita context. Conversely, if [P, U, V, Q] is a Morita context, and we put Reil = P, Rei2 = U, Re21 = V, Re22 = Q, then R = ®s^B2Rs is a contracted 5 2 -graded ring. Thus the definition of a Morita context is equivalent to the definition of a contracted i?2-graded ring. Morita contexts play crucial roles in the investigations of categories of modules over rings.

3.12

Rees Matrix Rings

Let R be a ring, I and A two sets, and let P be a A x /-matrix with entries in R. A Rees matrix ring over R with sandwich matrix P is the set M(R; I, A; P) consisting of all J x A matrices over R, with finite numbers of nonzero entries, equipped with the usual addition and multiplication • defined by A • B = APB, for A,B £ M(R; I, A; P). Row finite (column finite, row and column finite, row bounded, column bounded) Rees matrix

34

Examples of Ring

Constructions

rings are defined similarly. These rings are graded by groups, rectangular bands, and Brandt semigroups, see Section 3.9.

3.13

Smash Products

Let G be a finite group, and let R be a G-graded ring. Then the smash product R#Z[G*] of R and Z[G*] is the collection of all matrices M with rows and columns indexed by the elements of G and such that Mg>h € Rgh-\, for all g,h £ G. The ring R embeds in i?#Z[G*] by the mapping •K such that n(r)gth = rgh-i, for all g,h £ G. The whole smash product is generated by R and the standard matrix units pg = e 5;5 , for g € G. If R is an algebra with 1 over a field F, then we can also define the smash product R#F[G*} of R and G as the free left and right i?-module with a basis {pg \ g £ G} consisting of pairwise orthogonal idempotents whose sum is 1 and with multiplication given by the rule (apg)(bph) = abgh-iph.

(3.9)

Each smash product is a generalized matrix ring, and therefore it is graded by a group, a rectangular band, and a Brandt semigroup.

3.14

Structural Matrix Rings

Let R be a ring with 1, and let D = (V, E) be a directed graph with the set V = { 1 , 2 , . . . , n) of vertices and the set E of edges without multiple edges but possibly with loops. Edges of D correspond to the standard matrix units of the algebra Mn(R) of all (n x n)-matrices over F. Namely, for (i,j) £ E C V x V, let e (»>j) = e*>J — eiJ ^ e the standard matrix unit. Denote by MD(F) = ®weEFew the set of all matrices with nonzero entries corresponding to the edges of the graph D, and zeros in all entries for which there are no edges in D. It is well known and easy to verify that MD{F) is a subalgebra of Mn(F) if and only if D satisfies the following property (x,y),(y,z)

€ E =*• (x,z) £ E,

(3.10)

Incidence

Algebras

35

for all x,y,z € V. In this case the MD(F) is called a structural matrix ring. Structural matrix rings have been investigated by a number of authors, and many interesting results have been obtained (see, for example, [Dascalescu et al. (1999)], [Dascalescu and van Wyk (1996)], [Green and van Wyk (1989)], [van Wyk (1996)], [van Wyk (1996)], [van Wyk (1999)], [Veldsman (1996)]). Note that every structural matrix ring can be regarded as a semigroup ring. A graph D = (V, E) defines a structural matrix ring if and only if the set SD =

{0}U{eij\(i,j)GE}

forms a semigroup, and both of these properties are equivalent to condition (3.10). Then the structural matrix ring MD(F) is isomorphic to the contracted semigroup ring -POISD]The following definitions are used in the investigations of structural matrix rings. Let D = (V, E) be a graph. The in-degree and out-degree of a vertex v &V are defined by indeg(w) = \{w e V \ (w,v) € E}\, outdeg(u) = \{w G V | (v,w) € E}\. A vertex of D is said to be a source if indeg(u) = 0 and outdeg(v) > 0. A clique of D is a maximal complete subgraph of D. A graph is acyclic if it has no directed cycles.

3.15

Incidence Algebras

Let R be a ring, and let X be a partially ordered set. For x, y e X, the interval between x and y is the set [x,y] = {z \ x < z < y}. A partially ordered set is locally finite if every interval [x, y] is finite, for all x, y S X. The set I(X,R)

= {f:XxX-+R\

f(x,y)

= 0 if x £ y}

with operations

(f + g)(x,y) =

f(x,y)+g(x,y),

36

Examples of Ring

fg(x,y)=

^2

Constructions

f(x,z)g(z,y)

and

z£[x,y]

(rf)(x,y)

=

rf(x,y),

for all f,g £ I(X,R), r £ R, x,y,z € X, is called the incidence algebra of X over R. We refer to [Spiegel and O'Donnell (1997)] for a detailed exposition on incidence algebras. The following definition is more general. Let R be a ring, X a finite set, and let p be any reflexive relation on X. The incidence ring I(X,p,R) of X with coefficients in R is the free left .R-module with basis consisting of ordered pairs {(x,y) \ xpy}, where multiplication is defined by the distributive law and si \ ( (x:w) •v (z, w) = { \ ' ' ' \ 0

i

v(x,,yjy)

if y = z and xpw, F , . otherwise.

A relation p is said to be balanced if, for all x\,xi,xz,x\ X2pX3,

X3PX4,

,_ , , .

v (3.11) ;

€ X with x\px2,

X1PX4, X\pXz

•& X2PX4.

It is proved in [Abrams (1994)] that I(X, p, R) is an associative ring if and only if p is balanced. If X be a finite set with a preorder <, i.e., a reflexive and transitive binary relation, and R is a ring, then the above definition simplifies. In this case the incidence ring of X with coefficients in R is the free left .R-module with basis consisting of ordered pairs (x,y), for all x,y £ X,x < y, where multiplication is defined by the distributive law and the rule /

^

/

\

f (X, W)

(x.y) v yj • v (z,w) = < \ ' ' ' ' ' \ 0

if y = Z,

f . otherwise.

/n i n \

v (3.12) ;

Chapter 4

The Jacobson Radical

4.1

The Jacobson Radical of Groupoid-Graded Rings

Let S be a finite set, and let R = ®s&sR3 be a homogeneous sum. We say that R is a cancellative homogeneous sum if, given any homogeneous elements x £ Rs, y £ Rt, z £ H(R) and any u £ S, each of the conditions 0 ^ xz,yz £ Ru or 0 ^ zx, zy £ Ru implies s = t. This is equivalent to saying that the partial groupoid induced by R is cancellative. Considering S as a partial groupoid induced by R, we denote by m(R) the least common multiple of the orders of all subgroups of S. Theorem 4.1 ([Kelarev and Plant (1995)]) Let R be a cancellative homogeneous sum of a finite number of additive subgroups, and let n = m(R). Then, for every r £ J(R), all homogeneous components of nr belong to J(R). If G is a finite group, then the least common multiple of all subgroups of G is equal to \G\, and so m(R) divides |G|, for every G-graded ring R, and so Theorem 4.1 means that \G\H(J(R))

C

J(R).

Let R = (B™j=1Rij be a generalized matrix ring. Considering the induced partial groupoid on the indexing set s

= {(hj)

I hj = 1, • • • , " } ,

we see that S is cancellative and has only subgroups of order one. Therefore m(R) = 1, and so Theorem 4.1 shows that H(J{R)) C J(R), that is the 37

38

The Jacobson

Radical

radical of R is homogeneous. Now we prove Theorem 4.1. Proof. Suppose the contrary. Then we can find a minimal counter-example to the theorem, i.e., there exist a partial cancellative groupoid S with minimal \S\ and an 5-graded ring R such that m(R)H(J(R)) % J(R). Let k = 151 and n = m(R). First, consider the case where | supp(i? s i?)| < k, for some s £ S. Denote Rs by W. (This notation is needed so that we may refer back to this part of the proof later when we use the same reasoning with another set W. Our argument is valid for any set W contained in one homogeneous component of R and such that |supp(W.R)| < k.) Let K = RlWRl. Since K is a homogeneous ideal of R, evidently R/K is a homogeneous sum, too. We are going to show that R/K is also a counter-example to the theorem. This will allow us to factor out K and assume that W = 0, which will lead to a contradiction. Denote by A(R) the additive subgroup generated by the set nH{J{R)) of homogeneous components of elements of nJ{R). Take any x £ J{R) and y £ H(R). For any such y, there exists r £ R and b £ S such that y — rfe. Given that S is cancellative, all xay belong to distinct homogeneous components for different a £ S. Therefore (nxa)y = n(xr)ab £ nH(J(R)). Thus nH(J(R)) is an ideal of the multiplicative semigroup H{R). Hence A(R) is an ideal of R. Recall that K = R^WR1 and let us introduce / = R^WR, L = A(R) n K,F = A(R) n / , and P = IA(R). Then PCFCLC A(R) are ideals of R, and we claim that they are all quasiregular. We begin with P. Pick any e £ nH(J(R)). There exist r £ J(R) and g £ S such that e = nrg. Since P = IA(R) and A(R) is generated by nH{J(R)), in order to show that P is quasiregular it suffices to verify that Ie C J(P). Consider an arbitrary t in S. Since W is contained in one homogeneous component of R, obviously RtWR is a homogeneous right ideal of R. Quasiregularity is inherited by right ideals, and so we get RtWRr

C J(R) nRtWRC

J{RtWR).

Denote by T the partial groupoid induced on supp(RtWR) by RtWR. Let I = m{T). Since \T\ < k, the minimality of A; implies that W (J (RtWR)) C J(RtWR). If a product uv is defined in T for some u, v £ T, then the product uv is also defined in S. Therefore every subgroup of T is also a

The Jacobson Radical of Groupoid-Graded

Rings

subgroup of 5. It follows that t divides n. Therefore nH{J{RtWR)) J(RtWR). Since RtWRr C J(RtWR) and e = nrg, we get RtWRe

C nH(J(RtWR))

39

C

C J(iW#).

Given that e was an arbitrary generating element of A(R), it follows that RtWRA{R)

C

J(RtWR).

Since A(.R) is an ideal of H(R), we see that RtWRA(R) is a quasiregular right ideal of R. Therefore P = IA(R) = £ t e S RtWRA(R) is a sum of quasiregular right ideals, and hence it is quasiregular. Further, P = IA{R) D FA(R) D F2 implies (F/P)2 = 0, and so F/P is quasiregular. Since P is quasiregular and F/P is quasiregular, it now follows that F is also quasiregular. In order to show that L is quasiregular, consider the homogeneous ideal K = R1WR\ Clearly, K2 = R1WR1K C I. Therefore we obtain L2 = (A(R) n K)2 C (A(R) nl) = F. Since F is quasiregular, L is also quasiregular. Next, let R denote the S-graded quotient ring R/K. For X C R and r e R, denote by X and f the respective images of X and r in i?. Denote by V the partial groupoid induced on supp(i?) by R, and let m(V) = u. Then A(R) is the additive subgroup generated by uH(J(R)). All products defined in V are also denned in 5, because RsRt ^ 0 implies RsRt ^= 0. Hence all subgroups of V are also subgroups of 5. It follows that u divides n. This and J(R) C J(R) give us A{R) C A(fi). Suppose that i? is not a counter-example to our theorem. Then A(R) C J(R), and so A(i?) C A(R) C J ' ( S ) . Hence A(i?) is quasiregular and therefore A(R)/L = A(R) is quasiregular. Since L is quasiregular, this implies that A(R) is quasiregular. This contradiction shows that R is also a counter-example. Therefore without loss of generality we may assume that W = 0 from the very beginning. Finally, since Rs = W = 0, we see that R is a homogeneous sum of Rt, where t runs over S^js}. This contradicts the minimality of k and shows that the case when | supp(i? s i?)| < k, for some s € S, is impossible. Now, consider the case where | supp(i? s i?)| = k, for all s £ S. Then the product st is defined for all pairs s,t € S. First, suppose that S is not associative. Then there exist a,b,c £ S with (ab)c ^ a(bc). Thus RaRbRc C i? (ab ) c n i?0(bc) = 0. Therefore

40

The Jacobson

Radical

|supp(i2a.R&.R)| < k. As in the case when |supp(i? s i?)| < k, setting RaRb = W, we get W = 0. This means that RaRb = 0- As we have shown above this case is impossible. Second, suppose that S is associative. Then 5 is a semigroup. By Lemma 1.1, S is a group. This contradicts Theorem 2.1 and completes the proof. • If R = ®S£sRs is a homogeneous sum and S is the partial groupoid induced by R, then m(R) < m{S). Evidently, m(S) < \S\\. The exact upper bound for m(S) in terms of \S\ is given by the following. Proposition 4.1 Take a positive integer n and list all primes pi,P2, • • • ,Pk which are less than n. Consider the set Mn of products of the form p^1 • • • pf.k, where p"1 -f • • • + p^,k — k < n and define m(n) as the maximum product in Mn. Then, for every partial groupoid S, m(S) < m(\S\). Proof. Put n = \S\ and £ = m(S). Consider the prime decomposition of i = p"1 • • • p1k. Given that £ is the least common multiple of the orders of all subgroups of S, for every 1 < i < k, we see that S has a subgroup Ti such that p^ is a factor of | 5 | . By Theorem 1.3, Tj has a subgroup Hi of order precisely p"'. For any i ^ j , Hi n Hj = {1} contains only the identity element. Since the union of all the Hi is contained in S, we have | Uf=1 H\ = p" 1 + • • • + pakk - k + 1 < \S\ = n. This means that t belongs to the set Mn. Therefore £ is less than or equal to the maximum number m(n) in Mn. • The number m(R) in Theorem 4.1 cannot be replaced by smaller numbers. Example 4.1 Let 5 be a finite partial groupoid, and let n = m(S). Then there exists an 5-graded ring R such that m(R) = m(S) and, for any number £
% J{R).

Proof. Consider the prime decomposition n = p" 1 • • • pak of n. As above, denote by Hi a subgroup of order p" 4 in S. Let Fi = Zp^ be the ring of residues modulo p"\ Denote by R the direct sum of group rings Fi[Hi], for i = l,...,k. For s £ S, put Rs = 0 if s does not belong to the union of all

The Jacobson Radical of Groupoid-Graded

Rings

41

subgroups of S. If s is contained in some subgroup of S, then denote by Rs the sum of all sets Fts such that s G Hi and i = 1,... ,k. It is easily seen that R — (BS£sRs is -S-graded. Consider the augmentation ideal of Fi [Hi], that is the set w(Fi[Hi]) = { £ fjhj | ^ G F, ^ G Hi, £ j=i

/,- = 0}.

j=i

By Theorem 3.2, u^i^ifj]) is equal to the radical of Fi[Hi\. Obviously, the radical of R is the direct sum of all these augmentation ideals. Look at any number £ less than n. There exists i such that p" j does not divide £. Take any g € Hj\{e}, where e is the identity of Hi. The element l - o belongs to J ( f l ) . However, I ^ 0 in Fu and so 0 ^ tg <£ J(R). a If F is a field of characteristic zero or prime characteristic greater than \S\, then m(R) is invertible in F. Corollary 4.1 Let G be a cancellative partial groupoid, F a field with char(F) = 0 or chai(F) > \S\, and let R = (BgeGRg be a G-graded F-algebra inducing G. Then the Jacobson radical of R is homogeneous. The following example shows that Theorem 4.1 cannot be generalized to partial groupoids which are not cancellative. E x a m p l e 4.2 where

Consider the subring R of R 2 x R 2 given by R = RQ + Rx

R<

= {(»•

Ri =

{(

a b c d

"Or" 0 0

J

a, b, c, d € R

"Or" 0 0

reR

The set S = {0,1} with ordinary multiplication of integers is an idempotent semigroup. Clearly R is S'-graded and S is not cancellative. Consider the ideal 7 =

0 0

r 0

0

\r G R

42

The Jacobson

Radical

Since I2 = 0, I is a quasiregular ideal and since R/I ~ R2 is semisimple, it follows that / = J{R). Take any x G / , say " O r ' 0 0

J

' O r ' 0 0

) + (o,

"0 0

—r 0

where r G R. Evidently, R0 is an ideal of R, and so R0nJ(R) However, 0 ^ nx0 $

= J(RQ) = 0.

J(R0)

for all n £ J\f. Hence the analog of Theorem 4.1 does not hold without the assumption that the induced partial groupoid is cancellative. 4.2

Descriptions of the Jacobson Radical

This section contains a reduction of the Jacobson radical of a large class of semigroup-graded rings to radicals of group-graded rings, obtained in [Kelarev (1992e)]. For ring constructions graded by combinatorial semigroups this reduction gives a description of the radical. More generally, it can be combined with known facts on group-graded rings to provide useful information and various corollaries concerning the structure of the radical in many other ring constructions, too. Let R — (BsesR-s be an 5-graded ring. For e G S, recall that {s£S\e€S1sS1}.

e=

Let G be a subgroup of S with identity e. Define a mapping fa by the rule fa(r) =

TG

R —• Rr

if r G = rotherwise,

for r G JR. We say that a subsemigroup H of S is group-closed if, for each subgroup G of S and every g G G n H, the inverse element g~x belongs to H. In other words, H is group-closed in S if, for each group G, the intersection H fl G is a group, too. A subsemigroup H is said to have a finite length in S, if there exists a finite ideal chain 0 = S0 C Si C . . . C Sn = S°

Descriptions

of the Jacobson

Radical

in S° such that Hr\Si = 0, H C Sn, and each factor Sk/Sk-i 0-simple or nilpotent, where k = 1 , . . . , n.

43

is completely

Theorem 4.2 ([Kelarev (1992e)]) Let S be a semigroup such that every finite subset of S is contained in a group-closed subsemigroup of finite length, and let R = (BsesRs be an S-graded ring. Then the Jacobson radical J{R) is equal to the largest ideal among ideals I of R such that / G ( - 0 Q <J(RG) for every maximal subgroup G of S. An analog of Theorem 4.2 is valid even for a slightly larger class of semigroups if we replace the Jacobson radical J by the Levitzki radical C. It is also valid for Baer radical B, if we require that every Rees factor of S have a nonzero ideal which is minimal or nilpotent. The class of semigroups mentioned in Theorem 4.2 includes all locally finite semigroups, completely regular semigroups, and matrix epigroups. Now we prove Theorem 4.2. Proof. The inclusion fG(J{R)) C J(RG) follows from Lemma 2.2. Therefore it remains to consider any ideal J of R such that / G ( J ) C J(RG), for all groups G in S, and to show that / C J(R). To this end we choose any x in I and claim that x is right quasiregular. Consider the group-closed subsemigroup H of finite length in S containing supp(a;). Look at the shortest finite ideal chain 0CSocSiC...c5nCS° such that So n H = 0, H C Sn, and each factor S^/Sk-i is completely 0-simple or nilpotent. We proceed by induction on n. The case of n = 0 is trivial, since then x = 0 is quasiregular. Further, assume n > 1. Put T = Sn and N = Sn^i. Since J is hereditary and RT is an ideal of R, we get x £ J{RT)We show that there exists y € IH = I n ®h£HRh such that x + y + xy £ R^. Consider two cases. Case 1: Let F = T/N be nilpotent. Then xm £ R?j for some m > 1. A routine verification shows that we can take y = —x + x2 + . . . + (-l)™-1!"1-1. Case 2: Let F be completely 0-simple. For any r 6 RT, Q C RT, and V C T, put r = r + RN eR,Q = {q\q£Q},&ndV= (VuN)/N C T/N. The quotient ring R = RT/RN is -F-graded in a natural way. In view of Theorem 1.6, F is isomorphic to a Rees matrix semigroup M°(G; I, A; P) over a group G° with zero and sandwich-matrix P. We use symbols Ft\ and Fit introduced before Theorem 1.8, where i £ I, A e A.

44

The Jacobson

Radical

Let i?o = 0 and Ri\ =

®feFiXRf,

Ri* = ©AeA-RiAClearly, R = ®f£FRf, R = ®iejRu, and R = ©ig/.AeA-Ra- Theorem 1.8 tells us that Fit is a right ideal of F, and Fix is a left ideal of F^. Therefore Ru is a right ideal in R, and Ri\ is a left ideal in Ru. For any subring K of R, put Ku = K n i?j* and ifjA = -K" n i^AConsider the subring if = RJJXRH of iZ. We say that an element of R is homogeneous if it belongs to some .RJA- Evidently, K is equal to the set of all finite sums of all elements of the form uxv, where u, v are homogeneous elements of R-JJ. If u £ Ri\, v £ Rj^, then uxv C i?jM. Therefore # = ®it\Ki\. Now we claim that every Ki\ is quasiregular. This is obvious when 2

p\i = 0, because then KiX = 0. Assume p,\i ¥" 0- Theorem 1.8 shows that Fi\ is a maximal subgroup of F. Obviously, /G(K) C J{RG) for every maximal subgroup G of P , because K C. I. Since every nonzero maximal subgroup G of F is also a maximal subgroup of S, passing to R we get IG{K) Q J{RG) f° r every maximal subgroup G of F. Put G = Fi\. Then Rix = RG, and so Ki\ C JG{K) C J(Rix)- Evidently, H is a, group-closed subsemigroup of i*1. Therefore P = ff n G is a group. By Lemma 2.2, J7(flp) D ^ p f l J(RG); whence i f a C J ( # P ) . Clearly, K~ is an ideal of R-jj, and so if »A is an ideal of Rp. Therefore Kix is quasiregular. Further, K^ is quasiregular as a sum of quasiregular left ideals Ktx, where A 6 A. Since if is a sum of right ideals Ki*, we see that K is quasiregular, as well. Evidently, x3 £ K. Therefore x3 + w + x3w — 0 for some w £ K. Putting ~z = w — x + x2 — x~w + x2W, we get x + J + x~z = x3 + w + x3w = 0. Since w, x £ RJJ, it follows that z £ RJJ. Therefore z = —x — x~z belongs to the ideal generated in Rjj by x, and we can write ~z = J2eaex~bi, for some a,£, be £

a

RlH.

Consider the element y = 2_, ixbt. Clearly, y = z and y £ Rjj- We e _ get d = x + y + xy £ RN, because d = x + y + x~y = Q. Thus in both cases d = x + y + xy£ R^. Besides, d £ IH- Therefore supp(rf) C H U N. Since \H U N\ < \H\, by the induction assumption d is quasiregular, that is d + e + de = 0, for some e. Putting x' = y + e + ye, we

Semisimple

Semigroup-Graded

Rings

45

get x + x' + xx' = d + e + de = 0. Thus x is quasiregular, which completes the proof. Q

4.3

Semisimple Semigroup-Graded Rings

Let K. be a class of rings. A ring R is said to be K.-semisimple if no nonzero ideal of R belongs to K. Theorem 4.3 ([Teply et al. (1980)]) Let S be a semilattice, R = ®sesRs an S-graded ring, and let K, be a class of rings closed for ideals and homomorphic images. If all homogeneous components Rs are K,-semisimple, then R is /C-semisimple. Theorem 4.4 ([Munn (2002)]) Let e and f be V-equivalent idempotents of a semigroup S, and let R = © s € si? s be a T>-faithful S-graded ring. (i) / / Re is semiprime (respectively, prime, semiprimitive, primitive) then Rf is semiprime (respectively, prime, semiprimitive, primitive). (ii) If RHC is semiprime (respectively, prime, semiprimitive, primitive) then Rnf is semiprime (respectively, prime, semiprimitive, primitive). Theorem 4.5 ([Munn (2000a)]) Let S be a bisimple inverse semigroup with a maximal subgroup G, and let R = (BsesRs be a faithful S-graded ring. (i) / / RG is prime, then the whole R is prime. (ii) If RG is a primitive ring such that a € aRc, then the whole R is primitive. Theorem 4.6 ([Kelarev (1998c)]) Let S be an inverse semigroup, and let R = ®sesRs be a faithful S-graded ring. Then (i) / / all group-graded subrings RG are semiprimitive (semiprime) for all maximal subgroups G of S, then R is semiprimitive (semiprime). (ii) / / all group-graded subrings RG have no nonzero nil (locally nilpotent) ideals, for all maximal subgroups G of S, then R has no nonzero nil (locally nilpotent) ideals.

46

The Jacobson

Radical

(iii) / / all group-graded subrings RQ have no nonzero right nil ideals, for all maximal subgroups G of S, then R has no nonzero right nil ideals. It follows from the results of [Munn (2002)] that this theorem generalizes to 23-faithful 5-graded rings. Besides, by Theorem 4.4 it suffices to assume that RG is semiprimitive (semiprime) for one maximal subgroup G in every £>-class of S. Theorem 4.6 does not extend to arbitrary semigroups. Indeed, if S = {a, b} is a left zero band, and F a field, then it is well known and routine to verify that the Jacobson radical of the semigroup ring F[S] is equal to

{fa-fb\feF}, and so FS is not semisimple. However, its subrings Fa and Fb are semisimple. To illustrate the methods, we include a proof of Theorem 4.6. The following lemma was proved by Munn in the case of semigroup algebras. Lemma 4.1 Let S be an inverse semigroup, R = ®sesRs a faithful S-graded ring. If I is a nonzero ideal of R, then there exists an idempotent e G S and an element r € I such that e € supp(r) C He U (eSe\Pe). Proof. Take any b G I\0. Let supp(6) = { s i , . . . ,sn}. Choose a maximal idempotent e in {s,s~ | i = 1 , . . . , n}. Without loss of generality we may assume that e = s i s ^ . Since R is a faithful S-graded ring, there exist elements c G Re and d G R -i such that cbs.d ^ 0. Clearly, a = cbd G / . We claim that e G supp(a) C eSe. For i = l , . . . , n , put ti = esjSJ-1. Then supp(a) C esupp(6)sj" 1 = {t\,... ,tn}. Further, eij = ti = esisj" 1 sis]~ 1 = t^e for i = 1 , . . . ,n. Therefore supp(a) C eSe. Since a has a nonzero summand cbSld which is in Re, in order to show that e G supp(a) it remains to prove that this summand does not cancel with other summands cbsd of a. To this end it suffices to verify that all other summands belong to different homogeneous components of R. We show that if tk = e, then k = 1. Let t^ = e. Clearly, tk = esk{esk)~l• By the choice of e, it is maximal in {esi(es,) _ 1 | i = 1 , . . . , n}. Lemma 4 of [Munn (1987a)] implies that esu = es\ = s\. Hence esks^1 = esk{esk)~l = sis^1 = e, which means that e < SfcS^T • By the maximality of e, we get

Semisimple

Semigroup-Graded

Rings

47

Sfcsj" = e. Therefore Sfe = esk = s\, and so k = 1. Thus cbSld G Re is a homogeneous component of a, and so e € supp(a). We have shown that the set B = {a € I \ e G supp(a) C eSe} is not empty. It remains to prove that B contains an element a such that supp(a) C He U (e5e\F e ). Suppose to the contrary that supp(a) D (Pe\He) ^ 0 for all a G B. We show that this leads to a contradiction. Choose b 6 B such that the positive integer m = | supp(6) D (Pe\He)\ is minimal. Let supp(6) = { e , x i , . . . , x „ } , where n> m and supp(6) D (Pe\He) — {x\,... ,xm}. Given that R is a faithful S'-graded ring, we can find an element r e J J - i such that br ^ 0. In order to get a contradiction we are going to show that br G B and | supp(6r) n (Pe\He)\ < m First, we see that supp(br) C {ex^jXiX^ - 1 ,... . i , , ! ^ 1 } . Obviously, br £ I since b G I. For i — 1 , . . . , n, we get XiX~[l G eSe, because every Xi is in eSe and x]^ 1 = ( e x i e ) - 1 = ex^le G eSe. Further, exj~ = xj~ ^ e, XixJ-1 = e and since e is the greatest element in {x;X~l | i = 1 , . . . , n}, as in the beginning of the proof we see that x^xj- — e only if i = 1. Therefore the element bXlr is the homogeneous component (br)e of br. Hence e G supp(6r) C eSe, as claimed. Second, we note that, for any a; G eSe, x $ Pe\He is equivalent to l x~ x = e or xx~l < e. We shall use this several times. Suppose that k is a positive integer such that m < k < n. Then Xk £ Pe\He by the choice of m. Therefore either x^1Xk = e or x^1Xk < e, as noted in the preceding paragraph. If x^ Xk = e, then (xfcx]~ )_1XkxJ = 1 1 1 1 x\(x'^ Xk)x^ = Xiex^ = Xix^ = e, and so XfcXJ~ ^ Pe\He as noted above. If, however, x^x^ < e then XfcX^^XfcX^-1)-1 < x^xj - 1 < e, and so again XfcxJ"1 £ Pe\He. Further, x^1 = ex]"1 £ Pe\He, because xi G Pe\He. It follows that if m = 1, then (supp(6r))n(P e \i? e ) = 0. On the other hand, if m > 1, then (supp(6r)) n (Pe\He) C {X2XJ1,... . i n i j " 1 } , a set with at most m — 1 elements. Thus | supp(br) D (Pe\He)\ < m. We have obtained a contradiction to the minimality of m. It shows that supp(a) n (Pe\He) = 0 for some a € B. This completes the proof. • L e m m a 4.2 Let S be an inverse semigroup, R = (BsesRs a faithful S-graded ring. If I is an ideal of R, then there exists an idempotent e G S and an element r £ I such that e G esupp(r) C He U (eSe\Pe). This lemma is similar to Lemma 4.1 and we omit the proof.

48

The Jacobson

Radical

Lemma 4.3 Let S be an inverse semigroup, R = ©Sgs-Rs o,n S-graded ring, e an idempotent of S, and let G = He, P = Pe, T = eSe\P. Suppose that x G ReSe and x = xG + I T - Then, for an arbitrary element y G Rese, {xy)G =

xGyG.

Proof. Put L = P\G. Clearly, eSe = G U L U T and y = yG + yL = yT. It follows from the definition of T that T is a right ideal in eSe, and so xTy € RTFor any g G G, £ G L, there exist g~x e G, £' e L such that p y - 1 = e, ££' = e. Hence g££'' g~x = e, which means that g£ e P . If g£ e G, then we would get £ = e£ = g~xg£ G G, because e is an identity of eSe. This contradiction shows that g£ € L. Therefore GL C L and XG2/L & RLSimilarly, GT C T and xcyx £ -RT- Thus we get (xy)L = xoyL, (xy)T = xTy + xGyT, and (xy)G = xGyG, as required. • Lemma 4.4 Let S be a semigroup, R = ®sesRs an S-graded ring, e an idempotent of S, and let G = He. Suppose that the Jacobson radical J{R) contains an element r such that e G supp(r) C He U (eSe\Pe). Then rG G

J(RG).

Proof. We verify that the right ideal / generated by rG in RG is quasiregular. Indeed, take any element x € I. There exists y € RG such that x = rGy. Consider the element ry G Rese- Since ry G J(R) D Res and Res is a right ideal of R, it follows that ry G J(Res)Similarly, we get ry G J(ReSe), because ReSe is a left ideal of Res- Therefore ry has a right quasi-inverse element z G ReSe, i-e., ry + z + ryz = 0. Given that y G RG, we get supp(ry) C G U T, where T = eSe\Pe, and so ry = (ry)G + (ry)x and (ry)G = rGy. Lemma 4.3 tells us that (ryz)G — (ry)GzG = rGyzG. Hence we get 0

=

(ry + z + ryz)G

=

(ry)o + zG + (ryz)G

=

rGy + zG + rGyzG.

It means that rG generates a quasiregular right ideal in RG. Therefore rG belongs to the Jacobson radical of RG. •

Semisimple

Semigroup-Graded

Rings

49

Lemma 4.5 Let S be a semigroup, R = ®s^sRs an S-graded ring, e an idempotent of S, and let G = He. Suppose that the Baer radical B(R) contains an element r such that e G supp(r) C He U (eSe\Pe). Then rG G B(Ra). Proof. It suffices to prove that rG is a strongly nilpotent element of RG. Take any sequence xi,X2,. • - G RG, where Xi = re, and xi+\ = XiUiXi for i = 1,2,... and some yi G RGConsider an auxiliary sequence r\, r% • • • £ ReSe denned by r\ = r and n+i = riy^i for i = 1, 2, Since r G B(R), there exists n > 1 such that r„=0. Given that y, G i?o, it follows by induction on i that all elements x^ satisfy the following equality x, = (x^a + (XJT- Therefore Lemma 4.3 yields that ( X J + 1 ) G = {xi)GVi(xi)G- Since Xi = (ri)c, it follows that x% = (^I)G for all i > 0. Therefore x„ = (rn)c = 0G = 0. Thus r e is a strongly nilpotent element of RG, and so it belongs to the Baer radical of RGE Lemma 4.6 Let S be a semigroup, R = ®sesRs an S-graded ring, e an idempotent of S, and let G = He. Suppose that the Levitzki radical C(R) contains an element r such that e G supp(r) C He U (e5e\P e )- Then rG G

C{RG).

Proof. Let I be the ideal generated by x = re in RG- We are going to prove that / is locally nilpotent. Take any finite set of elements F = { x i , . . . , x m } C I. For i = 1 , . . . , m, there exist a^, bi G RG such that Xi = Oixbi- Consider the elements r, = airbi, where i = 1 , . . . ,m, and put Q = { r i , . . . ,rm}. Given that r G £(R), we see that Q C C{R), and so Qn = 0 for some positive integer n. We claim that Fn = 0. Indeed, pick any elements yi,---,yn € F. Clearly, (r,)c = Xj for i = 1 , . . . , m. Hence there exist z\,...,zn G Q such that j/i = {zi)G for all i Since z = zG + ZT for every z G Q, Lemma 4.3 implies yi---yn

=

(zi)a

=

(zi

•••

=

0 G = 0.

•• •

(zn)G zn)G

50

The Jacobson

Radical

Thus / is locally nilpotent, which completes the proof.

•

an

Lemma 4.7 Let S be a semigroup, R = (BsesRs S-graded ring, e an idempotent of S, and let G = He. Suppose that I is a nil ideal of R and r an element of I such that e G supp(r) C HeU (eSe\Pe). Then re generates a nil ideal in RGProof. Denote by N the ideal generated by ro in RG- We claim that TV is a nil ideal. Indeed, take any element x G N. There exist a,b € RG such that x = arab. Given that a,b G RG, we get x = (arb)a and arb — (arb)a + (arb)xHence Lemma 4.3 implies that [(ar6)™]c = [(arb)n]a for all positive integers n. Since arb G / , we see that (arb)n = 0 for some n. Therefore xn = [(ar&)c]" = 0. Thus N is a nil ideal, as claimed. • Lemma 4.8 Let S be a semigroup, R = (BsesRs an S-graded ring, e an idempotent of S, and let G = He. Let I be a right nil ideal of R and let r be an element of I such that e G esupp(r) C He U (eSe\Pe). Suppose that there exist t G Pe and a € Rt such that e G supp(ar). Then (ar)a = arG generates a nil right ideal in RG. Proof. Denote by N the right ideal generated by r g in RG- Take any x G N. There exists b G RG such that x = arab. As above x = (rb)G and rb = (rfo)c + (rb)r- Since rb G / , we see that (arb)n = 0 for some n. Therefore xn = [(rb)G]n = [(rb)n]G = 0 by Lemma 4.3. Thus N is a nil right ideal. • Now we can prove Theorem 4.6. Proof, (i) : Suppose to the contrary that R is not semiprimitive (not semiprime). Lemma 4.1 tells us that there exists an idempotent e G S and an element r G J{R) (respectively, r G B{R)) such that e € supp(r) C He U {eSe\Pe). Put G = He. By Lemma 4.4 (or Lemma 4.5) rG G J(RG) (respectively, rG € B(RG))Since e G supp(r), we get re ^ 0. This contradicts the semiprimitivity (semiprimeness) of RG(ii) : Similarly, suppose that R has a nonzero nil ideal (locally nilpotent ideal) I. Lemma 4.1 shows that e G supp(r) C He U (eSe\Pe), for some e G E(S) and r G I. Hence re ^ 0. By Lemma 4.7 (respectively, Lemma 4.6), rG generates a nonzero nil (locally nilpotent) ideal in RG, a contradiction. (iii) : Suppose that R has a nonzero right nil ideal I. Lemma 4.2 shows that e € esupp(r) C He U (eSe\Pe), for some e G E(S) and r e I. By

Homogeneous

Radicals

51

the nondegeneracy, there exists a homogeneous element a € Re such that e € supp(ar). Lemma 4.8 tells us that arG generates a nonzero right nil ideal in RQ- This contradiction completes our proof. • 4.4

Homogeneous Radicals

This section is devoted to the following general problem. Problem 4.1 Describe all semigroups S such that the Jacobson radical is homogeneous in every 5-graded ring. The first positive results on homogeneity of the Jacobson radical were obtained by Bergman [Bergman (1973)] and Cohen, Montgomery [Cohen and Montgomery (1984)]. Properties (a) and (b) of the following theorem are due to Bergman [Bergman (1973)] in the special case where G is torsion-free abelian. The case where G is free is due to Jespers and Puczylowski [Jespers and Puczylowski (1991)]. Theorem 4.7 Let R be a G-graded ring, where G is a group of one of the following types: (a) G is abelian and the orders of finite subgroups of G are units in R; (b) G is locally finite and the orders of finite subgroups of G are units in R; (c) G is locally free, or residually free, or free solvable, or torsion-free nilpotent; (d) G is a subdirect product of the groups Gi,i £ I, where each Gi is of one of the types (i), (ii), or (in). Then the Jacobson radical of R is homogeneous. It was proved in [Jespers et al. (1982)] that the Baer and Levitzki radicals are homogeneous in every ring graded by a u.p.-semigroup. An analogous question for the Jacobson radical is well-known and still remains open. A few generalizations of homogeneity were introduced in [Clase and Kelarev (1994)]. They apply, in particular, to certain types of inverse, regular and commutative semigroups. Theorem 4.8 ([Kelarev (1996b)]) Let S be a cancellative linear semigroup, and let R be an S-graded algebra over a field of characteristic zero. Then the Jacobson radical of R is homogeneous.

52

The Jacobson

Radical

T h e o r e m 4.9 ([Kelarev and Okninski (1996)]) Let S be a u.p. -semigroup, R an S-graded ring. If R is semilocal, or has right Krull dimension, or is right Goldie modulo the prime radical, then the Jacobson radical of R is homogeneous. T h e o r e m 4.10 ([Clase and Kelarev (1994)]) If the Jacobson radicals of all S-graded rings are homogeneous, then S is cancellative. Theorem 4.11 ([Kelarev and Okninski (1996)]) Let S be a cancellative semigroup, and let R be an S-graded Pi-algebra over afield of characteristic zero. Then the Jacobson radical of R is homogeneous. This theorem is proved in Section 4.6. The rest of this section contains proofs of several of the main theorems stated above. First, we collect a few lemmas needed for the proof of Theorem 4.8. Lemma 4.9 Let S be a cancellative semigroup, R an S-graded algebra, and I ^ 0 an ideal of R. Let P = P(l) be the set of all elements of the minimal positive length in I, and let Min(7) be the linear span of H(P) in R. Then Min(7) is an ideal of R. Proof. Take any x £ H{P) and y £ H(R) such that xy ^ 0. Clearly, x = rs for some r £ P,s £ S. Given that S is cancellative and xy ^ 0, we get ry £ P. Hence xy £ H(P). It follows that Min(7) is a right ideal of R. Similarly, it is a left ideal of R. • Lemma 4.10 If G is a linear group and R is a G-graded algebra, then J{R) is homogeneous. Proof. By Lemma 2.2, J(RT) 2 RT<~\J(R), for any subgroup T of G. Let J- be the set of all finitely generated subgroups of G. Then R = \JT€:F RT and all J(RT) 2 RT^J{R) imply \JTeJ:J(RT) 2 J(R)- Hence it suffices to show that J{RT) is homogeneous for every T £ T. So we may assume that G is finitely generated itself. Then G has a normal subgroup N of finite index such that N is residually finite ([Karpilovsky (1991)], Lemma 49.8). Therefore G itself is a residually finite group. For each 0 ^ r £ J{RG), there exists a normal subgroup K of finite index in G such that gK ^ hK, for all g, h £ supp(R), g ^ h. By Lemma 2.1, R is G/if-graded, and Theorem 2.3 shows that J(R) is homogeneous in this gradation. Since rgK = rg for every g £ supp(r), we get H(r) C J{RG). Therefore J(RG) is homogeneous. D

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The proof of Theorem 4.8 follows from Theorem 1.13 and Lemma 4.10: Proof. Let S be a cancellative subsemigroup of Mn(F). Suppose to the contrary that there exists an S'-graded algebra R such that J{R) is not homogeneous. Factoring out the largest homogeneous ideal of J{R) we may assume that J{R) ^ 0 has no nonzero homogeneous elements. Let L = Min(J'(R)) be the homogeneous ideal denned in Lemma 4.9. For k = 1,... ,n, let Ik be the set of all matrices of rank < k in Mn(F), and let Rk = Rik. Consider the minimal positive integer k such that L has a nonzero intersection with ideal Rk introduced in Theorem 1.13. Then K = Rk n L is a homogeneous ideal of R. Using the sets Gap, a, /3 £ Afc, defined in Theorem 1.13, we put Rap = RGa/3,Ka0 = K fl Rap, and Ka* = K n RGa,- By the choice of A;, we get Kjkl = 0. Therefore Theorem 1.13 gives the following: (i) either Gap is a linear group and Kap is a Gap-gra,ded algebra, or ^ = 0; (ii) Raf}KRaj3 C Ka/3] (iii) Kat is a right ideal of R; (iv) Kap is a left ideal of Ka*. If Gap is not a group, then K%p = 0, and so Kap is quasiregular. Next, suppose that Gap is a group. Let P = P(J(R)) be the set of all elements of the minimal positive length in J{R), and let Q = H(P) PI Rap- Then Ka/3 is the linear span of Q (see Lemma 4.9). Take any q £ Q. There exist r S P and g £ Gap such that q = rg. For any a,b £ H(Rap), it follows from (ii) that a r t € if a /3- By (iii) and (iv) we get arb £ J{Kap). Lemma 4.10 shows that J{Kap) is homogeneous. Since S is cancellative, aqb = (arb)h for some h £ Gap. Therefore aqb £ J(Kap). It follows that K^p Q J{Kap), and so Kap is quasiregular, again. Since Ka% is the sum of quasiregular left ideals Kap, and K is the sum of right ideals Katt, evidently K is quasiregular. Therefore J{R) contains a nonzero homogeneous ideal K, a contradiction. • The condition that the Jacobson radical be •S'-homogeneous determines a class of semigroups, which will be referred to as the class of JH-semigroups. We say that S is a JGH-semigroup if J{R) is a homogeneous ideal, for every .S'-graded ring R such that J{RG) is homogeneous for all subgroups GoiS. Lemma 4.11 Let S be a JGH-semigroup. Then S does not have two distinct elements e and f such that ef = f2 and fe = e 2 (or ef = e 2 and

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fe = f2)Proof. We prove the first case, the other being similar. Suppose to the contrary that S does have such elements e and / . Let V be the subsemigroup generated by e and / ; then V = { e " , / n | n > l } . Let F be a field of characteristic 0 and consider the semigroup ring R = F[V]. If we put Rv = Fv for v £ V and Rs = 0 for s £ S \ V, then R becomes an S'-graded ring. Now take any subgroup G of S. If G does not intersect V, then RQ = 0 and J{RG) is certainly homogeneous. Otherwise, P = G n V is not empty. Since Rs = 0 for s $. V, we see that RQ = Rp- Choose p to be the smallest power of e or / in P ; without loss of generality, we may assume that p = em for some m > 1. Denote by E the subsemigroup generated by e. We claim that P C E. Indeed, suppose that fk £ P for some k > 1. Then k > m by choice of p. Let 1 be the identity of the group G. Since em £ P C G, there is a u in G such that emu = 1. Also, e m l = e m and / f c l = fk because e " \ / f e € G; the former implies that ekl = ek since k > m. But / e = e 2 implies ekem = fkem. k k k m k m k k fe Hence f = f l = f e u = e e u = e l = e , so that / £ E. If e is not periodic, then E is an infinite cyclic semigroup and so P C E is isomorphic to a subsemigroup of Z. By Theorem 4.7, J{RG) = <J(Rp) is homogeneous. If e is periodic, then ife = Rp = F[P] and P = G C\ E is a finite group (since it is finite and cancellative). Then J(RG) = 0 by Theorem 3.1, and therefore J{RG) is certainly homogeneous. So J{RQ) is homogeneous for all subgroups G of S. Since S is a JGH-semigroup, it follows that J{R) is homogeneous. Put d = e — f. For any u £ V the equality ei> = fv holds, implying dR = 0 and d £ J(R). The homogeneity of J{R) implies e e J"(.R). This is not possible since R — F[V] and e has augmentation 1. • Lemma 4.12 Let S be a JGH-semigroup. If u,v £ S, u ^ v, then for any w £ S, the equalities uw = vw and wu — wv are equivalent. Proof. Suppose to the contrary that uw = vw but wu ^ wv. Putting e = wu and f = wv we get ef = f2 and fe = e 2 , a contradiction to Lemma 4.11. • Proposition 4.2

Let S be a JGH-semigroup.

Then S is cancellative.

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Proof. Suppose that S is not cancellative. Then there exist u,v,w £ S such that u ^ v but uw = vw. (By 4.12, uw = vw is equivalent to urn = wv.) We will construct an S-graded ring R for which J{RG) is homogeneous for all subgroups G of S but J(R) is not homogeneous. Denote by M the ring of 2 x 2 matrices over the complex numbers C. Let ei2 = I

I be the standard matrix unit and put N = Cei2- Then

JV2 = 0. Let W be the ideal of S generated by w. Define a subring R of the semigroup ring M[S] by R — NS + MW; this is a subring, because W is an ideal of S. Since R is a homogeneous subring of M[S] it inherits the natural 5-gradation from M[5]. Let G be a subgroup of S. Since W is an ideal of S, two cases are possible: G n W = 0 or G C W. In the former case, RG C iVS so i? G 2 = 0 and therefore J(RG) — RG- In the latter case, RG is the group ring M[G], which is isomorphic to the ring of 2 x 2 matrices over the group ring C[G]. But J(C[G\) = 0 (see [Passman (1977)]) so J(RG) = J(M[G}) = 0 (by [Karpilovsky (1991)], Proposition 6.13). In either case, J{RG) is homogeneous. Consider the element d = e ^ w — v) of the ring R. Take any s G W, say s = awb where a, b £ S1. Since uwb = vwb, 4.12 yields wbu = wbv; then su = awbu = awbv = sv and us — vs again by 4.12. Therefore dRs = 0 for s£W. If s£S\W then dRs C e12NS = 0. So dfl = 0 and d £ J(R). If J{R) were homogeneous, we would have e^u £ J{R). But the augmentation of e\2u (as an element of the semigroup ring Mf.!?]) is ei2, a contradiction, since the image of R under the augmentation map is M and

J{M) = 0 .

•

Theorem 4.12 Let S be a cancellative semigroup, and let R be an S-graded PI-algebra over a field of characteristic zero. Then the Jacobson radical of R is homogeneous. Easy examples of group algebras of finite groups show that the restriction on characteristic cannot be removed from Theorem 4.12. Theorem 4.13 Let S be a u.p.-semigroup, and let R be an S-graded ring such that at least one of the following conditions is satisfied: (i) all nil subsemigroups of H(R/J'gI(R)) are locally nilpotent; (ii) every nil subsemigroup of every right primitive homomorphic image of R is locally nilpotent;

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(iii) for every minimal prime ideal P of R, the ring R/P or embeds into a matrix ring over a division ring.

is a domain

Then the Jacobson radical of R is homogeneous. The class of rings satisfying (i) contains all rings R such that in all homomorphic images of R all multiplicative nil subsemigroups are locally nilpotent. This applies to all Pi-rings, left or right Noetherian rings and, more generally, all rings with left or right Krull dimension (see [McConnell and Robson (1987)], 6.3.5,13.4.2). The class of rings satisfying (ii) contains, beyond the classes mentioned above, all semilocal rings. Condition (iii) concerns all rings which are 'nice' modulo the Baer radical. In particular, this covers the case where R/B(R) is a right Goldie ring. If J C J(R), then J(R) is homogeneous if and only if J(R/I) is homogeneous. If / C B(R), then B(R) is homogeneous if and only if B(R/I) is homogeneous. Proposition 4.3 Let S be a u.p.-monoid with identity e, R an S-graded ring, r a rigid element of R, and let M be the multiplicative semigroup generated by H(r). Then (i) if r 0 Re and M^rM1 consists of quasiregular elements, then M is nilpotent; (ii) ifH(R)1rH(R)1 consists of quasiregular elements, thenre £ J(Re), and if, additionally, r 0 Re, then re belongs to the nilradical of Re. The following lemma allows us to consider rings and semigroups with identities. Lemma 4.13 (i) Let S be a u.p.- or t.u.p.-semigroup without identity element. Then the semigroup Se with identity e adjoined is also u.p. or t.u.p., respectively. (ii) Let R = (BsesRs be an S-graded ring. Denote by Re the subring generated in R} by 1. Then R1 = © ^ e s 1 ^ *s S1-graded. Proof. The assertion (ii) is obvious. In (i) we only consider the case where S is a t.u.p.-semigroup, since the proof for u.p.-semigroups is similar. Take two nonempty subsets X,Y £ Se with \X\ + \Y\ > 2, and any elements a,b £ S. The sets aX and Yb are contained in S, and \aX\ + \Yb\ > 2, because Se is cancellative. Therefore there exist distinct elements u', v' uniquely expressed in the form u' — axyb, v' = aztb, where ax, az £

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aX, yb,tb £ Yb. Put u — xy, v = zt. By the cancellativity of S these representations of u and v as products of elements from X and Y are unique. Thus Se is a t.u.p.-semigroup. • Lemma 4.14 Let S be a cancellative semigroup, R an S-graded ring, r a rigid element of R, and let M be the multiplicative semigroup generated by H(r) in R. If M contains 0, then M is nilpotent. Proof. Suppose that M contains 0. This means that y\... ym = 0 for some 2/i, • • •, ym e H(r). Choose m and yi,...,ym such that q = yx... y m _ x ^ 0, and consider the product z = qr. Since q is homogeneous, z is also a rigid element of R. Given that S is cancellative, we see that qym is a homogeneous component of z. Therefore qym = 0 implies qrs = 0 for all s £ S; whence z — 0 and, moreover, yi... ym-iH(r) = 0. Similarly, considering 2/i • ..ym-2rH(r) = 0, we get yx.. .y m _ 2 (i?(r) 2 ) = 0. Repeating this m times, we conclude (H(r))m = 0. Thus M is nilpotent. • Lemma 4.15 Let S be a cancellative semigroup, R an S-graded ring, I the ideal of nonunits of S and G = S\I (if S has no identity, then S = I). Assume that x + y = yx for some x 6 Ri,y £ R. Then y € K, where K is the subring generated by H(x). Proof. We will show that yg e K, for each g e I. The case where yg = 0 is trivial, and so we assume g £ supp(y). First, note that y £ Rj, because Rj is an ideal of R. If two elements s and t of I generate the same right ideal, then s = t. Therefore there exists a maximum positive integer n such that supp(^i)/ 1 D ... D supp(2:„)/ 1 , for some z\,...,zn £ H(y), where zn = yg. We call n the depth of yg. We proceed by induction on the depth of yg. Assume first that the depth of yg is 1. This means that gl1 is maximal in the set of right ideals of I generated by the elements of supp(y). Hence g = st implies ys = 0, and so yg £ H(x) C K. Next, assume that the depth of yg is n > 1. Since x + y — yx, we get Vg ~ Y^st=gysxt — xg. If st = g and ys ^ 0, then the depth of y3 is less than n, and by the inductive assumption yg £ K. It follows that yg £ K, as claimed. • Now we can prove Proposition 4.3. Proof. By Lemma 4.13, we may assume that R has an identity 1. Every u.p.-group is t.u.p. (see [Okniiiski (1991)], Chapter 10), and so the group

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of units G of S is a t.u.p.-group. It is routine to verify that I = S \ G is an ideal of S. Assume that r $ Re and M1rM1 consists of quasiregular elements. By Lemma 4.14, in order to show that M is nilpotent it suffices to prove that OGM. First, consider the case where r £ RG- In view of [Karpilovsky (1991)] Corollary 22.9, all elements of M1rM1 are quasiregular in RG- Since G is t.u.p., we can follow the argument used in [Jespers et al. (1982)]. Replacing r by rgrrh for some g,h £ supp(r), without loss of generality we may assume that all elements of M1rM1 are quasiregular, and that e ^ supp(r). Then |1 — r\ > 1 and 1 — r is a unit of RG- Therefore (1 — r)b = 1 for some b £ RG, and so ck(l — r)b = ck for any c £ H(r) and any k. Choose c £ H(r), k > 1, and b such that |6| is minimal among the lengths of all elements b satisfying ck{\ —r)b = ck for some c, k. lib = 0, then 0 = ck £ M and we are done. Suppose b ^ 0. Since S is t.u.p. and |1 — r\ > 1, we can find an element w £ S,w ^ supp(ck), uniquely expressed in the form w = uv, where u £ supp(ck(l — r)), v £ supp(b). Then ck{\ — r)b = ck implies [c^l — r)]ubv = 0. Since r is rigid and c £ H(r), it follows easily that ck+1{l-r)bv = 0. Therefore c fe+1 (l -r)(b-bv) = 0, contradicting the choice of b, and proving the claim. Second, suppose that r g- RG- Then there exists h £ I such that r^ ^ 0. Given that / is an ideal of S, we get b = rhr £ Rj. Obviously, b inherits the hypothesis imposed on r, and so we may assume that from the very beginning r £ Rj. Suppose that the semigroup M does not contain 0. Denote by T the subsemigroup generated in S by supp(r). Then, for every t £ T, we have

M n Rt ± 0. Suppose that T is not a right Ore semigroup. Let p be the left reversive congruence on T defined in [Okninski (1993a)]. Then there exists t £ T such that the set isupp(r) is /9-separated in the sense of [Okninski (1994)] and (h, 1) ^ p for every h £ isupp(r). Choose a nonzero element b £ M n Rt. We know that x = br ^ 0 and x + y = xy = yx for some y £ Rj. By Lemma 4.15, y £ RTThe left cancellativity of p implies that x' + y' — x'y' = y'x' for every p-class in T and the corresponding p-components x', y' of x, y. Let X be the semigroup generated by the support of x. Replacing x by any cxd, where c,d £ M, we see that the element cx'd has a quasi-inverse u £ Ri- Again,

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we know that u £ Rx- Therefore x' and Rx inherit the hypotheses on x and R. Proceeding in this way we eventually come to an element z whose support is in a single /o^-class, where pz is the left reversive congruence on the semigroup Z generated by the support of z. Prom [Okninski (1993a)], Lemma 3, we know that Z is a right Ore semigroup. Hence Z is a t.u.p.-semigroup (see [Okninski (1991)], Theorem 10.6). Since z inherits the hypotheses on r, we can apply the first paragraph of the proof to see that 0 is contained in the semigroup generated by the components of z. But z £ M, so this contradicts the assumption that 0 £" M. If T is a right Ore semigroup, then it is t.u.p., and we get a contradiction again. It follows that 0 £ M, which completes the proof of (i). Further, we show that (ii) easily follows from (i). Indeed, assume that H(R)1rH(R)1 consists of quasiregular elements. If r £ -Re, then RerRe C H(R)1rH(R)x consists of quasiregular elements. By [Karpilovsky (1991)], Corollary 22.9, all these elements are quasiregular in Re, as well. Therefore re £ J{Re). Assume that, in addition, r 0 Re. Since all elements in H(R)lrH(R)1 are quasiregular, clearly (i) applies to every nonzero xry, where x, y £ H(R)1. Therefore L = L U s ^ C R ) 1 ^ . ^ ) 1 is a nil ideal of H(R). In particular, re generates a nilideal of Re. This completes the proof. • L e m m a 4.16 Let R be an S-graded Pi-ring, and let T be a multiplicative subsemigroup of R. If T does not contain zero, then supp(T) is a permutational subsemigroup of S. Proof. Let H = supp(T). Every Pi-ring (or Pi-algebra) satisfies a multilinear identity. Let n be the degree of a multilinear identity (1.5) satisfied ini?. Take any elements t 1 , i 2 , • • • ,*n in T. Suppose that ti £ R^, for i — 1 , . . . , n. Applying (1.5) to the elements ti,...,tn we get h ...tn

£ Rht...h„ n

22

R

hal...han-

Given that T does not contain 0, it follows that t i . . . tn ^ 0. Therefore 0^t1...tn€Rhl...hnnRh<7l...hr7n, for some a ^ 1. Hence h\...hn = ha\... group H is permutational, as claimed.

han. This means that the semiD

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Lemma 4.17 Let S be a permutational cancellative semigroup, and let R be an S-graded Pi-algebra over a field of characteristic zero. Then J{R) is homogeneous. Proof. Theorem 1.18 says that S has a permutational group of fractions Q. Put Rg = 0 for q £ Q \ S. Then R is Q-graded and R = RS. It suffices to prove that J{RT) is homogeneous for every finitely generated subgroup T of Q (because then it will follow that J{R) is homogeneous, too, see [Karpilovsky (1991)], Lemma 30.27). Theorem 1.18 shows that T is abelian-by-finite. This means that T has a normal abelian subgroup A such that T/A is finite. Hence A is also finitely generated ([Robinson (1982), 1.6.11]), and so it contains a torsion-free subgroup of finite index ([Robinson (1982), 4.2.10]). Therefore we may assume that A is torsion-free itself. By Theorem 4.7, J(RA) is homogeneous. It is easily seen that RT is T/yl-graded. Theorem 2.1 implies J{RA) = RA ^J{RT)Therefore J{RA) generates a homogeneous ideal / in RT contained in J(JIT)We can factor out / and assume that J(RA) = 0. Take any element x in H(J(R)). There exists r e J{R) such that x — rg. Given that T/A is a finite group, it follows from Theorem 4.7 that J{R) is T/A-homogeneous. Therefore we may assume that supp(r) is contained in one coset gA of T/A (otherwise we could replace r by its T/A-homogeneous component involving rg). Let n be a positive integer such that gn e A. Then g"-'1 supp(r) C gnA = A, and so xn~1r e RA- It follows from [Karpilovsky (1991)], Corollary 22.9, that xn~lr e J{RA) = 0; whence xn~1r = 0. Given that G is cancellative, xn is a homogeneous component of xn~1r, and so xn = 0. Thus H(J(R)) consists of nil elements. By [Rowen (1980)], Theorem 1.6.36, H(J(R)) C B(R) C J(R). It follows that J(RT) is homogeneous, as required. • We are now ready for the proofs of the main theorems. First, note that the hypotheses of Theorems 4.12 and 4.13 (i) and (ii) are inherited by the ring R/JgT(R). Hence, for these proofs one can factor out Jgv{R) and assume that J(R) has no nonzero homogeneous elements. In the proofs we will suppose to the contrary that the Jacobson radical J{R) is not homogeneous. Then we can choose an element r of minimal length in J(R) with r s # J(R) for some s e S. Let W = HiR^rHiR)1, and let V = H{W). Denote by A the additive subgroup generated in R by

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V. Since 5 is a cancellative semigroup, it is routine to verify that V is an ideal of H(R), and so A is an ideal of R. First, we prove Theorem 4.12. Proof. We know that rs 0 P, for some right primitive ideal P of R and some s G S. Let W be the image of W in R' = R/P. Since S is cancellative, every element of W is a homogeneous component of an element of J{R). We know that R/P = Fn for a division ring F. Choose a G W such that its image a' G W is of minimal positive rank as a matrix in Fn. We can choose a that is not nilpotent, since otherwise a'H(R)' is a right ideal of H(R) that consists of nilpotents, so it is nilpotent, and therefore a'R' is a nilpotent right ideal of R', contradicting the fact that a' ^ 0. Theorem 1.13 tells us that a'H(R)'a' C G U 0 for a maximal subgroup of the multiplicative semigroup of Fn. Therefore it has no zero divisors. Replacing a by some aN we can also assume that every projection of a onto a right primitive homomorphic image of R lies in a maximal subgroup of this image. It is easily verified that J{R) HaRa C J (aRa). The choice of a implies that the image of aRa in every right primitive homomorphic image of R is a matrix ring over a division ring. Therefore it follows that J{R) D aRa = J {aRa). Note that aRa is an 5-graded subring such that H(aRa) C (PnH(R))U B (a disjoint union) for the subsemigroup B = {x G aH(R)a | x' G G}. Let K be the additive subgroup generated by P n H(R). Clearly, if is a homogeneous ideal of R and a S" K. Moreover, R/P is a homomorphic image of R/K. Hence, to get a contradiction, we may assume that K = 0. Then H(aRa) = B U {0}. From Lemma 4.16 we know that supp(aPui) is a permutational subsemigroup of S. Therefore Lemma 4.17 implies that J{aRa) is homogeneous. Since aza G J{R) for some z G R (because a G W), it follows that a3 G J (aRa) C J(R). Thus, o 3 G P, which contradicts the choice of a. • Similarly, we get the following Proposition 4.4 For every cancellative semigroup S, the Baer radical of every S-graded Pi-algebra over afield of characteristic zero is homogeneous. Next, we prove Theorem 4.13. Proof. Let S be a u.p.-semigroup, and let R = ®slzsRs ring.

be an S-graded

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(i): Given that 5 is a u.p.-semigroup, [Jespers et al. (1982)], Theorem 2.2, tells us that the Levitzki radical £(R) is homogeneous. Since J^gr(R) = 0, we must have C(R) = 0. Every nonzero w in W = H(R)1rH(R)1 also belongs to J{R) and is of the same length. Let M be the multiplicative semigroup generated by H(w). Obviously, M'^wM1 C J(R), and so all elements of M1wM1 are quasiregular. Proposition 4.3 shows that M is nilpotent. In particular, all the elements of H{w) are nilpotent. Therefore the set V = H(W) consists of homogeneous nilpotent elements. As above, V is a multiplicative ideal oiH(R). Suppose that all nil subsemigroups of H(R/JgT{R)) are locally nilpotent. Then V is locally nilpotent, and so the ideal A of R generated by V is locally nilpotent. Therefore V C C{R) = 0. This contradicts the choice of r ^ 0 and shows that J{R) = 0, as desired. (ii): Suppose that every nil subsemigroup of every right primitive homomorphic image of R is locally nilpotent. As above we see that V is a nil semigroup. By the assumption, every image of V in a right primitive homomorphic image of R is locally nilpotent. Hence, the corresponding image of A is a locally nilpotent ideal, so it is zero. It follows that A C J(R) and therefore A C JgT(R) = 0. This contradicts the fact that 0 ^ r e A. (iii): Suppose that, for every minimal prime ideal P of R, the ring R/P is a domain or embeds into a matrix ring over a division ring. We must prove that J(R) = J&(R). If J is a right primitive ideal of R, then J contains a minimal prime ideal P of R. Moreover, P is a homogeneous ideal by [Jespers et al. (1982)]. Therefore it is enough to show that the radical of every R/P is homogeneous. So, we may assume that R is a domain or a subring of Fn, for a division ring F and some n > 1. As above, suppose that the Jacobson radical J{R) is not homogeneous, choose an element r of minimal length in J{R) \ Jgr(R) and introduce W = HiR^rHiR)1, V = H(W) and the additive subgroup A generated in R by V. We are going to prove that A is quasiregular. This will imply H(r) C A C J{R), and will give a contradiction with r ^ J&{R). Fix any element c in W, say c = arb, where a, b e H(R)1. Choose any t ^ e in S and consider the homogeneous component v = ct of c. Clearly, R/JgT(R) is graded by the u.p.-semigroup S. By the choice of r, the image

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d of c in R/Jer(R) is a rigid element. Since t ^ e, Proposition 4.3 shows thatrf™= 0 for some n > 1. Hence vn £ J{R). Given that S is u.p., we get tn ^ e. Therefore applying Proposition 4.3 to vn £ J{R) we see that vn is nilpotent. Hence ct is nilpotent. First, we consider the easier case where R is a domain. Then we get ct = 0. This implies that c — ce. Hence W = We, where We = {ce | c € W } . Denote by Ae the additive subgroup generated in R by W e . Then A = Ae. Proposition 4.3 implies that We C J(Re), and so J4C C J(Re). Clearly, Ae is an ideal of Re. Therefore A = Ae is quasiregular, as claimed. Second, consider the case where R is embedded in a matrix ring Fn, for a division ring F and a positive integer n. Assume that n > 1, since otherwise F n = F is a domain again. For i = 0 , 1 , . . . , n, denote by Mj the set of all matrices of rank < i in Fn. Given that V is an ideal of H(R), it follows from Theorem 1.13 that V* = V n M j is an ideal of H(R). Let At be the additive subgroup generated by V* in R. Then AQQ A\ Q • • • Q An = A is an ideal chain of the ring R. We use induction on k to prove that all Ak are quasiregular. Obviously, Ao — 0 is quasiregular. Assume that A^-i is quasiregular for some 0 < k
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c = aub. Choose any t in supp(c) \ {e}. We have seen above that ct must be nilpotent. On the other hand, in view of Theorem 1.13, H{£) C DL)Eap. Since 0 £ Gap D Eap, we see that Eap has no nilpotent elements. Therefore it € D for all t ^ e. It follows that s = e and the images of c and £ = ce in Q coincide. Hence c-l G N = J(N). Given that c G J(R), we get € € J(R). Therefore L C J"(i?). But Q 3 is the additive group generated by L (viewed as a subset of Q). Hence Q3 C J(I/N). Since Q is a left ideal of a right ideal of I/N, it follows that Q3 Q J{Q)- So Q is quasiregular, as desired. We have proved that all Iap/N are quasiregular, for all a/3 6 A, and hence I/N is quasiregular. As we have seen, this implies that all Ak are quasiregular. Thus A = An is a quasiregular ideal of i?, as claimed. This contradicts the choice of r and completes the proof. D 4.5

Radicals and Homogeneous Components

This section is devoted to the question of how the radical of the whole ring R = ®sesRs depends on the radicals of the components Re, where e runs over the set of idempotents of S. This problem has been considered in the literature for various types of gradings (see, in particular, [Bell et al. (1996)], [Chick and Gardner (1987)], [Clase and Jespers (1994)], [Karpilovsky (1991)], Chapter 4, [Munn (1992)], [Nastasescu and Van Oystaeyen (1982b)], [Teply et al. (1980)], [Wauters (1986)], [Wauters and Jespers (1989)], and [Weissglass (1973)]). We say that the radical g is determined by the idempotents of a semigroup S if, for every S'-graded ring R, the radical g(R) is equal to the largest ideal I with the property that fe{I) C g(Re) for all idempotents e in S. The following theorem was proved in [Kelarev (1990b)] (see also [Kelarev (1991a)], [Kelarev (1992e)], and [Kelarev (1993c)]). Theorem 4.14 ([Kelarev (1990b)]) The Jacobson and Levitzki radicals are determined by the idempotents of a semigroup S if and only if S is a locally finite combinatorial semigroup. More is known for special band-graded rings. If R is a B-graded ring, b G B and x G R, then we put x =£•£=

2_^ aeB\beBaB

Xa

~

Radicals and Homogeneous

65

Components

Theorem 4.15 ([Munn (1992)]) Let B be a band, and let R = ®beBRb be a special B-graded ring, where lb is the identity of Rb- Then the Jacobson radical J{R) is given by J(R)

lbxblb e J(Rb)

= {xeR\

for all b € B}.

A different description of the radicals of special semilattice-graded rings was given by Ponizovskii [Ponizovskii (1984)]. Bell, Stalder and Teply [Bell et al. (1996)] obtained a description of the radicals, analogous to that given in Theorem 4.15, for a larger class of graded rings: Theorem 4.16 ([Bell et al. (1996)]) Let S be a semigroup with a preorder -< such that a -< ab and b -< ab for all a, b G S. Suppose that every prime ideal of S is completely prime, and that, for any prime ideal P of S and for any a,b € S\P, there exists c £ S\P such that c'ac" = c'bc" for all c', c" € (c)\P. Let R = ®sesRs be an S-graded ring. Suppose that for any a ~< b there exist homomorphisms 4>a,b '• Ra -^ B-b such that xy = 4>a,ab{x)4>b,ab{y) for all a,b € S, x e Ra, y G Rb, and b,c(a,b{x)) = <j>a,c(x) f°r a^ a -< b -< c. Then J{R) = {reR\

Y,

Mr

a) € J{Rb)

for all

beS}.

a with ab=b

This theorem is proved in [Bell et al. (1996)] even for a large class of socalled directed hereditary radicals (and so it is also true for Levitzki radical). Theorem 4.16 covers all commutative combinatorial periodic semigroups and all finite bands. A description of prime ideals of the rings satisfying the hypotheses of Theorem 4.16 is given in [Bell et al. (1996)]. Theorem 4.17 ([Kelarev (1990b)]) The Baer radical is determined by the idempotents of a semigroup S if and only if S is a locally finite combinatorial semigroup and every Rees factor of S has a nilpotent or minimal ideal. The main theorem of [Kelarev (1991a)] divides all bands into 18 classes, and, for bands B of each class, describes all radicals determined by the components of B-graded rings. Let us include only one corollary. Corollary 4.2 ([Kelarev (1991a)]) Let B be a band, and let Y be the largest semilattice homomorphic image of B. Then the following conditions are equivalent:

66

The Jacobson

Radical

(i) for every B-graded ring R = ©bgB-Rfc; the Baer radical B(R) is equal to the largest ideal I such that fs3\(x) £ B(RS), for all x € I,seS; (ii) S satisfies the descending chain condition for idempotents. The next interesting question is that of when there exists a nice relation between the radical of the whole ring R and those of homogeneous components Re, where e = e 2 . The investigation of radicals which have a good structure in graded rings was initiated by Gardner [Gardner (1975)]. Recall that a mapping g is called a radical if, for every ring R with ideal / , (i) g(R) is an ideal of R;

(ii) g(R/g(R)) = 0; (iii) g(R/I)2g(R)+I. All of these properties hold for the Jacobson, Baer, and Levitzki radicals. A radical g is strict if g(T) C g(R) for every ring R with a subring T. All strict radicals were characterized by Stewart [Stewart (1973)]. A radical is said to be weakly hereditary if g(I) 2 g(R)I + Ig(R) for each ring R and every ideal / of R. T h e o r e m 4.18 ([Kelarev (1989b)], [Kelarev (1992c)]) Let S be a nontrivial semilattice, and let g be a radical. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

g(®sesRs) = ®SGSQ(RS) for every S-graded ring R; g((BsesRs) C ®sesg(Rs) for every S-graded ring R; g(Re) = Re n Q(R) for any S-graded ring R and any e £ S; g is strict and weakly hereditary.

Theorem 4.19 ([Kelarev (1992c)]) Let S be a nontrivial semilattice, and let g be a radical. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

g is homogeneous in every S-graded ring; g(®sesRs) 2 ®s€SQ(Rs) for every S-graded ring R; if R = (BstsRs is g-semisimple, then all Rs are g-semisimple; g is strict.

T h e o r e m 4.20 ([Kelarev (1995b)]) Let S be a semigroup with zero, E the ideal of S generated by all idempotents. Then the following are equivalent: (i) J {Re) = J{R)C\Re for each contracted S-graded ring R = ©Sgs-Rs and every idempotent e in S;

Nilness and

Nilpotency

67

(ii) the largest nil ideal N of E is locally nilpotent, and if E ^ 0, then E/N is a 0-direct union of inverse completely 0-simple semigroups with locally finite maximal subgroups. Let S be a semigroup with zero. A radical g is said to be invariant in S-graded rings if and only if Rxg(Re)Ry C g(Rf) for every contracted S-graded ring R = (BstsRs, where e, / are any nonzero idempotents of S and x, y are any elements of S such that xey = / . Since this definition involves only nonzero idempotents, it applies to graded rings and contracted graded rings in exactly the same way. Sands [Sands (1989)] proved that a radical is invariant in a group-graded ring or a Morita context if and only if it is normal. Normal radicals were described by Sands [Sands (1975)]: A radical is normal if and only if it is left strong and principally left hereditary. A radical is said to be left strong if every radical left ideal of each ring R is contained in the radical of R. A radical is principally left hereditary if, for every radical ring R and element a £ R, the left ideal Ra is also radical. For any ring R, the ring with zero multiplication and the same additive group as R will be denoted by R+. A radical g is called an A-radical if g(R) = g(R+) for every R. Theorem 4.21 ([Gardner and Kelarev (1997)]) Let g be a radical, S a semigroup, E the ideal generated by all idempotents in S, and let N be the union of all ideals of S which are contained in E, but do not contain nonzero idempotents. Then the radical g is invariant in S-graded rings if and only if at least one of the following conditions holds: (i) g is an A-radical; (ii) the quotient semigroup E/N is a semilattice of height < 2; (iii) all principal factors of the quotient semigroup E/N are singletons and g is strict and weakly hereditary; (iv) the radical g is normal and the quotient semigroup E/N is an inverse semigroup in which every nonzero idem/potent is primitive.

4.6

Nilness and Nilpotency

A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [Karpilovsky

68

The Jacobson

Radical

(1991)], [Rowen (1991)]). The following new results have been obtained recently. Theorem 4.22 ([Clase and Jespers (1994)]) Let S be a finite (locally finite) semigroup, and let R = © geC ;i? g be an S-graded ring. If the radicals J{Re) are nilpotent (locally nilpotent) for all e £ E(S), then J{R) is nilpotent (respectively, locally nilpotent), too. Theorem 4.23 ([Kelarev and Okniriski (1995)]) Let G be a group with identity e, and let R = (BgeGRg be a G-graded Pi-ring. If J{Re) is nil, then J{R) is nil, too. Theorem 4.24 ([Kelarev (1993d)]) Let G be a non-torsion group with identity e. Then there exists a strongly G-graded ring Q such that J(Qe) = B(Qe) butJ{Q)^C{Q). The theorem above answered a question asked in [Puczylowski (1993)]. Theorem 4.25 ([Kelarev (1993d)]) Let S be a commutative The following are equivalent:

semigroup.

(i) for every strongly S-graded ring R = ®sesRs, if all the Jacobson radicals J{Re) are nilpotent for all e £ E(S), then the radical J{R) is nilpotent, too; (ii) S has an ideal I such that S/I is nilpotent and I is a finite union of groups with finite torsion parts. Theorem 4.26 ([Kelarev (1993d)]) Let S be a linear epigroup over any field, and let R = ®s&sRs be a strongly S-graded algebra over a field of characteristic zero. If all radicals J{Re) are nilpotent for e £ E(S), then J{R) is nilpotent, too. Lemma 4.18 Let R be a G-graded PI-ring, I a homogeneous ideal of R contained in J{R). If Ie is nil, then I is nil. Proof. Take any element r in H(I). Since I is homogeneous, r £ I. Let r £ J(R) n Rg, where g £ G. If g is a periodic element, then there exists a positive integer n such that rn £ lDRe — Ie, and so r is nilpotent. Further, assume that g is not periodic. Denote by T the infinite cyclic group generated in G by g. Lemma 2.2 shows that r £ J(RT), and therefore r is nilpotent again in view of [Karpilovsky (1991)], Theorem 32.5. Thus H(I) is a multiplicative nil subsemigroup of R. Since R satisfies a polynomial

Nilness and

Nilpotency

69

identity, it follows from [Rowen (1980)], Theorem 1.6.36, that / is nil, as required. • Lemma 4.19 Let G be a permutational group, R a G-graded Pi-ring. J(Re) is nil, then J{R) is nil.

If

Proof. By Theorem 1.18, G is finite-by-abelian-by-finite. Take any r £ J (Re)- Denote by S the subgroup generated in G by the support of r. It is easily seen that S is also finite-by-abelian-by-finite. Lemma 2.2 implies r € J{Rs), and so without loss of generality we may assume that G is finitely generated itself. By Theorem 1.18, G is abelian-by-finite, that is G has an abelian normal subgroup A of finite index. Then A is finitely generated, too (see [Kargapolov and Merzljakov (1979)]). Therefore G contains a torsionfree abelian subgroup T of finite index. By Lemma 2.1, R is graded by the finite group G/T with the identity component RT- Therefore [Okniiiski (1986a)], Lemma 1.1(1), shows that it suffices to prove that J{RT) is nil. However, J(RT) is homogeneous by Theorem 4.7, because T is torsion-free abelian. Lemma 4.18 completes the proof. • Lemma 4.20 Let G be a permutational group, R a G-graded Pi-algebra over a field of characteristic zero. If J{Re) is nil, then J(R) is homogeneous. Proof. We verify that H{J(R)) consists of nilpotent elements. Then [Rowen (1980)], Theorem 1.6.36, will show that H(J(R)) generates a homogeneous nil ideal I in R, and so J(R) = I is homogeneous. • Pick any r G J{R) and g e supp(r). We claim that rg is nilpotent. As in the beginning of the proof of Lemma 4.19, we may assume that G has a torsion-free abelian subgroup T of finite index. If we look at the natural G/T-gradation of R and apply Lemma 2.1, Theorem 4.7 and the fact that our field has characteristic zero, then we conclude that J{R) is G/T-homogeneous. We may assume that the whole supp(r) is contained in one T-coset of G (otherwise we would pass to the G/T-homogeneous summand of r corresponding to the coset containing g). Since G/T is finite, there exists a positive integer n such that rr™ £ RT. Given that J(Re) is nil, [Karpilovsky (1991)], Theorem 32.5, implies that all the homogeneous summands of rr™ are nilpotent. Therefore rg is nilpotent, as required. Now we can prove Theorem 4.11.

70

The Jacobson

Radical

Proof. By Lemma 4.18, the largest homogeneous ideal I of R contained in J(R) is nil. Obviously, R/I is a G-graded ring, and J(R/I) = J(R)/I. Therefore it suffices to prove Theorem 4.11 for R/I. To simplify the notation we may assume from the very beginning that 1 = 0. Suppose that J{R) ^ 0. Choose a nonzero element r with a minimal length in J(R). Denote by T and S the subgroup and, respectively, subsemigroup generated in G by supp(r). Let M = M{r) be the multiplicative subsemigroup generated in R by H(r). We claim that H (r) consists of nilpotent elements. If S is permutational, then the group T is also permutational by Theorem 1.18, and so all elements in H(r) are nilpotent in view of Lemmas 4.19 and 4.20. Further, consider the case where S is not permutational. Let n be the degree of a multilinear identity (1.5) of R. There exist elements sx,...,sn in S such that Si... sn ^ s CT i... sCT„, for all 1 ^ a € Sn. Clearly, there exist xi,..., xn € M such that Xi £ RSi for alii = 1 , . . . , n. Applying (1.5) to the elements x\,... ,xn we get

xi...xneRSl...Snn

^2

Rs„i...s„n,

(4-1)

whence xi...xn = 0. It follows that yi... ym = 0 for some yi,...,ym € H(r). Then we can choose m and 2/1,..., ym such that yi... ym-i ¥" 0Now look at the product y\... ym-ir. It also belongs to J(R) but has fewer homogeneous summands than r. Hence j / i . . . ym-ir = 0 by the choice of r. Since G is a group, we get y\... ym-\H{r) = 0. Further, we can look at yi... ym-2'>'H(r) = 0 and infer j / i . . . ym-2{H(r)2) = 0. Reasoning like this m times, we conclude (H(r))m = 0. In particular, all elements in H{r) are again nilpotent. Denote by L the ideal generated in H(R) by H{r). Each nonzero element of L is a homogeneous summand of a certain element of positive minimal length in J{R). It follows from what we have proved, that L is a nil ideal of H{R). Hence L is locally nilpotent by [Rowen (1980)], Theorem 1.6.36. Therefore L generates a nil subalgebra K in R. Evidently, K is a homogeneous ideal of R, and so K C I, contradicting the fact that 7 = 0. Thus J{R) = I, and so J{R) is a homogeneous nil ideal of R. D Corollary 4.3 Let G be a group with identity e, and let R = ©sgG-R9 be a strongly G-graded PI-algebra over a field of characteristic zero. IfJ{Re)

Nilness and Nilpotency

71

is nilpotent, then J(R) is nilpotent. Proof. By Theorem 4.11, we get J(R) = ®g
h

Then x^yji

e

^ e ' ^ or a ^ •?!• Further, suppose

that for some i = 2 , . . . , n elements yj*~ duced such that x2i-2 = Y~] Vj*~, ^ ' J i

an<

and z?~_ ' have been intro^

au

^jlli

e

-^si-szi-z- Then

ji-i

£2i S -^32, = ^(gi...g2i-i)~1^9i--92i' anc ^ s o there exist homogeneous elements y£ £ •R(Sl...S2i_1)-i and z]' 5 e Rgi...g2i such that x 2 i = ^ Z ^ j i £ and all z^J^a^i-iyj* belong to 7 e . Therefore X\...X2n

= I («-l)

which completes the proof.

(ri)\

(n)

D

Next, we prove Theorem 4.23. Proof. We prove that J(R) = B(R). To this end we show that R/B(R) is semisimple. Suppose to the contrary that J{R/B{R)) ^ 0. Since R/B(R) is a subdirect product of prime PJ-rings, there exists a prime Pi-ring R and a homomorphism / of R onto R such that f(J(R)) ^ 0. By Posner's theorem (see [Rowen (1980)]) R is contained in a matrix ring Dm, where D is a division ring. For any r £ R and T C P , put r = f(r), T = f(T). Consider the set L of all x £ H(R) such that either x = 0 or x has the smallest nonzero rank in Dm. By Theorem 1.13, all nonzero elements of L lie in the same completely 0-simple factor F of the multiplicative semigroup Dm. Obviously, L is a multiplicative ideal of H(R), and so L is a multiplicative ideal of H(R).

72

The Jacobson

Radical

Put M = LJ(R)L = {xay | a € J(R),x,y £ L}. Clearly, M is a multiplicative subsemigroup of R, because J{R) is an ideal of R. Denote by / the subring generated in R by L. It follows that / is an ideal of R and I is an ideal of R. Since 1^0 and R is prime, we get IJ(R) =fc 0, and so IJ{R)I ^ 0. Since / consists of all finite sums of elements of L, we get M ^ O . We show that every element in M is nilpotent. Fix any nonzero w £ M. There exists w = axb, such that a,b £ L, 3

x £ J{R),

x = y_]xk,

and all x^ are homogeneous. Elements a and b

fc=i

belong to the same completely 0-simple factor F of Z) m . We can represent F as a Rees matrix semigroup F = M(G°;I,A;P). Let a = (i,g,X) and b = (j, h, fi), where i, j £ I, A, /i € A, g,h £ G. It is routine to verify that axkb £ (i,G°,fi) for every k = l,...,s. If Pui = 0, then (i,G°,(i)2 = 0 in F, and therefore (axb)2 = 0 in F. Hence every element (ax^b)(lixib) has a smaller rank in Dm than a. Therefore (ax^b)(ax~ib) = 0, for all 1 < k, I < s. Thus w2 = 0, as required. Further, consider the case where p^i ^ 0. By Theorem 1.13, P = (i, G, fi) is a multiplicative subgroup of Dm. Put T = { 5 e G | Rg n P ^ 0}. Clearly, T is a subsemigroup of G. Let n be the degree of a multilinear identity (1.5) satisfied in R. For any g\,...,gn € T, we can choose r^ e i?Sfc such that ¥k £ P, where k = l , . . . , n . Applying (1.5) we get (4.1), which implies that T is permutational. By Theorem 1.18, T generates a permutational subgroup Q in G. Lemma 4.19 shows that J(RQ) is nil. Since all nonzero summands axkb belong to P, where k = 1 , . . . , s, we get axkb £ RQ, and so axb £ J(RQ) by Lemma 2.2. Therefore w = axb is a nilpotent element. Thus M is a multiplicative nil subsemigroup of Dm. Hence M = 0 for some q > 1 by [Rowen (1991)], Proposition 2.6.30. Since M is, evidently, closed under multiplication by elements of H(R), it follows that M generates a nilpotent ideal in R. This contradicts the primeness of R and completes the proof. •

Chapter 5

Groups of Units

If R is a ring with identity element 1, then denote by U(R) the group of units of R, i.e., the set of left and right invertible elements. Every ring is a monoid with respect to the circle composition o defined by xoy = x+y—xy. The group of units of this adjoint monoid (R, o) is called the adjoint group of R and is denoted by V(R). If R has an identity element, then it is easy to verify that x £ U(R) & 1 — x £ V(R). Moreover, the group of units is isomorphic to the adjoint group. Theorem 5.1 ([Kelarev (1994d)]) Let R = (BbeBRb be a special bandgraded ring, and let x £ R. Then x £ V{R) if and only if ltaJr-lb £ V{Rb) for each b £ subs(:r). By supp B (r) we denote the set of all b £ B such that rj, ^= 0. Let subs(r) be the subsemigroup generated in B by supp B (r). Since every band is locally finite, subs(r) is a finite set. The next lemma was obtained in [Chick and Gardner (1987)] and in [Teply et al. (1980)]. Lemma 5.1 Let C be a semilattice, R = ®cecRc a semilattice graded ring, c £ C, and let f : R —> R- be the mapping defined by f(r) = r-~ for r £ R. Then f is a homomorphism of R onto R-. Now let C be a semilattice, R — (BcecRc a semilattice graded ring, i V € V(R)t a n d let xoy = yox = 0. It is easy to give an example showing that supp c (a;) and supp c (y) may happen to be distinct. However, we show that subc(a^) = subc{y)x

Lemma 5.2 If C is a semilattice, R = ®cecRc o- semilattice graded ring, x,y £ V(R) and xoy = yox = 0, then subc(x) = subc(y)73

74

Groups of Units

Proof. Suppose to the contrary that there exists an element m belonging to only one of the sets subcOc) and subc(y)- Choose a maximal element m with this property. We may assume that m £ subc(y)- Put A = sut>c(x)nm and B = subc(y)nm. By the choice of m, we get B = {m}UA. Denote by p the product of all elements of A. Since subc(^) is a subsemigroup of C, it is clear that A is also a subsemigroup of C, and so p £ A. Therefore x~ = x~ and y— = ym + y~. By Lemma 5.1, we get x c + V x~y^ = x^ + y~— x^y^ = 0, whence ym + x^ym = 0. Since m

a

m

m^m

p

a

p

p P

m^'

l

x £ V(R), Lemma 5.1 shows that x~ £ V(R), and so 1 + x~ £ [/(.R1). Therefore ym = 0, a contradiction. • Every band B is a semilattice C of rectangular bands Hc, where c £ C. For b £ B, denote by b the image of b in C. Let R = (BbeBRb be a band graded ring, r = ^2beB rb £ R. If we put Rc = X^= c Rt>, t n e n R — ®cecRc is a semilattice graded ring, and r c = Yjb=crb- ^ e n e e d t n e following Lemma 2.3 of [Munn (1992)]. Lemma 5.3 If R — (BbeBRb is a special band-graded ring, b £ B and x, y £ R^, then xlby = xy, Now let R = (BbeBRb be a special band-graded ring. For a,b £ B with a > b, define fg:Ra—> Rb by fb(x) = hxlb (where x £ Ra). The following basic properties of these mappings were established in [Munn (1992)], Lemma 2.2. Lemma 5.4 Let R = (BbeBRb be a special band-graded ring, and let a,b £ B with a >b. Then (i) fjj is a homomorphism; (") fbfa =_fb f°r anyceB with c > a; (hi) if a = T> then fjj is an isomorphism, with inverse

f\.

Lemma 5.5 Let R = (BbeBRb be a special band-graded ring, b £ B, and let f : RT- —> Rb be defined by f(x) = lbxlb for x £ R~. Then f is a homomorphism of R?- onto RbProof. The equality f(x + y) = f(x) + f(y) is obvious. Hence it suffices to check f(xy) = f(x)f(y) for x £ Rg,y £ Rh, where g,h £ b. Let B be a semilattice C of rectangular bands Hc, and let b £ Hc for c £ C. Put

75

H = Hc. Then lbX,ylb £ RH, and so Lemma 5.3 yields f(xy) = lbxylb = lbxlbylb = f(x)f(y), completing the proof. • Lemma 5.6 Let R = (BbeBRb be a special band-graded ring, and let B be a semilattice C of rectangular bands Hc. For each c £ C choose an element e(c) in Hc and define a mapping f : R —> r i c e c -^e(c) by the rule f(x) = ^2C£c le(c)a;'7~\le(c) for x £ R. Then f is a homomorphism of R into the direct product D = Ilcec ^e(c)> and the kernel of f is a nil ideal of R. Besides, if C is finite, then f(R) = D. Proof. Lemmas 5.1 and 5.5 show that / is a homomorphism. Now take any x £ Ker(/). We claim that x is nilpotent. Let m be a maximal element in subc(a;). Look at any a,b £ supp B (a;). If a ^ m or b ^ m, then agb ^ m for any g £ supp B (x). Further, if a = b = m, then Lemma 5.3 implies xaxmxb = xale{m)xml^m)Xb = a; a l e ( m ) a;---l e ( m )a; 6 = 0. Therefore (x3)m = 0, and so subc(a; 3 ) C sub(x)\{m}. Hence the induction on |subc(x)| shows that x is nilpotent. Now assume that C is finite and take any c € C. Put z

= l[0-e(c) ~ le(d)), d
where d £ C. It is routine to check that f(zR) = Re(c)- Since C is finite, we get f(R) 2 Ylcec -^e(c) = Elcec -^e(c)i which completes the proof. • We also need the following well-known and easy Lemma 5.7 Let R = (BbeBRb be a ring, I a nil ideal of R. Then x € V(R) if and only if x +1 £ V(R/I). Now we can prove Theorem 5.1. Proof. Since the element e(c) in Lemma 5.6 was taken arbitrarily in Hc, Lemma 5.6 shows that x £ V(R) implies lbX^lb £ V(Rb), for any b £ B. Now suppose that lbX^h S V(Rb), for any b £ subs (a:). Put A = subs {x). Then A is a finite band and RA is a special band-graded ring with the same components. Let / be the mapping from RA into P = YlceC Re(c) defined as in Lemma 5.6. Given le^x-~~le^ £ V(Re(C)), it follows that fix) £ V(P). By Lemma 5.6, f(RA) = P. Therefore RA/Ker(f) ^ P, and so Lemmas 5.6 and 5.7 imply that x £ V(R). This completes the proof. • By Theorem 5.1, in order to verify that x £ ViR) it suffices to look at a finite number of the components Rb, because subs (a;) is finite. At

76

Groups of Units

the same time, the description of the Jacobson radical J(R) shows that x £ J{R) often depends on infinitely many of the R), (see Theorem 4.15). This is the key reason why V(R) admits a less complicated characterization

than J(R). The main theorem of this chapter was used in [Kelarev (1994d)] to reduce the group of units of commutative monoid rings to the groups of units of group rings. Note that the Baer radical B(R[S]) of commutative semigroup rings has been completely described in terms of separative congruences of S by Munn (see [Munn (1983a)], [Munn (1985)], or the survey [Kelarev (1994c)]). The group of units U(R[S]) of the commutative semigroup ring R[S] is isomorphic to the adjoint group V(R[S]) with respect to circle composition. For r £ R[S], by cliff(r) we denote the subsemigroup of S generated by all subgroups intersecting supp(r). It follows from Theorem 1.15 that cliff (r) is a disjoint union of a finite number of groups. Therefore cliff (r) contains a finite number of idempotents. For each idempotent e £ S, denote by Ge the largest subgroup containing e. Theorem 5.2 ([Kelarev (1994d)]) Let R be a commutative ring, S a commutative semigroup, G the union of all subgroups of S. Then V(R[S]) = V(R[G]) + B(R[S)). Besides, an element r £ RG belongs to V(R[G]) if and only if er7£V(R[Ge}), for all idempotents e £ cliff (r).

Chapter 6

Finiteness Conditions

6.1

Groupoid-Graded Rings

Several interesting results establish connections between the properties of a graded ring R and homogeneous components Re, where e is an idempotent of S. Denote by E(S) the set of idempotents of S. Let K be a class of rings. The following implication Re G K for all e G E(S)

=^> R e tC

(6.1)

Re G K for all e G E{S) -£=> R G K

(6.2)

and equivalence

have been considered. The main theorem of this section shows that, for ring classes K. with certain natural closure properties, as soon as the relation (6.1) or (6.2) has been verified for rings graded by finite groups, it immediately holds for rings graded by finite groupoids (Theorem 6.1). For rings graded by finite groups results of this sort are deducible with the use of duality theory of Cohen and Montgomery [Cohen and Montgomery (1984)]. A class K. of rings is said to be closed under finite sums of one-sided ideals if and only if, for every ring with right (or left) ideals A,B G /C, it follows that A + B G tC. Our proof applies to both (6.1) and (6.2), and so we combine "if and only if" and "provided that" parts in one theorem. Theorem 6.1 ([Kelarev (1995c)]) LetK be a class of rings which contains all rings with zero multiplication and is closed for homomorphic images, 77

78

Finiteness

right and left ideals, ring extensions. equivalent:

Conditions

Then the following assertions are

(i) for each finite groupoid S, an S-graded ring R — ®s£sRs belongs to K. provided that (if and only if) Re belongs to K for every idempotents e of S; (ii) for each finite semigroup S, an S-graded ring R = ®sesRs belongs to K. provided that (if and only if) Re belongs to K, for every idempotent e of S; If, moreover, K, is also closed for finite sums of one-sided ideals, then the following is equivalent to the above assertions: (iii) for every finite group G with identity e, a G-graded ring R = ®geGRg is in K provided that (if and only if) Re G K. Proof. The implication (i)=>(ii) is trivial. Assume that (ii) holds. We claim that then K is closed for finite sums of one-sided ideals. Clearly, it suffices to consider a ring A which is the sum of its two right ideals M and N from K. Let L = {a, b} be the two element semigroup such that ab = a = a2, ba = b = b2. In the semigroup ring A[L] consider subrings Ra — Ma and Rb = Nb. Then R = Ra + Rb is L-graded. Since Ra = M and Rb = N belong to AC, it follows from (ii) that R G K,. It is easily seen that I = {k(a — b) | k G M n N} is an ideal of R and R/I = A. Given that K is closed for homomorphic images, we get A G K, as required. Since (ii)=4>(iii) is also easy, it remains to prove (iii)=>(i) with the extra hypothesis that K. is closed for finite sums of one-sided ideals. Let S be any finite groupoid, R = ®szsRs a n -S-graded ring and / a homogeneous two-sided ideal of JR. Given that K, is closed for ideals and homomorphic images, it is easy to prove that R is a counterexample for (i) if, and only if, either / or R/I is a counterexample for (i). Suppose now that R = ®s&sRs is a counterexample for (i) with \S\ minimal. We claim that, given any s G 5 and any additive subgroup A of Rs such that supp(yl.R) ^ S, there is a two-sided homogeneous ideal I of R such that A C 7, I G K, and Ie G K. for every idempotent e € E(S). Obviously, I is not a counterexample for (i). As a consequence, we shall be able to factor out I, and get a new counterexample to (i) with A = 0. If AR = 0, we take I = RlA. Then I2 = 0, and so I G K and It G /C, for every t G S.

Groupoid-Graded

Rings

79

If AR = P ^ 0 then, by the minimality of \S\, P cannot be a counterexample to (i). In the "provided that" part of our theorem, R being a counterexample implies that Re is in /C, for every e £ E(S). Then Pe £ K. for every e £ E(S), because Pe is a right ideal in Re. Since P satisfies (i), we get P £ fC. (In the "if and only if" version of the theorem, R being a counterexample implies that either R £ K or Re is in fC, for every e £ E(S). If R £ JC, then P £ K. because P is a right ideal of R. If i? e is in /C, for every e e JS(S'), then Pe is also in /C, and so P £ /C, again.) But supp(i? x P) ^ S1, for every x £ S, and the same argument applied to the additive subgroup RXA contained in Rxs tells us that RXA and (RxA)e are in K, for every idempotent e £ S. Given that fC is closed for finite sums of two-sided ideals, it follows that I = RlP = P + Y^xes RxP is a homogeneous two-sided ideal with the desired properties. If now s £ S and sS ^ S, then supp(RsR) C sS =£ S and, putting A = Rs above, we get a counterexample for (i) whose s-th homogeneous component is zero and hence can be graded by the set T = S\{s}. We can introduce a multiplication on this set, make it a groupoid and get a contradiction to the minimality of \S\. Therefore sS = S and, by changing sides in this argument, we get Ss = S. Thus S is a left and right simple groupoid. We claim that it is also associative and it is thereby a semigroup. Indeed, if (st)x / s(tx) then RsRtRx Q R(st)x^Ps(tx) implies that RsRtRx = 0 and so s\ipp(RsRtR) ^ S, because S is finite. By applying the above paragraph with A = RsRt, we may assume RsRt = 0. Again using the same reduction with A = Rs, we can also assume Rs — 0, because supp(RsR) ^ S. This yields a contradiction with the minimality of | 5 | . So our claim has been established. By Lemma 1.1, 5 is a group, and we get a contradiction with (iii), which completes the proof. • The following example shows that the closedness restrictions on K are essential in Theorem 6.1. Example 6.1 The class A4 of Brown-McCoy radical rings satisfies implication (6.1) in (iii), but not in (ii). It is well-known that M. contains all rings with zero multiplication and is closed for ring extensions, homomorphic images and ideals ([McCoy (1964)], § 37). Theorem 5 of [Grzeszczuk (1985)] says that, for every finite group G with identity e, each G-graded ring R belongs to M provided that Re £ Ai. Take a simple non-Artinian domain R with unity (for example, the Weyl

80

Finiteness

Conditions

algebra Ai). Pick two different maximal right ideals M and N in R. It is proved in [Beidar (1982)], Lemma 2, that M and N are simple. Since R has no nonzero idempotents, M and N are rings without identities, and so they are Brown-McCoy radical rings. However, R = M + N is BrownMcCoy semisimple. Therefore M is not closed for sums of two right ideals. It follows from the second paragraph of the proof of Theorem 6.1 that M. does not satisfy (ii). For groupoid gradings the terms "graded ring with finite support" and "ring graded by a finite groupoid" are the same. As a consequence, conditions (i), (ii) and (iii) of Theorem 6.1 are equivalent, under the given hypotheses, to the corresponding statements obtained by replacing "ring graded by a finite groupoid (semigroup, group)" by "groupoid (semigroup, group)-graded ring with finite support," and so the theorem can be rewritten for graded rings with finite supports. In the "if and only if" version we can slightly weaken the closedness restrictions imposed on the class in the hypothesis of Theorem 6.1. Indeed, the example in the last paragraph of the proof of necessity of [Kelarev (1991a)], Theorem 6.1, shows that every class K, which contains all rings with zero multiplication, is closed for ring extensions and homomorphic images and satisfies the "if and only if" version of (ii) is also closed for one-sided ideals. For completeness we include a proof of this fact. Consider a ring A G K with a right ideal / . Let L = {c, d} be a semigroup such that cd = c = c 2 , dc = d = d?. Then R = Ac + Id is L-graded. It is readily verified that N = {i(c — d) \ i G 1} is an ideal of R and iV2 = 0. Since ./V G K and R/N =* A G /C, we get R G K. The "if and only if" version of (ii) yields I = Rd G /C, as required. Thus, in the "if and only if" case of Theorem 6.1 the closedness of K. for right and left ideals can be moved from the hypothesis of the theorem to the extra restrictions before (iii). Suppose that R is an 5-graded ring and Rs is a subring of R. If s 0 E(S), then fi2 C Rs n Rs2 = 0. Since every class of Theorem 6.1 contains all rings with zero multiplication, this theorem and several corollaries can be rewritten by replacing "every e in E(S)" by "every Rs which is a ring". For the class of all quasiregular (locally nilpotent, Baer radical) rings the properties in the hypothesis of Theorem 6.1 are well-known. In particular, it is known that Jacobson (Levitzki, Baer) radical of a ring contains all quasiregular (locally nilpotent, Baer radical) left and right ideals of the ring, and therefore these classes are closed for sums of one-sided ideals. For

Groupoid-Graded

Rings

81

Pi-rings all the necessary closedness properties are obvious except the very difficult fact that every sum of two right (or left) ideals satisfying polynomial identities is a Pi-ring. This was proved by Rowen [Rowen (1976)]. Jaegermann and Sands established that the class of Jacobson rings is an iV-radical class, and that all such classes are closed for left and right ideals (see [Jaegermann and Sands (1978)], p. 348 and Theorem 11). For semilocal, right or left perfect, semiprimary rings all properties were proved by Clase and Jespers [Clase and Jespers (1993)]. For the classes of semilocal, right or left perfect, semiprimary, nilpotent, locally nilpotent, T-nilpotent, Baer radical, quasiregular and P.I. rings the "if and only if" versions of (iii) are also known: see [Jensen and Jondrup (1991)] for right or left perfect and semiprimary rings; [Beattie and Jespers (1991)] for right or left perfect, semilocal, and semiprimary rings; [Cohen and Rowen (1983)] for nilpotent rings; [Cohen and Montgomery (1984)] for quasiregular, Baer radical, and locally nilpotent rings; [Kelarev (1993e)] for Pi-rings. Note that for nilpotent and T-nilpotent rings assertion (iii) (and even Corollary 6.1 below) follows from Ramsey theorem (see [Kelarev (1994a)]). Combinatorial proofs are also possible for the classes of locally nilpotent and Baer radical rings ([Kelarev (1994a)]). "Provided that" version of (iii) for Jacobson rings is contained in [Passman (1989)], see also [Clase and Jespers (1993)]. Corollary 6.1 Let K, be the class of all semilocal (right perfect, left perfect, semiprimary, nilpotent, locally nilpotent, T-nilpotent, Baer radical, quasiregular) rings, S a finite groupoid, R — ®s^sRs an S-graded ring. Then R 6 K if and only if all Re belong to K, for all e £ E(S). Besides, R is a Jacobson ring provided that Re is a Jacobson ring for every e £ E(S). Corollary 6.2 Let K be a class of rings which contains all rings with zero multiplication and is closed under subrings, homomorphic images, ideal extensions and finite sums of one-sided ideals. Then, for any finite groupoid S, each S-graded ring R = ®s^sRs belongs to K. if and only if Re is in K. for every e in E(S). Proof. The result is true for S a group by [Cohen and Montgomery (1984)], Theorem 3.5, as was noted in [Kelarev (1993e)], Lemma 4. Hence the corollary follows from Theorem 6.1. • Corollary 6.3 ([Kelarev (1995c)]) Let AC be the class of all semilocal (right perfect, left perfect, semiprimary, nilpotent, locally nilpotent, T-nilpotent,

82

Finiteness

Conditions

Baer radical, quasiregular) rings, S a semigroup, R = ®s^s^s an S-graded ring with finite support. Then R 6 K. if and only if Re £ fC for every e in E(S). Corollary 6.4 Let S be a semigroup, R = (BsesRs o,n S-graded ring. If Re is semilocal for every e in E(S) and if Rs C J{R) for all but finitely many s in S, then R is semilocal, too. Let us include only one corollary concerning rings with not necessarily finite supports. Corollary 6.5 Let S be a periodic semigroup with a finite number of idempotents and only finite subgroups, and such that all nil factors of S are nilpotent. Let R = (BsesRs be an S-graded ring. Then R is semiprimary if and only if Re is semiprimary for every e in E(S). Proof. By Theorem 1.9, S has a finite ideal chain with finite or nilpotent factors. If / is an ideal of S, then R is an extension of Rj by R/Ri, and R/Ri is graded by S/I. Hence it follows by induction on the length of the ideal chain of S that it suffices to prove the corollary for all factors of S. For finite factors Theorem 6.1 gives the result. For nilpotent factors the claim is obvious. • Similar corollaries can be written for other properties mentioned in Corollary 6.1. Known examples of semigroup rings show that all restrictions on semigroup S in Corollary 6.5 are essential. 6.2

Structural Approach of Jespers and Okninski

Let us begin this section with two best results on finiteness conditions achieved so far. Theorem 6.2 ([Jespers and Okninski (1995)]) Let G be a group, and let R be a G-graded ring with J{Re) Q Jgr{R) ^ R- Then the G-graded ring R/Jgr{R) is semilocal (respectively, left perfect, semiprimary, left Artinian) if and only if the following conditions are satisfied: (i) Re/J(Re) = M n i ( D ( 1 ) ) x •••Mnr(D(r)), where every D(i) is a division ring. (ii) For any complete set of orthogonal idempotents eu of Re/J(Re), where \
Structural Approach of Jespers and

83

Okninski

the direct product of matrix rings over crossed products over some periodic subgroups Hi of G Mmi(D{1)

* H^ x • • • x Mmt(Dw

* He),

with q = mi + • • • + mi, and for each 1 < i < I, Mmi(D(i)

* Hi) = RejR,

D(i) =

ei{RejJ{Re))ei,

for some 1 < j < q. In particular, these matrix rings are homogeneous subrings. (iii) Every crossed product D^*Hi is semilocal (respectively, left perfect, semiprimary, left Artinian). Theorem 6.3 ([Jespers and Okninski (1995)]) Let S be a semigroup, and let R be an S-graded ring with J{R) nil. If R is semilocal, then there exist finitely many subgroups G\,..., Gn of S, with respective idempotents e\,..., en, and there exist homogeneous elements fi £ Rei such that

R = J{R)+

J2

Y,

a

i,jfiRGjibi,j,

l
for some finitely many homogeneous elements aij,bij S R. Furthermore, each fiRdfi *s a semilocal Gi-graded ring. If, moreover, R is left perfect (respectively, semiprimary, left Artinian), then each fi can be chosen to be an idempotent and fiRdfi is left perfect (respectively, semiprimary, left Artinian) with an identity element. Proofs of these theorems are rather technical. In order to introduce the reader to the methods used here, we include a few corollaries, a proposition and lemma of independent interest. After that a complete proof is given for an easier related result. Corollary 6.6 ([Jespers and Okninski (1995)]) Let G be a group, and let R be a G-graded ring. Suppose that the support of R does not contain infinite periodic subgroups. If R is left Artinian, then the following conditions are satisfied: (i) Re is left Artinian; (ii) R2 is a finitely generated Re-module; (iii) the additive group R/R2 is Artinian.

84

Finiteness

Conditions

Corollary 6.7 ([Jespers and Okninski (1995)]) Let G be a u.p.-group, and let R be a G-graded ring. If R is semilocal, then R/J&T{R) has finite support. Proposition 6.1 ([Jespers and Okninski (1995)]) Let G be a subgroup of a semigroup S, and let R be an S-graded ring. If R is semilocal (left perfect, semiprimary), then so is RGLemma 6.1 ([Jespers and Okninski (1995)]) Let G be a group and let R be a G-graded ring with J{Re) nil. If L is a homogeneous left ideal jof R with LnReC J(Re), then L C J{R). Lemma 6.2 ([Jespers and Okninski (1995)]) Let S be a semigroup and let R be a left perfect S-graded ring. Then the set of homogeneous elements of R is an epigroup. Theorem 6.4 ([Clase et al. (1995)]) Let S be a semigroup without infinite subgroups and let R be a right Artinian S-graded ring. Then the support of R is finite. Note that the converse statement fails even for semigroup ring; the rational semigroup ring of a two-element right zero semigroup is neither Artinian nor Noetherian. The example due to Passman [Passman (1971)] of a field which is a twisted group ring over an infinite group shows that it is impossible to remove the restriction on infinite subgroups from Theorem 6.4. Lemma 6.3 Let S be a semigroup and let R be an S-graded ring. Suppose K. is a family of right ideals of R satisfying the following: (i) there is a natural number k such that | supp(7)| < k for all I 6 K;

(ii)

U/GK:

SU

PP(-0 * S

infinite.

Then R is not right Artinian. Proof. It suffices to find a sequence I\, 1%, I3, •.. of right ideals in K. such that supp(/„) £ U m > n suPP(-fm) for all n; then OO

OO

OO

i=l

i=2

i=3

is an infinite strictly descending chain of right ideals.

Structural

Approach of Jespers and

Okninski

85

We proceed by induction, and given a family K.n satisfying (i) and (ii), we find an In+i G K,n and a subfamily /C n + 1 C /C„ satisfying (i) and (ii) and such that supp(7 n + i) ^{JIeKn l supp(7). Since IJ/eiC su PP(-0 i s infinite, there is a finite set {Hi, H2,..., 77/} C Kn such that | U»=i s u PP(#i)l > ^. Let X = \Ji=1 supp(77j); then X is finite. For x G X, let £§ be the subfamily of /Cn consisting of all 7 such that x £ supp(7). Note that for any 7 G fCn, we have X C^supp(7) because |supp(7)| < k, so I G £§ for some x. Hence, /C„ = Uxex^§> anc ^ there must be an x such that U/e£ SUPP(-D ^s infinite. Choose i such that x G supp(i7j). Then 7 n + i = 77; and ICn+i = £§ have the required properties. To begin the induction, put Ko = K. It is easy to see that the sequence so constructed has the desired properties. • Lemma 6.4 Let S be a semigroup and let R be a right Artinian S-graded ring. Let I be a nilpotent homogeneous ideal of R such that there are only finitely many s G S with Rs C7. Then supp(7) and hence supp(i?) are finite. Proof. Let n be the index of nilpotency of 7. We proceed by induction on n. Suppose n — 2. For x G supp(7), let Ax = Ix + IXR. Let m be the number of s G S with Rs CJ7. Because 7 2 = 0, it follows that | supp(A :r )| < m + 1 . Now each Ax is a right ideal of R and x G supp(yl x ). If supp(7) were infinite, then applying Lemma 6.3 to the family {Ax}x€supp^ yields that R is not right Artinian, a contradiction. So supp(7) and hence supp(7?) are finite. Suppose now that the nilpotency index of 7 is n > 2 and the result holds for smaller nilpotency indices. Consider the ring R = R/I2 with ideal 7 = 7/7 2 . R and 7 satisfy the hypotheses of the lemma and I2 = 0. Hence, by the n = 2 case, supp(T^) is finite. This implies that there are only finitely many s £ S such that Rs CI2. Since 7 2 has nilpotency index less than n, we conclude by the induction hypothesis that supp(i?) is finite, as required. • Lemma 6.5 Let S be a semigroup and let R be a right Artinian S-graded ring. Let s be a non-periodic element of S. Then Rs consists of nilpotent elements.

86

Finiteness

Conditions

Proof. Let Se be the semigroup with an identity e adjoined. Denote by R1 the ring obtained by adjoining an identity 1 to R in the usual way. If we put Re = Z, the subring generated by 1, then R1 = (BteS'Rt is an 5 e -graded ring. Suppose to the contrary that there exists an r £ Rs which is not a nilpotent element. For any non-negative integer m, denote by Im the right ideal of R1 generated by 1 — r2 . Then Im D Im+i because 1 — r 2 = (1 — r 2 )(1 + r 2 ). Given that R is right Artinian, there is some k > 0 such that IknR = Ik+i tlR = .... However, R1 = Ik + R = 7fc+1 +R= .... It follows that Ik = Ik+i = Since r is not nilpotent, neither is r 2 . Replacing s2 by s and r2 by r we may assume that k = 0 and that IQ = I\ = .... Since 1 — r2 £ I\ = IQ, there exists an element w £ R1 such that 1 — r = (1 — r2)w. Denote by B the subsemigroup generated in Se by e and s, and let C = S \ B. Note that R1 = RB © Re, s o for any x € R1, we can write x = XB +%C uniquely with XB € RB and xc G Re- Consider the equation 1 — r = (1 —r2)(wB+wc). Since 1, r, and % are elements of the subring RB, if we take components in Re, we see that wc = (r 2 toc)c^ But r2 is a homogeneous element of R, so | supp(r 2 wc)c| < | suppr 2 ?!^! < |suppwc|- This and the previous equality imply that wc = r2wc, and so 1 — r = (1 — r2)wBNow write WB = WQ + wx + W2 + • • • where wo £ Re and wn £ Rsn for n > 0. Comparing the homogeneous summands in the equation 1 - r = (1 - r2)(w0 + wi + w2 + ...) we see that WQ = 1 and w„ = ± r n for n > 0. Since w„ = 0 for n sufficiently large, this contradicts the assumption that r was not nilpotent. • Proposition 6.2 ([Clase et al. (1995)]) Let S be a semigroup with no infinite subgroups and let R be a right Artinian S-graded ring. Then there exist finitely many elements xi, x^, • • •, xn £ S such that R = ^T(R) +RXl + H-X2

I • • • 1 Kxn

•

Proof. Since R/J{R) is semisimple Artinian, there are ideals W\, W%, . •., Wi of R, containing J(R) such that each Wi/J(R) is simple Artinian and R/J(R) = ®\=1Wi/J(R). It suffices to show that each W* C J(R) + Rx for some finite set X C S. Fix W = Wi, and let W = W/J(R). We will denote the image of an element r £ R (or a subset Y C R) under the composite of the natural

Structural Approach of Jespers and

Okniiiski

87

maps R —> R/J(R) -+ W by r (or y ) . Since VK is simple Artinian, it is isomorphic to a ring of p x p matrices over some division ring. Notice that R = W. Consider the right ideals of R of the form RxSi for x G S. Since R is right Artinian, we can choose x such that Rxs^ is minimal subject to the condition Rxsi ^ 0. We claim that there is a u G xS1 and an a G Ru such that a is not nilpotent. For otherwise, the semigroup H of homogeneous elements of Rxsl is such that its image H in W is nil. By Theorem 1.12, Rxs^ is nilpotent, each element being a sum of elements of H. But this is not possible, since RxSi is a nonzero right ideal of the simple Artinian ring W. By Lemma 6.5, u must be a periodic element of S. So some power of u is an idempotent e. Note that Re / 0 since it contains a power of a. But eS1 C xS1. So by choice of x, we must have Resi = RxS1- Replacing a by the appropriate power, we may assume that a G Re and a is not nilpotent. Regarding a £ W as a, matrix, there must be a q such that aq and a2q have the same rank. Replacing a by a 9 , we still have a G Re and we may further assume that o and a2 have the same rank as matrices. This means that there is an idempotent / in W such that faf = a and / has the same rank as a (that is, a lies in the maximal subgroup of W containing / ) by Theorem 1.13. Hence, aRa = fRf = fWf is a corner of the matrix ring

W. We claim that it is sufficient to find a finite set X C S such that fWf C Rx- For we may choose a matrix basis e^ for W in such a way that / = e n + e^2 + • • • + e-u (where i is the rank of / ) , and then W = Y^lj=i enfWfeij. Ifr,j is an element of R such that fij = e^ for each i and j , then eufWfeij C RY(i,j) where Y(i,j) = supp(rji)Xsupp(rij) is a finite set. Hence, W C RY where Y = UI\'=i ^(*> j ) is finite, and it follows that W C J(R) + RY as desired. So we must find such a finite set X. In fact, we will take for X the subgroup G = W(]5]) of units of the monoid eSe. By the hypothesis on S, such a G must be finite. Let z G S and let b G Rz be a homogeneous element such that aba ^ 0. We claim first that there is a y G S and a homogeneous element c G Ry such that abaaca is not nilpotent. For otherwise, as before, abaaRa is a nilpotent right ideal of aRa, and therefore abaaRa = 0. Since aRa is simple Artinian, this implies aba = 0, a contradiction.

88

Finiteness

Conditions

Now, ezeye = ezeeye = supp(abaaca) is periodic by Lemma 6.5 since abaaca is not nilpotent. Hence, there is a v > 1 such that e' = (ezeye)v is idempotent. Note that e' = ee' = e'e. We have e'S1 C e-S1. But 0 ^ (abaaca)v € i?e'i s o by the minimality of Res^, we must have i?e<si = i? e gi. Because 0 ^ o € i? e , we must have eS'1 = e'5 1 , and it follows that e = e'e = e'. Hence eze has a right inverse in the monoid eSe. Similarly, eze has a left inverse in eSe, and so eze € G. We have shown that for all homogeneous b G R, if a6a ^ 0, then aba € i?G. Hence, / W / = ai?a C RQ. By the assertions above, this completes the proof. D Now we can prove Theorem 6.4. Proof. By [Murase and Tominaga (1978)], Theorem 8, the ring R2 is both right Artinian and right Noetherian. Since R/R? is a nilpotent right Artinian .S-graded ring, it follows from Lemma 6.4, that supp(i?/i? 2 ) is finite. So it suffices to prove that R2 has finite support. Replacing R by R2, we may assume that R is right Artinian and right Noetherian. By Proposition 6.2, there is a finite subset XQ C S such that R = J{R) + Rx0- Since R is right Artinian, J{R) is nilpotent; let m be its index of nilpotency. Fix k, with 1 < k < m. Since R is right Noetherian, the quotient J(R)k/J(R)k+1 is finitely generated as a right i?-module, so there are elements b\, 62, • • •, bi G J{R)k such that l

J(R)k = J2 (btZ + hR) + J(R)k+1. i=\

Note that btR = biJ(R) + hRXo. Since biJ(R) C J(R)k+1 and since k+1 the support of each bi is finite, we see that hR C J(R) + RXk. where Xk,i = supp(bi)X0 is finite. Therefore, J(R)k C RXk + J(R)k+1 where Xk = Ui=i (Xk,i U supp(fcj)) is finite. Combining these sets for 1 < k < m, we see that J{R) C RXl + RX2 + ... i?x m _i) a n < i therefore, the support of R is contained in the finite set X0l)XlU...UXm-i. • An examination of the proof shows that we only require that each subgroup G of S intersects the support of R finitely. Since bands have only trivial subgroups, we get

Finiteness

Conditions

and Homogeneous

Components

89

Corollary 6.8 ([Kelarev (1995a)]) Let B be a band and let R be a right Artinian B-graded ring. Then the support of R is finite.

6.3

Finiteness Conditions and Homogeneous Components

Let us begin with the following generalization of a well-known theorem by Chin and Quinn. Theorem 6.5 ([Dascalescu and Kelarev (1999)]) Let S be a semigroup which is a disjoint union of polycyclic-by-finite groups, satisfies d.c.c. for principal ideals, and does not contain nontrivial left zero bands. Let R = 0 s e S Rs be an S-graded ring, and let M = © s £ 5 Ms be an S-graded R-module. Then M is graded Noetherian if and only if it is Noetherian. The remaining main theorems of this section show that, for several interesting ring properties, if a ring R is graded by an epigroup S with certain restrictions (like the finiteness of the number of idempotents, etc.), then the property of R is determined by the group-graded subrings RQ or homogeneous components Re. All these theorems are complete in the sense that all the restrictions on the underlying semigroup S are essential. The following technical concept plays key roles in these results. Let K. be a class of rings, S a semigroup. We say that K is S-closed if /C contains each 5-graded ring R — ®sSsRs, provided that all subrings Re belong to K, for all idempotents e € S. Also, we include analogous results relating the properties of an S-graded ring R to the properties of subrings RQ of R graded by the maximal subgroups G of S. The theorems of this section have two equivalent conditions (i) and (ii) for a semigroup S. Condition (i) is always a ring property satisfied in 5-graded rings, and condition (ii) gives a clear description of a certain type of semigroups. Prom the point of view of graded ring theory, implication (ii)=S>(i) is more interesting. It shows that whenever a ring can be graded by a semigroup with (ii), a certain ring property of the graded ring can be determined as in (i). On the other hand, implication (i)=^(ii) shows that the class of semigroups included in (ii) cannot be extended, and for more general gradings (i) is not applicable. Theorem 6.6 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent:

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(i) Every S-graded ring R = © s € gi? s is right Artinian (Noetherian), provided that all group-graded subrings RQ are right Artinian (respectively, Noetherian) for all maximal subgroups G of S; (ii) S is the union of a finite number of groups and S does not contain the two-element right zero band. A few lemmas are required for the proof. Lemma 6.6 / / S is not a union of groups, and if a class K contains 0 and every S-graded ring R such that all RQ belong to K, for all maximal subgroups G of S, then K, contains all rings with zero multiplication. Proof. Take any ring R with zero multiplication. Pick an element t in S which does not belong to any subgroup of S. If we put Rt = R, and Rs = 0 for all s £ S, then R becomes an S-graded ring. Since RQ = 0 € /C for all subgroups G of S, it follows that Re K.. • Lemma 6.7 Let S be a semigroup such that the class K. of all semilocal (semiprimary, right or left perfect, right or left Artinian, right or left Noetherian, nilpotent, PI) rings (or the class of all rings with nilpotent Jacobson radicals) contains each S-graded ring R, provided that RQ G K. for every subgroup G of S. Then the number of idempotents of S is finite. Proof. Suppose to the contrary that S has pairwise distinct idempotents ei, e-i, It is known that every class K, in the hypothesis contains rings Pi,P2,... such that the direct sum © ^ . P * does not belong to K. For n = 1,2,..., let R€n = Pn. For all the other s £ S put Ra = 0. Then R = ®sesRs is an S-graded ring. Fix a subgroup G of S. Clearly G can contain at most one of idempotents e i , e 2 , . . . . Hence either RG = 0 or RQ = Re„ for some n. Therefore RQ 6 /C. However, R £ K,. This contradiction completes the proof. • Lemma 6.8 If L is a right (left) zero band, and K is an L-closed and closed for homomorphic images class of rings, then K, is closed under finite sums of left (right) ideals. Proof. Take any ring R which is the sum of a finite number of left ideals Ra, s £ S, belonging to fC. Introduce an operation on S such that S becomes a right zero band. Consider the semigroup ring RS. Clearly, the subring T — ®3&sRss is S-graded with components Rss isomorphic to Rs.

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Given that K is 5-closed, we get T G K. Therefore R € K., because R is a homomorphic image of T. D Lemma 6.9 Let IC be a class of rings closed under ideal extensions, and let S be a semigroup such that S° has a finite ideal chain 0 = S0 C Si C S2 C • • • C Sn = S°. Then K. contains all S-graded rings with subrings of the form RQ in IC, where G runs over all maximal subgroups of S, if and only if the same can be said of every factor of the chain. Proof. The 'if part. Suppose that, for all i = 1 , . . . , n, the class K contains all 5j/5j_i-graded rings with subrings of the form RQ in K. Consider an S-graded ring R = ®s^sRs s u c h that RQ G fC for all maximal subgroups G of S. For 1 < i < n, the quotient ring RsJRsi^! is an S^/S^i-graded ring with homogeneous components isomorphic to the components of R. For every maximal subgroup G of S, either G C Si-i and Ro/Rsi-i — 0; or G n Si-i = 0 and Rc/Rsi-! — RG € IC. Hence our assumption implies that Rsi/Rsi-i € IC- Since the class IC is closed for ideal extensions, easy induction on n shows that R G IC. The 'only if part. Consider an Si/St-j-graded ring R with homogeneous component Ro. It is an ideal extension of RQ by a contracted S'j/5'j_1graded ring. It remains to note that every contracted S , j/S , i _i-graded ring is S-graded. • Now we can prove Theorem 6.6. Proof. The implication (i)=>(ii) is easy. By Lemma 6.6, S is a union of groups. Lemma 6.7 tells us that S has only a finite number of idempotents, and so it is a union of a finite number of groups. Since A is not closed under finite sums of left ideals, Lemma 6.8 shows that S does not contain the two-element right zero band. Conversely, if S is the union of a finite number of groups and S does not contain the two-element right zero band, then S has a finite ideal chain whose factors are unions of their right ideals which are groups with the same zero adjoined. Given that RG & A for all such subgroups, we get R € A by Lemmas 1.5 and 6.9. • Corollary 6.9 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent:

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(i) Every S-graded ring R = (BsesRs is right Artinian provided that all components Re are right Artinian for all idempotents e of E(S); (ii) S is a finite left regular band.

(Noetherian), (Noetherian)

Lemma 6.10 If S is not a band, and K, is an S-closed class of rings containing 0, then K. contains all rings with zero multiplication. Proof. Take any ring R with zero multiplication. Pick t e S such that t T^ t2. If we put Rt = R, and Rs = 0 for all s £ S, then R becomes an S-graded ring. Since Re = 0 € K. for all e e E(S), it follows that R e K,. • Now we prove Corollary 6.9. Proof. Assume that (i) holds. Lemma 6.10 (i) shows that S is a band. By Lemma 6.7 S is finite. Since A is not closed under finite sums of left ideals, it follows that S does not contain the two-element right zero band. By Theorem 1.16 S is a left regular band. Conversely, let 5 b e a finite left regular band. Then all subgroups of S are singletons, and Theorem 6.6 completes the proof. • Corollary 6.10 ([Kelarev (1994b)]) Let B be a semilattice, R = (BbeBRb a special band graded ring. Then R is right Artinian (right Noetherian) if and only if B is finite and all the R/, are right Artinian (right Noetherian). Corollary 6.11 ([Kelarev (1994b)]) Let R = (BbeBRb be a special bandgraded ring. If R is right Artinian (right Noetherian), then B is finite and all the Rf, are right Artinian (right Noetherian). Theorem 6.7 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent: (i) each S-graded ring R is semilocal (semiprimary, right perfect, left perfect) if and only if all group-graded subrings RQ are semilocal (semiprimary, right perfect, left perfect) for all maximal subgroups

GofS; (ii) S is an epigroup with a finite number of idempotents and every nil factor of S is locally nilpotent (nilpotent, right T-nilpotent, left T-nilpotent). Lemma 6.11 ([Kelarev (1989c)], Lemma 4) Let S be a semigroup which is not locally finite. Then there exists an S-graded ring R such that R\ = 0 for every e € E(S), and R is not quasiregular.

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Lemma 6.12 Let S be a semigroup such that the class of all semilocal (semiprimary, right perfect, left perfect, nilpotent, right T-nilpotent, left T-nilpotent, Jacobson, quasiregular, Baer radical) rings (or the class of all rings with nilpotent Jacobson radicals) contains each S-graded ring R, provided that RQ = 0 for every subgroup G of S. Then S is an epigroup. Proof. Suppose to the contrary that S is not an epigroup. It means that S has an element t generating a subsemigroup T = {t,t2,...} which does not intersect any subgroup of S. Consider the ring R of polynomials over C (in the case of the class of all rings with nilpotent Jacobson radicals, we consider polynomials over any locally nilpotent but not nilpotent ring) with non-commuting variables x\,X2,... without free terms. For n = 1,2,..., denote by 7?t" the set of all homogeneous polynomials of degree n. Put R3 = 0 for s G S\T. Then R = ®sesRs is S-graded. Clearly, RQ = 0 for every subgroup G of S, and so R belongs to the class in the hypothesis of this lemma. However, it is well-known that all the classes included in lemma do not contain R. (Note that R is not Jacobson. For example, it can be homomorphically mapped onto the Golod ring, that is the first example of a nil ring which is not locally nilpotent.) • Lemma 6.13 Let S be a semigroup such that the class of all semilocal rings contains each S-graded ring R, provided that RQ = 0 for every subgroup G of S. Then every nil factor of S is locally nilpotent. Proof. Suppose that a nil factor F of S is not locally nilpotent. There exist ideals I C J of S such that F = J/I. Consider the contracted semigroup ring R = CQF. By Lemma 3.1, R is not semilocal. Put Rs = Cs for s G J\I, and Rs = 0 for s G JU (S\J). Then R = ®s€SRs is S-graded and every Re = 0 is semilocal for each subgroup G of S. This contradiction completes the proof. • Lemma 6.14 Let S be a semigroup such that the class of all semiprimary (or nilpotent) rings (or the class of all rings with nilpotent radicals) contains each S-graded ring R, provided that RQ = 0 for every subgroup G of S. Then every nil factor of S is nilpotent. Proof. Suppose to the contrary that a nil factor F of S is not nilpotent. There exist ideals / C J of S such that F = J/I. Consider the contracted semigroup ring R = C0F. Put Rs = Cs for s € J\I, and Rs = 0 for s G IU(S\J). Then R = ®seSRs is S-graded. Since J/I is nil, clearly J\I

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has no idempotents, and so Re = 0 for every e e E(S). The choice of 5 forces R to belong to the class of rings in the hypothesis. Since F is not nilpotent, evidently, R is not nilpotent, either. Lemma 3.1 shows that R is not semiprimary. Further, take any locally nilpotent ring L which is not nilpotent. Consider the contracted semigroup ring R = LQF. Put Rs = Ls for s € J\I, and Rs = 0 for s e / U (S\J). Then R = ®sesRs is S-graded. Since Re = 0 for every e £ E(S), the radical ^(.R) is nilpotent. Obviously, R is locally nilpotent, and so J{R) = R. However, R is not nilpotent. This contradiction completes the proof. • Lemma 6.15 Let S be a semigroup such that the class of all right (left) T-nilpotent or right (left) perfect rings contains each S-graded ring R, provided that RQ = 0 for every subgroup G of S. Then every nil factor of S is right (left) T-nilpotent. Proof. Consider any nil factor F of S and introduce the same S-graded ring R as in the proof of Lemma 6.13. If F is not right (left) T-nilpotent, then clearly R is not right (left) T-nilpotent, and Lemma 3.1 implies that R is not right (left) perfect. • Lemma 6.16 Let S be a semigroup with a sequence of elements e\, e 2 , . . . such that, for any 1 < k < I < m, the products e^ . . . e; and e ; + i . . . em are not contained in the same subgroup of S. Then there exists an S-graded ring R such that RG is a ring with zero multiplication for every subgroup G of S, but R is neither right nor left T-nilpotent. Proof. We shall construct an 5-graded ring which is not left T-nilpotent. (The case of right T-nilpotency is dual.) Let F be the free semigroup with the set X = {#1, £2, • • •} of generators. Let I be the set of all products Xix... Xin such that e j , , . . . , ej n is not a subword of the infinite word ei,e2, Obviously, I is an ideal of S. Consider the contracted semigroup ring R = Co(F/I). Put R3 = Cs for s € F\I, and Rs = 0 for s e L Then R = (BszsRs is S-graded. For all n, x\... xn $ I, and so R is not left T-nilpotent. Fix a subgroup G of S. For any elements a, b 6 RQ of the form a = Xix ... Xim and b = x^x ... Xjn, where all Xik, Xjl £ X, we claim that ab = 0. Indeed, if ab ^ 0, then x^ .. .XimXj1 .. .Xjn does not belong to I, and so e , ! , . . . , ej m , ejx,..., ejn is a subword of the word ei, e2,.... Given that

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95

a,b £ RQ, we get e^ . . . ej m £ G, and ejl... e Jn G G, which contradicts the choice of the sequence e\, e 2 , Thus .RQ = 0 for every subgroup G of the semigroup S. • Lemma 6.17 Let S be a semigroup containing an infinite idempotent chain. Then there exists an S-graded ring R such that RQ is a ring with zero multiplication for every subgroup G of S, but R is neither right nor left T-nilpotent. Proof. Every infinite idempotent chain contains an infinite descending or ascending subchain. If S contains idempotents e\ > e% > ... or e\ < e-2 < ..., then it is routine to verify that both these sequences satisfy the requirements of Lemma 6.16, which completes the proof. • Lemma 6.18 Let S be a semigroup such that every S-graded ring R is left (right) T-nilpotent, provided that RG = 0 for every subgroup G of S. Then S satisfies the descending chain condition for principal right ideals (respectively, the descending chain condition for principal left ideals). Proof. Suppose to the contrary that S contains an infinite descending chain of principal right ideals siS1 D S2S1 D ..., for some si,S2, • • • £ S. Then, for n = 1,2,..., there exists en G S such that snen = s„+i. It easily follows that the sequence ei, ei,... satisfies the requirements of Lemma 6.16, which completes the proof. • Lemma 6.19 Suppose that the class of right or left T-nilpotent rings contains each S-graded ring R, provided that RG = 0 for every subgroup G of S. Then every completely 0-simple factor of S° has a finite sandwich matrix. Proof. Take any completely 0-simple factor F of 5°. There exist ideals / C J of S such that F = J/I. Represent F as a Rees matrix semigroup M°(G; I, A; P). Suppose to the contrary that P is infinite. Then / U A is infinite. We consider only the case where |7| > 00, since the case |A| > 00 is analogous. Choose pairwise distinct elements ii,i2,... in J. Since every column of P has a nonzero entry, for n = 1,2,..., there exists A„ £ A such that P\nin+i ¥" 0- F i x a n Y 9 € G and introduce elements e„ = (g,in,^n) for n = 1,2,.... For i G I, put Tt = U A e A (G,i,A). If 1 < m < n, then G Tim. Clearly, all T,TO are pairwise disjoint, and each nonzero

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subgroup of S is contained in one Tj m . Therefore the sequence e\, e2, • • • satisfies all requirements of Lemma 6.16, which completes the proof. • Lemma 6.20 Let S be a semigroup such that the class of all Baer radical (Jacobson) rings contains each S-graded ring R, provided that all groupgraded subrings Ra are Baer radical (Jacobson) for all maximal subgroups G of S. Then S has no infinite descending idempotent chains. Proof. We shall record the complete proof for Jacobson rings and point out the difference with the case of Baer radical rings. Suppose to the contrary that S contains an infinite descending chain of idempotents e\ > e^> Put Y = {ei, e2,...}. For n — 1, 2 , . . . , denote by Pn the ring of polynomials in n commuting variables over C. Let P^, be the ring of polynomials in commuting variables x\,X2,... over C. It is known that all Pn are Jacobson rings, but P^ is not Jacobson. Consider semigroup ring PooY. Put Rn = Pnen for n = 1,2,..., and Rs = 0 for s G S\Y. Then R is an S-graded ring. If G is a subgroup of S, then RQ = 0 or RQ = Pn for some n. Therefore R is a Jacobson ring. However, Poo is a homomorphic image of R. This contradiction completes the proof. For the case of Baer radical rings it suffices to replace each P„ by the quotient ring Rn/In, where Rn is the ring of polynomials in n commuting variables over C without free terms, and In is the ideal generated in P„ by all products x^Xj where 1 < i < j < n. • Lemma 6.21 If a ring R is the union of an ascending chain of Jacobson ideals R^, where \i < T, then R is a Jacobson ring. Proof. Evidently, J(R) = U / i < r i 7(P M ) and B{R) = U M
= S0.

Then K. is S-closed if and only if it is T-closed for every factor T of the chain.

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97

Proof. Let S be a semigroup with ideal / , and let R be an 5-graded ring. Putting Ro = 0, we can regard R as an ^-graded ring. It is easily seen that R is an ideal extension of Rj by the 5//-graded ring R/Rj. This observation and easy induction on the length of the chain completes the proof. • Lemma 6.23 Let S be a semigroup such that S° has a finite ideal chain with factors which are nil or completely 0-simple with finite sandwich matrices. Let K be a class of rings containing all rings with zero multiplication and closed under ideal extensions and finite sums of one-sided ideals. Let K. be T-closed for every nil factor T of S°. If all group-graded rings RQ belong to fC for all maximal subgroups G of S, then R G /C. Proof. Consider one of the factors F = S^/Sk-i of the ideal chain of S. If it is null, then i?sfc/i?sfc_j is a ring with zero multiplication, and so it belongs to /C. Next, assume that F is completely 0-simple. Then F = M°(G;I,A;P) is isomorphic to a Rees matrix semigroup. It follows from Theorem 1.8 that if p\i = 0, then Rp = 0, and so RFix G K,. If Pxt 7^ 0, then Theorem 1.8 tells us that Fi\ is a maximal subgroup of F, and so RpiX G /C by the choice of /C. Thus RFiX G ^ f° r all i G / , A G A. Now, i?j» is a direct sum of its left ideals Ri\, A G A. Hence i?;» G /C. Further, Rp is a direct sum of its right ideals Ri*, i G I, and so it lies in fC, too. The ring Rsk is an ideal extension of Rsk_1 by Rp- Therefore Rsk belongs to /C. Lemma 6.22 and induction give us R £ fC, as required. Easy induction on m + n, where m = l-E^S1)! and n is the number of factors in the ideal chain of S°, completes the proof (see also Lemma 6.22). • Lemma 6.24 Let K. be a class of rings closed under left and right ideals and homomorphic images. Let S be an epigroup, G a maximal subgroup of S, and let R be an S-graded ring. If R 6 K, then RQ G /C. Proof. Let R G K,. Since RGS is a right ideal of R, we get RQS G K,. Similarly, RGSG G /C, because RQSG is a left ideal of -RGSFurther, denote by H the ^7-class of S containing G. Let I be the ideal generated in S by G, and let N be the set of non-generating elements, that is N = I\H. Without loss of generality we may assume that N ^ 0. By Theorem 1.5, the factor I/N is completely 0-simple. It is isomorphic to a Rees matrix semigroup over G°, because G is a maximal subgroup of S. Hence GSG = G U (GSG n N). It follows that G° ^ GSG/N. Thus RG is a homomorphic image of RGSG- Therefore RQ G /C, as claimed. •

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Lemma 6.25 ([Okniriski (1991)], Corollary 2.12) Let S be a semigroup, R an S-graded ring, H(R) = (JSPS -^» ^he semigroup of homogeneous elements of R. Then R is right (left) T-nilpotent if and only if H(R) is right (left) T-nilpotent. Lemma 6.26 Let S be a nilpotent (locally nilpotent, right T-nilpotent, left T-nilpotent, Baer radical) semigroup, R an S-graded ring, and let RQ = 0. Then R is nilpotent (locally nilpotent, right T-nilpotent, left T-nilpotent, Baer radical). Proof. If S" = 0, then Rn C R0 = 0, and so R is nilpotent. Further, assume that S is locally nilpotent. Take any finitely generated subring P of R. Clearly, P C RT for a finitely generated subsemigroup T of S. Then T is finite by the local finiteness of S. As we have proved this implies that RT is nilpotent. Therefore R is locally nilpotent. Lemma 6.25 gives the conclusion in the right (or left) T-nilpotent case. Finally, assume that S is Baer radical. Then S has an ideal chain with nilpotent factors. Since for every nilpotent factor J / 7 we have already proved that RJ/RI is nilpotent, it follows by transfinite induction that R is Baer radical. • Now we prove Theorem 6.7. Proof. Assume that (i) holds. Then S is an epigroup by Lemma 6.12, and S has a finite number of idempotents by Lemma 6.7. In view of Lemmas 6.136.15 all nil factors of S are as indicated in (ii). Thus (i) implies (ii). Implication (ii)=>(i) follows from Theorem 1.9, and Lemmas 6.23, 6.24 and 6.26. D Corollary 6.12 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent: (i) each S-graded ring R is semilocal (semiprimary, right perfect, left perfect) if and only if all subrings Re are semilocal (semiprimary, right perfect, left perfect) for all idempotents e of S, (ii) S is a periodic semigroup with a finite number of idempotents, every subgroup of S is finite, and every nil factor of S is locally nilpotent (nilpotent, right T-nilpotent, left T-nilpotent). Proof. Assume that (i) holds. Then all subgroups of S are finite by Lemma 3.1. Every epigroup containing only finite subgroups is periodic. Therefore implication (i)=>(ii) follows from Theorem 6.7.

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Theorems 2.5 and 6.7 prove the converse implication.

99

•

Corollary 6.13 ([Kelarev (1994b)]) Let a band B be a semilattice Y of rectangular bands. A special band-graded ring R = (BbeBRb is semilocal if and only if Y is finite and all the Rb are semilocal. Corollary 6.14 ([Kelarev (1994b)]) A special band-graded ring R = ©bgB-Rb is local if and only if B is a rectangular band and all the Rb are local. Corollary 6.15 ([Kelarev (1994b)]) Let a band B be a semilattice Y of rectangular bands, and let R = (BbeBRb be a special band-graded ring. Then R is right perfect (semiprimary) if and only if Y is finite and all the Rb are right perfect (semiprimary). Theorem 6.8 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent: (i) each S-graded ring R is nilpotent (has a nilpotent Jacobson radical), provided that all group-graded subrings RQ are nilpotent (have nilpotent Jacobson radicals) for all maximal subgroups G of S; (ii) S is an epigroup with a finite number of idempotents and every nil factor of S is nilpotent. Proof. Lemmas 6.12, 6.7 and 6.14 show that (i) implies (ii). Implication (ii)=>(i) follows from Theorem 1.9, and Lemmas 6.23 and 6.26. • Corollary 6.16 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent: (i) each S-graded ring R is nilpotent (has a nilpotent Jacobson radical), provided that all subrings Re are nilpotent (have nilpotent Jacobson radicals) for all idempotents e of S; (ii) S is a periodic semigroup with a finite number of idempotents, every subgroup of S is finite, and every nil factor of S is nilpotent. Lemma 6.27 Let S be a semigroup which contains an infinite subgroup. Then there exists a locally nilpotent S-graded ring R such that Re — 0 for every e £ E(S), and R is neither right nor left T-nilpotent, nor Baer radical, nor Jacobson. Proof. Let G be the infinite subgroup of S, and let e be the identity of G. By induction we shall define an infinite sequence of elements gi, • • •

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in G. Choose any gi ^ e in G. Suppose that g\,... ,gn have been denned, for some k > 0 and n = 2k — 1, and suppose that the sequence gi,...,gn has no segment with the product of elements equal to e. Given that G is infinite, there exists gn+i such that (a) gn+i £ {gi, • • •,gn} and (b) <7i,... ,gn,gn+ii9ii • • • ,9n does not contain a segment equal to e. We shall also immediately put gn+2 = g\, ..., g2n+i = gn- Here 2 n + l = 2fc+! _ 1. Thus, by induction on k, we get an infinite sequence of elements g i , . . . , gn of G. Note that (c) every segment gi,.-.,gn repeats in the sequence infinitely many times. Denote by R the ring with generators xi,x2,... and relations xt = Xj if gi = gj, and xh---xin = 0 if gh,...,gin is not a segment of gug2,.... For s € S, let i? s be an additive subgroup generated in R by all elements £»!-•• x i n such that <7JJ • • • gin = s. Then R = ®seSRs is 5-graded. It follows from (b) that Rf = 0 for all idempotents / of S. Since xi • • • xn -^ 0 for all n, we see that R is not T-nilpotent. Further, (c) and X\ • ' • XnXn-\-iX\

• • • Xn

for all n = 2k — 1 imply that R does not contain nilpotent ideals, and so it is not Baer radical. However, R is locally nilpotent in view of (a). Therefore the Jacobson and Baer radicals of R are different, and R is not Jacobson. This completes the proof. D Now we can prove Corollary 6.16. Proof. Assume that (i) holds. Then all subgroups of S are finite by Lemma 6.27, and so S is periodic. Therefore implication (i)=>(ii) follows from Theorem 6.8. Theorems 2.6 and 6.8 prove the converse implication. • T h e o r e m 6.9 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent: (i) each S-graded ring R is a Baer radical (Jacobson) ring, provided that all group-graded subrings RQ are Baer radical (Jacobson) rings for all maximal subgroups G of S; (ii) S is an epigroup without infinite descending chains of idempotents and every nil factor of S is Baer radical.

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The following lemma was proved in [Clase and Jespers (1994)] for finite completely 0-simple semigroups with the use of the description of the Jacobson radical of band-graded rings (see Theorem 4.2). Lemma 6.28 ([Clase and Jespers (1994)]) Let S be a completely 0-simple semigroup, R an S-graded ring. Suppose that all group-graded subrings RQ are Jacobson for all maximal subgroups G of S. Then R is a Jacobson ring. Lemma 6.29 Let S be a semigroup such that the class of all Baer radical rings or all Jacobson rings contain each S-graded ring R, provided that RG = 0 for every subgroup G of S. Then every nil factor of S is Baer radical. Proof. Suppose to the contrary that a nil factor F = J/I of S is not nilpotent. Take any locally nilpotent ring L which is not Baer radical. Consider contracted semigroup ring R — LQF. Put Rs = Ls for s £ J\I, and Rs = 0 for s e / U (S\J). Then R = ®seSRs is 5-graded. Since RG = 0 for every subgroup G of S, the Jacobson and Baer radicals of every homomorphic image of R coincide. Clearly, R is locally nilpotent, and so J(R) = R. However, it is easily seen that R is not Baer radical. Therefore R is not a Jacobson ring, either. • Now we can prove Theorem 6.9. Proof. Assume (i). Then S is an epigroup by Lemma 6.12. Lemma 6.20 shows that S has no infinite descending idempotent chains. By Lemma 6.29 all nil factors of S are Baer radical. Thus (i) implies (ii). Now assume (ii). By Theorem 1.10 and Lemma 6.21 it suffices to prove (i) in the case where S is nil or completely 0-simple S. If S is nil, then (i) follows from Lemma 6.26. If S is completely 0-simple, then Lemma 6.28 completes the proof. • Corollary 6.17 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent: (i) each S-graded ring R is a Baer radical (Jacobson) ring, provided that all subrings Re are Baer radical (Jacobson) rings for all idempotents e of S; (ii) S is a periodic semigroup without infinite descending chains of idempotents, every subgroup of S is finite and every nil factor of S is Baer radical.

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Finiteness

Conditions

Proof. Assume that (i) holds. Then all subgroups of S are finite by Lemma 6.27, and so S is periodic. Therefore (ii) follows from Theorem 6.9. Theorems 2.6 and 6.9 prove the converse implication. D A semilattice B is an m.u.-semilattice if and only if (a) for any a,b £ B where a < b, there exists c which is maximal with the property that a < c < b; (b) the set {b G B \ b is maximal under a} is finite for every a € B. Theorem 6.10 ([Ho (1992)]) Let B be an m.u.-semilattice, and let R be a strong semilattice sum of rings R^, b £ B. If B is infinite, then R cannot be a right Goldie ring. If B is finite, then R is a semiprime right Goldie ring if and only if all the Rb are semiprime right Goldie rings.

6.4

Classical Krull Dimension and Gabriel Dimension

The main result of this section, for each ring R graded by a finite group, gives a bound on the classical Krull dimension of R in terms of the dimension of the initial component Re. It follows that if Re has finite classical Krull dimension, then the same is true of the whole ring R, too. It is interesting to note that the analogous assertion is not valid for Krull dimension defined on the lattice of right ideals. Indeed, if we take any group G with identity e and an element g ^ e in G, and take a ring R with zero multiplication which has no Krull dimension, then we can view R as a group-graded ring with Re = 0, Rg = R, and Rh = 0 for all h £ G\{e,g}. Example 2.4 in [Chin and Quinn (1988)] shows that our theorem does not transfer to rings graded by infinite groups, even in the case of the infinite cyclic group. For any ordinal a and positive integer n, we introduce ordinals an, setting ax = a + 1, a n + 1 = (a + l ) ( a „ + 1). Theorem 6.11 Let G be a finite group with identity e and \G\ = n, and let R = ©g6G-Rg be a G-graded ring. If Re has classical Krull dimension a, then R has classical Krull dimension, too, and cl—K—dim(i?) < an. Proof. Suppose to the contrary that R contains a strictly increasing chain of prime ideals Pi C Pi C . . . C Pa„- Theorem 2.4(i) tells us that, for each 7 < ctn, there exists a finite set 5 7 of prime ideals of Re minimal over

Classical Krull Dimension

and Gabriel

Dimension

103

Re fl P 7 and such that

f| P = flenP7 Pesy and | S 7 | < |G| = n. Put 5 = (J 7 < Q "S-r If a prime ideal contains an intersection of a finite number of ideals, then it contains at least one of them. Therefore, for any 5 < e < an and P £ Se, there exists Q £ Ss, such that Q C P. For S < e < an, Q £ Ss, and P £ Se, we shall write Q « P if and only if, for all fi, 6 < /i < e, we can fix 1% £ 5M so that 7M C 7V whenever 5 < fi < u < e, where Is = Q and 7e = P . We shall show by induction on 7 < an that, for each P £ S7, there exists Q £ S\ such that Q < c P . The case of 7 = 1 is trivial. Suppose that this has been proved for all S < 7. Take any ideal P £ Sy. If 7 is not a limit ordinal, then there exists 7 — 1 and we can take P ' € Sry-i such that P ' C P . By the induction assumption Q c F for some Q e 5 i . It follows that Q < P . Consider the case where 7 is a limit ordinal. Denote by L the set of all Q £ U<5<7 &8 s u c h that Q C P . By induction on 5 we shall define ideals Qs £ Ss, for all S < 7. Given that 5j is finite and every ideal Q, where Q £ Sv fl L ^ 0, 1 < v < 7, contains at least one ideal of Si fl L, it follows that there exists Qi £ SiD L such that for any /i < 7 we can find /u < v < 7 and Q £ Sv satisfying <3i
S

(J

S„|Qi«Q}.

We have verified that Li intersects all Sv, for all 1 < v < 7. Suppose that for some <5 < 7 all ideals Q e have been defined for all e < S, and suppose that these ideals form an ascending chain. In addition, assume that the sets

Lt = Ln{Q£

\J £
Sv\Qe
104

Finiteness Conditions

intersect all Sv for e < v < 7. Obviously, M C L and M = f]£
_

p(i)

_

_

p(i)

p

r

Put 4 1 } = 5 M \{P^ 1) } for (5 < /i < <5 + a „ _ i + 1, and

s(1) =

U 4 X) «5
For any < 5 < / u < i / < £ + a „ _ i + 1, and any ideal I G Su there exists Q G 5M such that Q C I. If Q £ S^ 1 ', then Q = P^ 1} = P ^ ; whence / D P ) J , a contradiction. Therefore Q e S^ '. Thus S^ satisfies the same property we used for S, but now | 5 7 '\ < n — 1 for all 7. Suppose that for some 7 such that 8 < 7 < 5 + a „ - i the set 5 7 is empty. Then, for any 7 < Li < <5 + a n _ i + 1 and Q G S^, we have p W = P ^ = P 7 C Q. Hence Q = PJP and so s£ x) = 0. Therefore P 7 = P 7 +i by Theorem 2.4(ii). This contradiction shows that all sets S\ are nonempty for all 8 < 7 < 8 + an-\. Let us apply the same argument as above to S^1'. Take an ideal P ' 2 ' in S ^ ^ . Find P ] ^ G S8+i with p g \ < P(2>. Take a chain p (2)

c

p (2)

c

c

p (2)

_ p(2)

c

„

where P 7 G S 7 for all 8 < 7 < 8 + an-\. Find a new ordinal 82 such that pV>+l = p £ 2 = • • • = P £ ; „ _ 2 + 1 . Put S f = ^ ) \ { P f } ,

5( 2 ) =

(J i52<7<<52+a„-2

S<2>.

Classical Krull Dimension

and Gabriel

Dimension

105

Then the set S^ satisfies the same property we used for S, but now \S\ | < n — 2 for all 7. As above, all sets S 7 ' will be nonempty for all 62 < 7 < 52 + an-2If we repeat this reduction n — 1 times, we get a set


(J

St""1)

5„_i<7<5„_i+ai

satisfying the same conditions and such that |S 7 | < 1 for all <5„_i < 7 < <5n-i + «i- As earlier we can show that all sets S\n are nonempty for all <5„_i < 7 < <5„_i + a i . Thus \S^ \ = 1 for all 7. Given that a^ = a + 1, we get S^ — S^\ for some 82 < 7 < <52+a:i. It follows from Theorem 2.4(h) that P 7 = P 7 +i. This contradiction completes the proof. • The Gabriel dimension of an P-module M is an ordinal a =G-dimflM defined inductively by the rules: (a) G-dim f l M = 0 if and only if M = 0. (b) Suppose that a is a nonlimit ordinal, and G-dim RM = f3 has been defined for all f3 < a. A nonzero .R-module S is said to be a-simple, if for every 0 ^ S' C S, G-dim RS' £ a and G-dim RS/S' < a. We put G-dim RM = a if G-dim RM •£ a and, for every M' C M, M/M' contains a /3-simple submodule for some (3 < a. (c) Suppose that a is a limit ordinal. Then G-dirriijM = a if G-dim RM •ft a and, for every M' C M, M/M' contains a /3-simple submodule for some /? < a. Let S be a partially ordered set. If a, b € S, then we put [a, b] = {c € S I a < c < b}. We say that S is discrete if a •£ f3 and /? ^ a for all a,/? € 5. The /frwZZ dimension of a partially ordered set is the ordinal a = K-dimS1 defined inductively by the following rules. First, K-dimS' = — 1 if and only if S is a discrete set. If 7 > 0 is an ordinal and K-dim S ft 7, then K-dim S = 7 if and only if, for any descending chain a\ > d2 > • • • of elements of S, there exists a positive integer no such that K-dim [ a n + i , a n ] < 7 for all n > UQ. Let B be a semilattice. A subset S of B is called convex if a, b £ S implies [a, b] C S. A semilattice B is said to be multidiamond if every convex subset S of B has a finite or infinite sequence of elements {bi}i such that for every b G S there exists i with 6 i + i
106

Finiteness

Conditions

Theorem 6.12 ([Stalder (1997)]) Let B be a semilattice, R = ®beBRb a strong semilattice sum of rings. If R has Gabriel dimension, then all Rb have Gabriel dimensions. If the Gabriel dimensions of all the Rf, exist, and B is a multidiamond semilattice with Krull dimension, then the Gabriel dimension of R exists. Theorem 6.13 ([Dascalescu and Kelarev (1999)]) Let S be a semigroup which is a disjoint union of polycyclic-by-finite subgroups, satisfies d.c.c. for principal ideals, and does not contain nontrivial left zero bands. If R = © s € s Rs is an S-graded ring, and M = © s £ i S Ms is a Noetherian S-graded R-module, then K-dim ( s > / e ) _ g r (M) < K-dimR(M) where (G\)\^F

< K-dim ( S i f l ) _ g r (M) + max/i(G A ) AG/*1

is the family of maximal subgroups of S intersecting supp(M)

If S is a (possibly, infinite) union of finite groups, then we can say more about the relationship between M having graded Krull dimension and M having Krull dimension as an .R-module. Theorem 6.14 ([Dascalescu and Kelarev (1999)]) Let S be a semigroup which is a disjoint union of finite groups, and has neither infinite chains of principal ideals nor nontrivial left zero bands. If R = @ses Rs is an S-graded ring, and M = ( B s e 5 Ms is an S-graded R-module, then M has Krull dimension as an object of the category (S,R)—gr if and only if it has Krull dimension as an R-module, and in this case K—dim^ /j)_ gr (M) = K-dimfl(M).

Chapter 7

PI-Rings and Varieties

Note that several theorems in Section 4.6 also deal with Pi-rings. The first main theorem of this chapter follows from Theorem 6.1. Theorem 7.1 ([Kelarev (1995c)]) Let G a finite groupoid, and let R = (BsesRs be an S-graded ring. Then R is a Pl-ring if and only if all Re are Pi-rings, for all e 6 E(G). In the special case of rings graded by finite groups this theorem was proved in [Kelarev (1993e)]. Corollary 7.1 ([Kelarev (1993e)]) Let S be a semigroup, and let R = @sesRs be an S-graded ring with finite support. Then R is a Pl-ring if and only if all component Re are Pi-rings, for all e G E(S). The next theorem deals with semigroup-graded case and handles graded rings with not necessarily finite number of nonzero components. Theorem 7.2 ([Kelarev (1993e)]) For every semigroup S, the following conditions are equivalent: (i) each S-graded ring R = © s € s Rs is a Pl-ring if and only if all the components Re are Pi-rings, for all e e E(S); (ii) S has a finite ideal chain
= S

such that each factor Si+i/S, (0 < i < n — 1) is finite or nilpotent. The following lemma is straightforward and well-known. 107

108

Pi-Rings

and

Varieties

Lemma 7.1 If a ring A for each n > 0 contains elements a\, a^-, • • •, an such that a\a2 . . . o„ ^ 0 and aa\aa2 ... aan = 0 for all 1 ^ a £ Sym(n), then A is not a Pi-ring. Lemma 7.2 semigroup.

If the class of Pl-rings is S-closed, then S is a periodic

Proof. Suppose that S has a non-periodic element x. Put X = {x, x2,...} and consider the free ring R of countable rank. For s = xn e X, by Rs we denote the set of homogeneous polynomials of degree n. For y £ S\X, set Ry = 0. Then R = ®s^sRs a nd all subrings among Rs are equal to zero; however, R is not a Pi-ring. The contradiction shows that S is periodic. • Lemma 7.3 If the class of Pl-rings is S-closed and S contains a subgroup G, then G is finite. Proof. Suppose that G is infinite. Denote the identity of G by e. We are going to find an infinite sequence of elements gi,gi,... in G such that 9m9m+i... fifn 7^ e for any natural numbers m
• • • 9n)~l | m < n | U{e}

is finite and we may choose gn+i not in M. Thus the sequence is defined. Consider the F-ring A with generators 01,02,... and relations a^aj = 0 for all natural numbers i, j , where j ^ i + 1. Denote by Ag (g £ G) the subspace of A spanned by all those products a m a m + i . . . an, m < n, such that gmgm+i.. .gn = g. For s e S \ G, set As = 0. Then A = (BsesAsBy the choice of g\,gi,..., we get Ae = 0. Hence all the components A3 which are rings are equal to zero. Evidently, aCTia,T2 • • • aan — 0 for any l / u 6 Sym(n). By Lemma 7.1, A is not a Pi-ring, a contradiction. • Lemma 7.4 nilpotent.

If the class of Pl-rings is S-closed and S is nil, then S is

Proof. Suppose that S is not nilpotent. It follows that for any n, there are s("\ 4 " \ .. •, snn) in S such that s^s^ ... s^ £ 0. Take the ring A with generators a™ (where i — 1 , . . . , n, and n runs over the set of integers) and relations a] 'ay = 0 whenever m^novj^i + 1. For any s € S, by As we denote the F-module generated by those products a*. o4+i • • • % , k <

109

m < n, such that s*. s ^ • • •s™ = s- ^ a P r o c m ct ajf a^x . •. a™ is in Ao, then s j f ^ S i ... sfi? = 0, whence s^s^ ... s^ = 0, a contradiction. Therefore, A0 = 0. If s =£ 0 and A s is a ring, then A\ C A s . Given that 5 is nil, we get s / s 2 and so A ' C A, fl As2 = 0. Thus all the rings As are equal to zero. On the other hand, for any 1 ^ a G Sym(n), evidently a£i^a£^ . . . aS„ = 0. By Lemma 7.1, A is not a Pi-ring. The contradiction completes our proof. • Now we prove Theorem 7.2. Proof. The 'only if part follows from Lemmas 1.9, 6.7, 7.2-7.4. Next, we prove the 'if part. Assume that S has a chain with the properties above, and R = (Bs^sRs- Set A; = ©sgSjflj, i = 0, . . . , n . The quotient ring T = Ai/Ai-i is Q^-graded, where Qi = 5i/5j_i; T = ©qgQ^q. Here the components Tq are isomorphic to the corresponding components Rs, s e Si\ 5j_i; and if Qi has a zero, then To = 0. Thus all the Tq, which are rings, belong to the class of Pi-rings. Lemmas 1.2, 6.9 and Theorem 2.7 show that the class of Pi-rings is closed under every finite semigroup. Therefore, if Qt is finite, then T is a Pi-ring. Further, if Qi is not finite, then it is nilpotent and so T is PI again. Thus every factor Ai/Ai-i is a Pi-ring, implying that R is PI, as required. • Let F be an associative and commutative ring with identity. To include both rings and algebras over fields in one theorem, we shall consider varieties of .F-algebras over a Dedekind ring F. If E is a set of polynomials in FLY*], then the variety defined by E is the set of all F-algebras satisfying all identities / — 0, for / £ E. A class of F-algebras is a variety if and only if it is closed for direct products, subalgebras and homomorphic images. If / is an ideal of F , then by Aj we denote the variety of all F-algebras R such that Ix = 0 for all x G R. Note that ^o is the variety of all F-algebras, and Ap contains only zero rings. A product of two varieties V, W is a class VW consisting of all F-algebras R with an ideal I £V such that R/I 6 W. A product of varieties is a variety (see [Maltsev (1967)]). A variety is said to be semisimple if it is generated by a finite (possibly empty) set of finite fields. It is known that a variety is semisimple if and only if it consists of Jacobson semisimple rings. Identities of semisimple varieties were described in [Gardner and Stewart (1975)]. Theorem 7.3 Let F be a Dedekind ring, and let S be a semigroup which is not a singleton. A variety V of F-algebras is S-closed if and only if either

110

Pi-Rings

and

Varieties

V contains all F-algebras, or S is a band and V = Aj for some ideal I of F, or S is a semilattice and V = AjW, where W is a semisimple variety. Proof. If S is a band, the assertion was proved in [Volkov and Kelarev (1986)]. Suppose that S is not a band. Then S has an element s such that s =£ s2. Let (s) be the monogenic semigroup generated by s in S. Denote by X the non-commutative free F-algebra with free generators £1,22, — For a positive integer k, let X^ be the F-submodule generated in X by all products x^x^ •••Xik, for all h,i2,---,ik G N. We shall show by induction on k that V contains all rings X/Xk. The induction basis is trivial, as X/X = 0. Next assume that X/Xk € V. For m e N,put R3m =

XW/Xk+1,

^ n

n6N;5 =s

m

and let Rt = 0 for all t £ S\{s). Then X/Xk+1 = ®t&sRt is an 5-graded ring. If (s) has no idempotents, then it follows that X/Xk+1 belongs to V. If, however, (s) has an idempotent s m , then m > 1, and therefore i?sm is a homomorphic image of X/Xk. Hence R3™ g V. Given that V is ^-closed, we see that X/Xk+1 G V. Since X is a subdirect product of the X/Xk, k = 1,2,..., it follows that X belongs to V. Therefore V contains all F-algebras. This completes the proof. • A class of rings closed for subrings and direct products is called a prevariety. We say that a class K. is an additive class if and only if, for any rings R,Q with isomorphic additive groups, Re JC implies Q € K. Theorem 7.4 ([Kelarev (1987)]) Let B be a semigroup which is not a semilattice, and let K. be a prevariety of rings. Then fC is B-closed if and only if it is additive.

Chapter 8

Gradings of Matrix Rings

8.1

Full and Upper Triangular Matrix Rings

Let F be a field, and let A — Mn(F) be the algebra of all n x n matrices over F. For each finite semigroup S, we describe all gradings A = © s e s ^ s of A by S such that all standard matrix units e^- of A are homogeneous elements of A. Let R = (BsessRs be an 5-graded subalgebra of the full matrix algebra Mn(F). Following [Dascalescu et al. (1999)], we say that the grading is good, if all standard matrix units e^ contained in R are homogeneous. Good gradings were studied in the group-graded case in [Dascalescu et al. (1999)] and in a different setting in [Green (1983)] and [Green and Marcos (1994)], where they were constructed from weight functions on the complete graph r on n points, using the fact that Mn(F) is a quotient of the path algebra of the quiver I\ If G is a group, then all good G-gradings of the F-algebra Mn(F) have been described in [Dascalescu et al. (1999)]. It is known that in several natural cases all gradings are isomorphic to good ones. For example, if the group G is torsion free, then [Dascalescu et al. (1999)], Corollary 1.5, tells us that each grading is isomorphic to a good grading. In this paper, for each finite semigroup S, we give a complete description of all good gradings of the full matrix algebra Mn(F) by S. For every completely 0-simple factor Q = M(G°; I, A; P) of S, denote by TQ the set of all triples { / and <£>A : {1>2,... ,n} —> A are functions such that p¥,A(j)¥>/(j) ^ 0 for all i = 1 , . . . , n, and (pa : { 1 , 2 , . . . , n — 1} —> G is an arbitrary function. A semigroup S may also have one completely simple factor, say Q = 111

112

Gradings of Matrix

Rings

M(G; I, A; P ) , which then coincides with the least ideal of S. In this case we denote by TQ the set of all triples {ipj, y>A, I, A, and ipG : { 1 , 2 , . . . ,n - 1} —> G are arbitrary functions. Fix a triple r = (
=


+

l)pVA{ic+2)v,I(k+2)---

•••P
(8.1)

» =


(8.2)

A =


(8.3)

Let D ^ T be the set of all pairs (fc, £) such that 1 < t < k < n and

» =

¥>/(*),

(8.5)

A =


(8.6)

Let C®T be the set of all pairs (fc, fc) such that 1 < A; < n and 5 = t = A =

?;!(i)W(n. ¥>/(*), VA (fc).

(8-7) (8-8) (8.9)

Finally, put pQr

=

U?T \J CfT U DfT.

(8.10)

Theorem 8.1 ([Dascalescu et al. (2002)]) Let F be a field, n a positive integer, and let S be a finite semigroup. Then there exists a oneto-one correspondence between all good S-gradings of the matrix F-algebra Mn(F) and the union of all sets TQ, where Q runs over all simple and 0-simple principal factors of S. Namely, for each simple or 0-simple principal factor Q = M ( G ; / , A ; P ) or Q = M ° ( G ; / , A ; P ) of S, every triple T = (ipj,
(8-11)

Full and Upper Triangular Matrix

113

Rings

where ]T

FeM

ifs =

(g-i,\)eQ\{0},

r

(8-12)

lk,t)€P?

0

ifs#Q\{0}.

The next theorem describes all S such that Mn(F) has a good grading as an F-algebra. We say that a good grading is trivial, if S has an idempotent e such that the only nonzero homogeneous component of the grading is Re. T h e o r e m 8.2 ([Dascalescu et al. (2002)]) The F-algebra Mn(F) has a nontrivial good S-grading if and only if S contains a homomorphic image of the Bn defined by generators x^j, 1 < i, j' < n and relations xfi= X{^, x x i,j i,k = Xi,k, for all 1 < i, j , k < n. The semigroup Bn is infinite, it has arbitrarily large homomorphic images, and therefore it is impossible to replace it by any finite semigroup in the theorem above. Let She a, semigroup with the set E(S) of all idempotents, and let n be a positive integer. Denote by Wn the set of all mappings from {1, 2 , . . . , n} to E{S). For any / in Wn, let Cf be the set of all mappings fl:{l,...,n-l}-+5

such that g(i) € f(i)Sf(i

+ 1) for all i = 1 , . . . , n - 1.

Denote by Dn the set of all pairs (/, g) such that / G Wn and g £ Cn. T h e o r e m 8.3 ([Dascalescu et al. (2001)]) Let F be a field, n a positive integer, and let S be a finite semigroup. Then there exists a one-to-one correspondence between all good S-gradings of the upper triangular matrix F-algebra Un(F) and the set Dn. Namely, each pair V> = (/,g) € Dn determines a grading Un(F)=®3£SRW,

(8.13)

where

53 FeM ifH?^
(M)etf? T 0

( 8 - 14 ) T

ifH^ =
and

114

Gradings of Matrix

HW

=

{(k,£)\l
U

{{k,k)\l
Rings

+ = s}.

l)---g(£-l)=s} (8.15)

First, we prove Theorem 8.1. Proof. First, we take any simple or 0-simple factor Q = M(G; I, A; P) or Q •= M°(G; I, A; P) of S, and any triple T = ( £, then we use the definitions of CfT and DfT, respectively, and conditions (8.7)-(8.9) or (8.1)—(8.3) show that s is uniquely determined again. Thus Mn(F) = (BsesRsT is a direct sum. Next take any s, s' G S. If s 0 Q \ {0} or s' g- Q \ {0}, then Rs = 0 or Rg> — 0 and so RsRa' Q RSs'Suppose that s = (g; i, A) G Q \ {0} and s' = {g';i',\') G Q \ {0}. Consider Ry' = ^2^k^ePQr Feu and R^' = Y,(k,e)ePQ,T ^eke- In order to show that RSRS' C R3S> it suffices to pick any (k,£) G P^T and (k',£') G P S , T and verify that e^eyii G R3JIf £ ^ k', then ekt^wv = 0, and so we may assume that £ = k'. Then ekiek't' = e^/ and we need to verify that (k,£') G P3JFirst, we consider the case where k < £ < £'. Since ss' = (gpxi'g'; i, A'), condition (8.2) given for s implies the same condition for ss', and condition (8.3) for s' yields the same condition for ss'. Now conditions (8.1) for s and s' give us the following equalities: 9 = ¥>G(fc)P¥>A(fc+l)¥,.r(fc+l)V:>G(fc + l)P ¥ . A (fc+2) V/ (fe+2) • ' •
115

Full and Upper Triangular Matrix Rings g' = fa(k')pv,A(k'+i)lpI(k'+i)LPG(k'

+ l)p
Combining these with conditions (8.2) and (8.3) we get gPXi'91

=


Thus condition (8.1) for ss' is satisfied, too. This means

that (MO e ^ Other cases are similar, and we omit them. Thus RSRS' Q Rss', a n d so Mn(F) = © s € si?i T ) is an S-graded ring. As we have shown above, for every 1 < i, j < homogeneous element of Mn(F), and so this grading is good. Conversely, take any good grading Mn(F) = ®S£sRs- For each 1 < i,3 < n, there exists an element Sij £ S such that e^ € RSi . We need to find a simple or 0-simple factor Q of S and a triple r = (ipj, tp\, cpa) € TQ which defines the same grading, i.e., Rs = Rs for all s S S. Since every unit e^ generates the whole ring Mn(F) as an ideal, it follows that all elements s^j generate the same principal ideal of 5. Denote by Q the principal factor of S containing all the sij. All su are idempotents, whence Theorem 1.13 shows that Q is completely simple or completely 0simple. We consider only the case where Q = M°(G; I, A; P) is completely 0-simple, since the second case is similar. For 1 < i < n, let su = (gu; A. Since Su is an idempotent, it follows that 9iiPVA(i)Vl(i)9ii = 9u whence gu = P~Aii)vi{i)9ii-

(8.17)

Therefore pVA(i)Vl(i) ^ 0, as required in the definition of TQ. Take any i G { 1 , . . . ,n — 1}, and suppose that e^j+i = (<^o(i); £(i),r](i)). This defines a function G. The equality e i i e i i i + 1 = e i ) i + i implies ((i) =
(8.18)

116

Gradings of Matrix

Rings

Next, we verify that the triple r = ( ^ C V / . V A ) determines the same good grading of Mn(F). Take any s e S and consider the component Rs. We are going to prove that it satisfies (8.12). Suppose that Rs ^ 0 and choose a nonzero element x £ Rs. Since x e Mn(F), we get

x

( H e" ) = ( 5Z e» ) x= x-

Hence there exists 1 < i,j < n such that sus = SSJJ = s. It follows that s generates the same principal ideal as all the su in S, and so s € Q \ {0}. Thus Rs = 0 for all s £ Q \ {0}. Now fix any s £ Q\ {0}, say s = (g; i, A). We have to show that

RM=

£

Feu.

{k,e)eP^

Since all elements eke are homogeneous, it suffices to prove that eke £ Rs if and only if (k,£) G P®T'. Take any pair (k,£) where 1 < k,l < n. First, consider the case where k < t. Then Zkt = ^k,k+l&k+l,k+2

'''

&i-l,t-

This and (8.18) give us Skt = (h;
• • •
- 1).

Thus (k,t) satisfies (8.1)-(8.3), and so (k,£) belongs to U$T C pQT. Second, if t < k, then (8.18) and eu = ek,k-iek-i,k-2

• • • ee+i,e

similarly show that all conditions (8.4)-(8.6), and so (k, €) belongs to DfT C pQr Finally, if k = i, then (8.17) implies (8.7) and (8.16) yields us (8.8) and (8.9). Thus (k,£) belongs to C$T C P®T. This completes our proof. • Further, we prove Theorem 8.2.

Full and Upper Triangular Matrix

Rings

117

Proof. Suppose that the F-algebra Mn(F) has a good 5-grading Mn(F) — © se s-Rs- As usual this defines the degree function deg : U s e S i? s —> S.

For i, j G {1,2}, put sitj = deg(ei:j), i.e., denote by sitj an element of S such that eitj £ RSij. Denote by H the subsemigroup generated in S by all Si j for 1 < i, j , k < n. Consider the homomorphism ip : Bn -+ H defined by ip(xij) = Si j for all 1 < i, j < n. For any 1 < i,j,k < n, e?^ = e M and e^e^k = ei(x? J =
Un(F) = Yl

Fe

™

l
and each nonzero component Rs

equals

(k,e)eH*

it suffices to show that the set of all pairs (k, (.) such that 1
l)-..g(e-l)=s.

118

Gradings of Matrix

Rings

Since g is fixed, it follows that s is uniquely determined by the pair (k,£). If k = £, then (k, k) is in Hf if and only if f(k) = s. Hence s is unique again. Thus Un(F) =

(BsesR^

is a direct sum. Next, take any s, t £ S. If Rs — 0 or Rt = 0, then RsRt — 0 is contained in Rst, and so further we assume that Rs,Rt ^ 0. Then Hf,H? ^ 0. Consider Ry' = J2(k,e)eH* ^eM anc *

(k,e)eHf In order to show that R3Rt Q Rst it suffices to pick any (k,£) £ Hf and (ifc',0 e f fif f and verify that

If £ 7^ fc', then ekiek't' = 0, and so we may assume that t = fc'. Then Zkie-k'V — efcf. Hence it remains to verify that ew £ R^l , which is equivalent to (fc,^') € Hft. First, consider the case where k < £ < £'. By (8.15) we have s = g{k)g(k + 1) • • • g(£ - 1) and t = g(£)g(£ +

l)---g(£'-l).

Hence st = g(k)g(k + i)...g(e-

l)g{£)g{£ + 1) • • • g{£' - 1).

This means that (k,£') £ E^. Second, consider the case where k = £ < £'. Then (8.15) gives us s = f(k) and t = g(£)g(£ +

l)---g(f-l).

By the definition of Dn, since ip = (f,g) £ Dn, we get f(k)g(k) Therefore st = g(k)g(k +

l)---g(£'-l),

= g(k).

Full and Upper Triangular Matrix

Rings

119

and so ( M O e E ^ . Third, suppose that k < I = £'. Then s = g(k)g(k +

l)---g(e-l)

and t = f{t') by (8.15). The definition of Dn implies that g{i - \)f{l) g{t-l). Therefore st = g(k)g(k +

=

l)---g(£-l).

This means that (fc,f) G E^. Finally, consider the case where k = £ = £'. Then (8.15) yields s — t — f(k). Hence s = t is an idempotent of S. Therefore st = s and

(k,f) = (k,£)eH* = Hft. Thus we have proved that RSR3> C Ras>, and so Un{F) = (BszsRs is an 5-graded ring. As we have shown above, for every 1 < i < j < n, e^ is a homogeneous element of Un(F), which means that this grading is good. Conversely, take any good grading Un(F) = © s € 5i? s . For each 1 < i < j < n, there exists an element sy G S such that e^- £ RSi • We need to find a pair tp = (f,g) G Dn which defines the same grading, i.e., Rs = Rs holds for all s G S. For i = 1 , . . . ,n, since en is an idempotent of Un(F) and e\ G RSiiRSii, it follows that su lies in E(S). This defines the function f:{l,2,...,n}^E(S) such that /(i) = Su for i = 1 , . . . , n. Also, we get the function g:{l,2,...,n-l}^S defined by g(i) = si(i + 1) for i = 1 , . . . , n — 1. For every i = 1 , . . . , n — 1, we have ei(i+i) = eiie i ( i + 1 ) e( i + 1 ) ( i + 1 ) G i? S i i fi S i ( i + 1 ) i? S ( . + 1 ) ( . + 1 ) . Therefore s i ( i + 1 ) = s i i s f ( i + 1 )S( i + 1 )( i + 1 ). Since e^ and e ( i + 1 )( i + 1 ) are idempotents, it follows that g(i) = s i ( i + 1 ) G f(i)Sf(i+l). Thus = (/, p) G L>„.

120

of

Gradings of Matrix Rings It remains t o check t h a t t h e pair ip determines t h e same good grading Un(F). Take any s £ S and consider the component Rs. We shall prove t h a t

Rs = Rf. Suppose t h a t Rs ^ 0. Given t h a t the grading is good, it follows t h a t eij G Rs for some 1 < i < j < n. Ui = j , t h e n su = s implies (i, i) G Hf. O n t h e other hand, if i < j , then ei(i+i)e(i+i)(i + 2 ) • • • e(j_i)(j) = etj implies 9i(i+i)9(i+i)(i+2) and so (i,j)

€ Hf.

T h u s Hf

• ••9(j-i)(j)

= s,

^ 0, and therefore (8.14) shows t h a t

I$»=

^ (fc/)Sif?

Feu. T

We have to show t h a t Rs = ^ , f c (\£H* Feke, too. Since all elements e^ are homogeneous, it suffices t o prove t h a t e ^ s R3 if and only if (k,£) G Hf. Take any pair (k,£) where 1 < k < i < n. First, suppose t h a t k = £. T h e n efcfc G R3 if and only if f(k) = Skk = s. By t h e definition of Hpsis this is equivalent t o {k,t) G Hf. Second, consider the case where k < £. T h e n eia = efc(fc+i)e(fc+i)(fc+2) • • • e(e-i)(£) shows t h a t eke. belongs to Rs if and only if s = 9k(k+i)9(k+i)(k+2)

• • • 9{t-i){i)-

By definition the latter is equivalent t o (k,£) proof.

8.2

G Hf.

This completes our •

Gradings by Two-Element Semigroups

Matrix rings play i m p o r t a n t roles in applications of superalgebras (see, for example, [Avan (1990)], [Celeghini et al. (1991)], [Dubois-Violette et al. (1991)], [Gade (1998)], [Saidi (1995)]). A n a t u r a l step is t o relax the notion of a superalgebra, and investigate matrix algebras graded by t h e two-element semigroups.

Gradings by Two-Element Semigroups

121

There exist five types of semigroups with two elements: the group Z 2 , the semilattice, the left zero semigroup, the right zero semigroup, and the null semigroup. The left zero semigroup and the right zero semigroup cases are dual, and so in the first theorem we consider only one of these cases. Theorem 8.4 ([Dascalescu et al.]) Let S = {s, r} be a left zero semigroup with two elements. Then any S-grading of the algebra A = M2(F) is of one of the following types: (i) Aa = A, Ar = 0; (ii) As = 0, Ar = A;

for some A, fi £ F such that Xfi ^ 1;

for some fi S F;

«*-{(? A ;)—4M(» o H e 4 for some A £ F. Apart from the gradings obtained for fi — 0 in type (iv) and A = 0 in type (v), there are no other identical gradings in the list. Any grading of type (Hi) is isomorphic to the grading As=

\Q

0 J'

Ar=

\F

Fj'

and any grading of type (iv) or (v) is isomorphic to the grading AS=

Ar=

FJ'

{F

{o

0J•

In particular there exist 4 isomorphism types of S-algebra gradings on M2(F). Proof. For any a,b £ A we have that (ab)s

=

asbs + asbr

=

asbs + as(b — b3)

=

asb.

122

Gradings of Matrix

Rings

Thus the map tp : A —> A,(p(a) = as, is a morphism of right A-modules. Therefore it is of the form
Then h2 = his equivalent to a 2 + /3 7 = a, /J(a + 8) = /?, 7 ( a + 8) = 7, <52 + £ 7 = 5. If a + 8 ^ 1, then /3 = 7 = 0, and a, 8 e {0,1}, thus either a = J = 0 or a = 8 = 1. In this case we obtain two solutions, h = 0 and h = I2. If /i = 0, then we obtain the trivial grading A s = 0, Ar = A. If h = I2, then we have the other trivial grading As = A,Ar = 0. If a + 7 = 1, the solutions are of the form

where a € F and /?, 7 € -F such that /3~f = a — a2. In this case, if

then the homogeneous component of degree s of a is _ / ax + f3z \ 7X + (1 — az)

ay + (3t \ 72/ + (1 - a)t / '

Note that j(ax + f3z) = 01(70; + (1 — a:)z), j(ay + (3t) = a(7j/ + (1 — a)t). Let us consider first the case when a ^ 0. For any u,v £ F choose some z,t £ F and x = %=&-, y = ^ ^ . Then ax + (3z = u and ay + (3t = v. It follows that

By a similar computation we find that

* = {(t t)=-«4

Gradings by Two-Element

Semigroups

123

We claim that the pair ( J , — f ) can take any value (A, /z) € F 2 with A/z ^= 1. Indeed, if a,/?, 7 e F such that a ^ O and /?7 = a ( l - a ) , then ^(—f) = ^ = i ^ 1. Conversely, if A, /z £ F satisfy A/z ^ 1, then take 1 ^ _ 1 — A/z'

-P a. _ 1 — A/z'

A

1 — A/z

Then clearly a ^ 0, (3j = a(l - a), and A = ^,/z = — ^ . Thus we obtain gradings of type (iii). If a = 0, then either /3 = 0 or 7 = 0. In the first case we obtain gradings of type (iv), while in the second case we find gradings of type (v). Let us show that a grading of type (iii), given by the parameters A,/z with A/z ^ 1, is isomorphic to the grading As=

{

0 0 J'

Ar=

\F

F J'

Take

-(AT). which is clearly an invertible matrix. Then the map / : A —> A defined by / ( a ) = XaX"1 for any a £ A is an algebra isomorphism, and it is straightforward to check that

Thus / is an isomorphism of 5-graded algebras. Similarly, the gradings of type (iv) and (v) are isomorphic to the grading AS =

[F

F)'

Ar=

{o

0 )• D

Proposition 8.1 ([Dascalescu et al.]) Let S = {l,s} be a semilattice, and let A = M2(F). Then there exist exactly two S-algebra gradings of A, namely (i) Ai = A, As = 0, and (ii) Ai=0,As=A.

124

Gradings of Matrix

Rings

Proof. For any structure of an S-graded algebra A, we clearly have that A3 is a two-sided ideal of A. Then either As = 0, and the grading is of type (i), or As = A, and the grading is of type (ii). • Finally, if S is the zero group, it is obvious what an S'-grading must be. Proposition 8.2 ([Dascalescu et al.]) Let S be a null semigroup (not necessarily with two elements) and A an F-algebra. Then there exists only one S-grading of A, this is AQ — A and As = 0 for any s ^ O . Proof. We have

A = AA = (£Aa)(52At) = £ AsAt C £ Ast = A0, s€S

tes

s,tes

s,tes

and so A = Ao- Then obviously As must be 0 for all s ^ 0.

•

Let us consider the .F-algebra of 2 x 2 upper triangular matrices

Theorem 8.5 ([Dascalescu et al.]) Assume that the field F has characteristic different from 2. Then any grading of the F-algebra T by the group Z 2 = {e, g} is of one of the following two types. (i) The trivial grading, i.e., Te =T,Tg

=0.

( )

M(o" r )— }'M° »)• for some a G F. Moreover, any grading of type (ii) is isomorphic to the grading F le g T-( °) J' T=(° ~\ 0 F \0

F ) 0 J'

In particular, there exist two isomorphism types of ^-graded ture on T.

algebra struc-

Lemma 8.1 Let : T —> T be an F-linear map. Then <j> is an algebra automorphism of T if and only if there exist a £ F and b € F* such that (e12) = be12, <^(e22) = - a e 1 2 + e 22 .

Gradings by Two-Element

125

Semigroups

In this case 4>2 = Id if and only if either 6 = 1 and a = 0 (when = Id) or b = — 1 and a S F. This lemma can be proved by a straightforward but tedious computation which is omitted. Finally, we prove Theorem 8.5. Proof. If T = Te ®Tg is a Z 2 -grading of T, let us define the map :T -> T by 0(A) = A e - Ag for any A £ T. Then {AB) =

(AJ5)e-(AB)s

=

J\.eJDe

"T" AglDg

=

4>{A){B)

A~eJDg

Ag t> &

and since 0 is clearly bijective we obtain that 0 is an algebra automorphism of T. Moreover, 0 2 (A) = (f>(Ae - Ag) = Ae + Ag = A, thus 0 2 = L I In terms of the automorphism (f>, the grading is Te = {A e |A e T} = { i ( A + 0(A))|A e T } ,

T9 = {A 9 |A e T} = {\{A - 0(A))|A e T}. Lemma 8.1 shows that either = Id or 0(en) = e n + aei2, (ei2) = —ei2,
we have

*W=(o

a(x

~zz)-y),

and then

'•-{(; , ( v " ) — 4 r - - ( S o ) which proves the first part of the statement. If for any a € F we denote by T(a) the algebra T with the grading

126

Gradings of Matrix

Rings

then the map ip : T(a) —• T(0) denned by

«(o 0 » - ( o ° (I 7 ) " V is an isomorphism of Z2-graded algebras.

•

Theorem 8.6 ([Dascalescu et al.]) Let S = {s,r} be the left zero semigroup. Then an S-grading of the F-algebra T is of one of the following types: (i) Ts=0,Tr (ii) Ts=T,Tr (hi)

= T. = 0.

r.-( '

F

0

F '

{(

°

0 0

cz z

:z£F

for some c € F. (iv)

r.-{ {/

0 cz

0

z

H4-U0

for some c £ F. Moreover, any grading of type (Hi) is isomorphic to the grading s

( F " ^ 0

F \ o j '

_( 0 0 V0 F

J r _

and any grading of type (iv) is isomorphic to the grading s

\ 0 F J'

l r

~ \ 0

0

In particular, there exist 4 isomorphism types of S-gradings of the F-algebra T. Proof. If T = Ts ©T r is an S-grading of T, define <j>: T -> T by (A) = As for any A£T. Then as in the proof of Theorem 8.4 we see that there exists h eT with h2 = h such that <j>(A) = hA for any A eT, and T. = {4>{A)\A G T } , T r = {A - «^(A)|A G T } .

Gradings by Two-Element

Semigroups

127

Straightforward computations show that h must be one of the following matrices

°' / 2 '(o

i )'(o o

for some c £ F. If h = 0 we obtain Ts = 0, Tr = T. If h = Ii we have Ts = T, Tr = 0. If h = I matrix A = (

AS = ^ )

I, the homogeneous components of the

J G T are

= M = ( *

y

~

C

")

andAr=A-
producing a grading of type (iii). Similarly, h = I

^

J produces a

grading of type (iv). Finally, if we denote by T{c) the algebra T with the grading

H o o ) ' M ( » : ) — }• we have that the map / : T(0) —> T(c) defined by

'<(o 0 ) = ( o

_C(l

"/) + !'

is an isomorphism of S'-graded algebras. The proof for the gradings of type (iv) is similar. • Theorem 8.7 ([Dascalescu et al.]) Let S = { l , s } be a semilattice. an S-grading of the F-algebra T is of one of the following types:

Then

(i) T 1 = T , r s = 0. (ii) T1 = Q,TS=T. (iii)T 1

=

{(°

" ) : i e F } , T . = ( J

(iv)T1=F/2,Ts=(J

(v)Tl =

F

oy

for some

ceF.

F Q

{ ( o Co):x€F}>Ts

=

{°o

F)'forsomeceF-

128

Gradings of Matrix

Rings

F

wT1=FI„Ts=(i (vi

.)Ti

=

^ x

Fy

«(*"») ) : a ! , y E F } > r . =

(S

F

Q),forsome

ceF. Moreover, a grading of type (Hi) is isomorphic to the grading Tl =

Ts =

o)'

(o

(Fo o ) '

a grading of type (v) is isomorphic to the grading Tl

=

( 0

0 ) '

Ts

= ( 0

FJ'

a grading of type (vii) is isomorphic to the grading

Tl =

( o °F)'

TS =

{°O

o )'

and the isomorphism types (i), (ii), (Hi), (iv), (v), (vi), and (vii) of Sgradings are different. In particular there exist 7 isomorphism types of S-gradings of the F-algebra T, five of them being good gradings, and the other two not isomorphic to good gradings. Proof. As in the proof of Proposition 8.1, Ts is a two-sided ideal of T. Thus Ts is one of

OT(F

F

)

(°

F

)

(°

F

)

If Ts = 0, we obtain Tx = T. If Ts = T, we get Tx = 0. If F T-( ls ~ \ 0

F

0

)

)'

then Ti has dimension 1 over F, more precisely

for some a,b,c £ F, with c ^ 0. Since T{Tx QT\, we see that either a = b and c = 0, or a = 0. In the first situation we obtain a grading of type (iv), in the second one a grading of type (iii).

Structural Matrix

Similarly, if Ts = (

„

129

Superalgebras

, then we obtain gradings of types (v)

and (vi). Assume now that Ts = I

I. Then it is easy to see that T\ has a

basis consisting of the matrices

1 a \1 , ( 0 ft and ' 0 0 J V ° ! for some a, ft G F. If we write that the product of these two matrices is in Ti, thus spanned by the two matrices, we obtain that ft = —a. Then

+

F

* - Hs;) »(; 7h>* ) x 0

a(x — y) ..

,\x,y&F

i.e., we have a grading of type (vii). The rest of the claim follows now as in the proof of Theorem 8.5. • In conclusion we note that gradings of T by the null semigroup have already been described in Proposition 8.2.

8.3

Structural Matrix Superalgebras

We say that the matrix superalgebra R = RQ + Ri C Mn{F) is homogeneous, if all the standard matrix units ey contained in R are homogeneous, i.e., belong to the union RQ\JR\. The special attention to this case is motivated by the results of [Dascalescu et al. (1999)]. This section contains a description of all homogeneous superalgebras represented by the matrix algebra MD(F), for a class of directed graphs D. In fact, the results of this section also describe all homogeneous group gradings of the matrix algebras, too. First, we introduce two technical concepts and show how they can be used to describe matrix superalgebras. Definition 8.1 Let D = (V,E) be a graph satisfying (3.10), where V = {1,2,... ,n}. A subset B of E will be called a grading basis if

130

Gradings of Matrix

Rings

there exists a one-to-one correspondence between the set of all homogeneous superalgebras represented by MD(F) and the set of all mappings / : B —> Z2, where the mapping / corresponds to a unique superalgebra MD(F) = L0 + LI such that ew £ £/(«,) for all w £ E. Definition 8.2 Let D = (V,E) be a graph, and let G be a group. A mapping h from E to G will be called a homomorphism if and only if H(i,j)) + h((j,k)) = h((i,k)), for all (i,j),(j,k)

(8.19)

£ E.

Theorem 8.8 ([Kelarev (2003)]) Every homomorphism f : E —> Z 2 defines a homogeneous matrix superalgebra MD(F) of the structural matrix ring

= ®geZ,Lf

MD{F),

(8.20)

where

Lg = ®w<=Ej(w)=gFew.

(8.21)

Conversely, for each homogeneous superalgebra represented by MD{F), there exists a homomorphism f : E —• Z2 which produces this grading via (8.20) and (8.21). Therefore, a subset B of E is a grading basis of D if and only if each mapping f : B —> Z2 uniquely extends to a homomorphism from E toZ2. Proof. Let / : E —» Z2 be a homomorphism. Consider the direct sum decomposition denned by (8.20) and (8.21). Take any g,h £ Z2 and two edges v = (v\,v2),w — (wi,wz) such that f(v) = g and f(w) = h. If i>2 ^ wi, then evew = 0, and so FevFew = 0. If, however, v2 = w\, then By the choice of / we get f({vi,v2)(wi,w2)) = f{{v\,v2)) + f((wi,w2)) = g + h; whence FevFew = Fe ( „ IiU , 2 ) C Lfg+h. Therefore L'gLh — ®veEj(v)=gFev

®weE,f(w)=h Few C Lg+h-

It follows from (2.1) that (8.20) is a superalgebra. Conversely, suppose that MD{F) = ®gez2Ijg is a homogeneous superalgebra. For each w G E, there exists g £ Z2 such that ew £ Lg. Define the value f(w) to be equal to g. This introduces a mapping f : E —> Z2. For any two edges v = (a, b) and w — (b, c) in E, let g = f(v), h = f(w). Since e„ £ Lg and ew £ Lw, we get e(a>c) = evew £ Lg+h. Hence f((a,c)) = g + h. This means that / is a homomorphism. If we define a components LQ and

Structural Matrix

Superalgebras

131

Li using / , (8.20), and (8.21), then it is easily seen that Lg = L?, for all g G Z2. Thus / defines the same superalgebra, as required. • Let M i , . . . , Mm be the set of all maximal cliques of D. Choose vertices ui,...,um so that «i G Mi, ..., um G M m . For i = 1 , . . . ,TO,denote by 5 t ' the set of all edges (u*, u;), where w runs over i?< \ {vi}, and let B0 = B I U B ^ U . . . U B ^ . Introduce a new graph A = (VA,EA), which is obtained from D by identifying all vertices of each clique M; with Mj, for all i, and then removing all loops. For each edge (u',v') of A, we choose one edge (u, v) in D such that (u',u') is the image of (u, v) in the natural mapping from D to A. Put ip((u,v)) = (u',v') and V'_1(('u'>'1'')) = ( u > w )- Clearly, A has no loops. Given that D satisfies (3.10), it follows that A satisfies (3.10), too. Therefore A is acyclic. The following theorem reduces the problem of finding a basis to the case of acyclic graphs. Theorem 8.9 ([Kelarev (2003)]) A set BA of edges is a grading basis of the A if and only if B = Bo U IP~1(BA) is a grading basis of D. Proof. The 'if part. Suppose that Bo U T](B') is a grading basis of D. Consider any mapping / ' : B' —» Z2. Define a mapping / : B —> Z2 by putting f ( A J[e

>-\

_\

/(e) =/'(V-(e)) 1

HeeV(B'), ifeGSo.

By Lemma 8.8, / is uniquely extended to a homomorphism / : E —• Z2. Obviously, the restriction / ' of / on E' is a homomorphism f : E' —* Z2 which extends / ' . It follows from Lemma 8.8 that B' is a grading basis of D'. The 'only if part. Suppose that B' is a grading basis of D'. Take any mapping / : (JB0 U n(B')) —> Z2. Define an induced mapping / ' : B' —» Z2 by putting /'(e) = f(r](e)), for all e G B'. By Lemma 8.8, / ' is uniquely extended to a homomorphism f : E' —* Z2. We define a mapping f : E —> Z2. First, put / ( e ) = f'(ip(e)), for all e G T](E'). Second, consider any edge e G E\T](E'). By the definition of the graph D', we see that e = (vi,Vj), where at least one of the vertices Vi,Vj belongs to the maximal clique {vi,..., Vk}. If vt belongs to the clique, then

132

Gradings of Matrix

Rings

1 < i < k, (vi,Vi) £ B, and we put 7(e) =

-f((v1,vi))+T((v1,vj)).

If Vj belongs to the clique, then 1 < j' < k, (v\,Vj) £ B, and we put J(e) = f((vi,v1))

+

f((v1,vj)).

It is routine to verify that / is a unique homomorphism from E to Z2, which extends the mapping / . It follows from Lemma 8.8 that B is a grading basis of D. D

Chapter 9

Examples of Applications

This chapter introduces several examples of applications, but it is far from a complete and systematic overview of all applied directions. Brief references to other applications have been mentioned earlier in the text. Here we also note that, for example, certain semigroup rings are used in number theory, where they are sometimes called Dirichlet algebras (see [Knopfmacher (1975), §2.1]).

9.1

Codes as Ideals in Group Rings

It is well known that introducing additional algebraic structure results in several advantages for coding applications. For example, linear codes are in general better than arbitrary ones, cyclic codes are better than linear codes. The additional algebraic structure makes it possible to use a small number of generating elements to store the whole code in computer memory, and to use these generators in faster encoding and decoding algorithms. These circumstances have motivated serious attention of several authors to considering ideals of various ring constructions from the point of view of coding applications (see the recent survey [Kelarev and Sole (2001)] and books [Pless, Huffman, Brualdi (1998)], [Poli and Huguet (1992)]). All cyclic codes of length n are ideals in the group algebra F[Z„], where Z n is the cyclic group of order n. For several types of well-known cyclic codes it has been shown that they are ideals in group algebras of other groups. This additional algebraic structure helps to find more efficient encoding and decoding algorithms for known codes, and to find new codes in the group algebras. On the other hand, various codes have been originally 133

134

Examples of

Applications

defined in terms of certain group algebras. A group-algebra code is a one-sided ideal in a group algebra F[G], where F is a finite field, and G is a finite group. If the group G is abelian, then the one-sided ideals of F[G] are called abelian codes. All elements of the field F are letters of the encoding alphabet. The order n = |G| of the group is the length of codewords. To illustrate we include the following easy example. Consider the 1-errordetecting (4,3) binary cyclic code which adds a parity-check digit C3 = C0 + C1+C2 to each message m = (CQ , c\, c-i). Denote by g a generator of the cyclic group C4 in multiplicative notation. If we identify each codeword c = (00,01,02,03) with the element coe + c\g + C252 + C3g3 of the group algebra i ^ C ^ , then the set of all codewords forms an ideal of i ^ I ^ ] - On the other hand, take the elementary abelian group G = C2 x Ci- In additive notation G = {(0,0), (0,1), (1,0), (1,1)}. If we identify the codeword c = (co,ci,c 2 ,c 3 ) with the element co(0,0) + ^ ( 0 , 1 ) + c 2 (l,0) + c 3 ( l , l ) of the group algebra i^fC], then the same code is embedded into another group algebra, and again it is clear that it forms an ideal in this algebra. Since every finite abelian group G is a direct product of its primary components, it follows that the group algebra F[G] is a tensor product of the group algebras of these components, and therefore every code in F[G] is a direct sum (i.e., a concatenation) of codes in the group algebras of p-groups, for all prime divisors p of the order of G. Taking this into account, further we consider only the case where G is a p-group. Let K be an extension of F, containing all n-th roots of unity. A character of G is a homomorphism of G into the multiplicative group of a given field K. The set G* of all characters of G becomes an abelian group under pointwise multiplication: {xiXi){g) = Xi(9)X2(g), for g S G and XiiX2 £ G*. Each element g £ G defines a character g* of G* by the rule g*(x) = x(g)- This gives a homomorphism of G into G**. If we look at every element g as the characteristic function g : G —> K given by

\ 0

otherwise,

then each character x of G can be regarded as an element X = Yl x(g)9 of the group algebra i^[G], and in this way extends to a linear function

Codes as Ideals in Group Rings

135

X : F[G] —> K. Each character x defines the idempotent

in the group algebra if [G]. Let C be an ideal in F[G]. The root set R{C) is the set of characters X such that x(C) = 0- I n the special case of cyclic codes, the characters which are roots correspond to the roots of the generator polynomial of the code. Theorem 9.1 Let G be a finite abelian group, F a finite field with characteristic not dividing \G\, and let K be an extension of F, containing all \G\-th roots of unity. Then (1) all three groups G, G* and G** are isomorphic; (2) the homomorphism g H-> g* is an isomorphism of G and G**; (3) if H is a subgroup of G, then every character of G/H gives us a character of G, and (G/H)* ^H°

= {XeG*\

X(g)

= 1 for all g e H}

(4) (5) (6) (7)

if we identify G and G**, then (H°)° = H; if H is a subgroup ofG, then H* 9* G*/H°; the characters of G form a basis of K[G]; the idempotents ex, x € G*, are mutually orthogonal and generate one dimensional ideals in K[G\; (8) each idempotent e in K[G) is equal to the sum of all ex for all x with x(e) = 1; (9) if C is an ideal in F[G], then e = X/vrfflrc) ex *5 ^ie idempotent generator of C.

Thus the identity of the abelian group algebra F[G] is equal to the sum of central idempotents e i , . . . , e m such that e^- = 0 whenever i ^ j , and each ei is a unique nonzero idempotent in the principal ideal it generates. Every ideal of F[G] is generated by the idempotent equal to the sum of all the e, which belong to this ideal. Let q = pm, where p is an odd prime, m is a positive integer, and let F be a field of characteristic s not equal to p. The generalized quadratic residue codes (or GQR codes) of length q are ideals of the group algebra F[G], where G is the additive group of Fq, defined as follows.

136

Examples

of

Applications

Denote by a a primitive p-th root of unity in the algebraic closure of F. A linear character of G is a homomorphism of G into the multiplicative group of nonzero elements of F(a). If x € G, then put Tr(:r) = X^=o xVFor g € G, define the character Xg by = a1*™

Xg(h)

for h € Fq.

The set {\g \ g € G} is the set of all characters of G. Let Q be the set of all nonzero squares in Fg, and let N be the set of nonsquares in Fq. The generalized quadratic residue codes CQ(F) and CN(F) (or GQR codes) of length q are defined by CQ(F)

= {ce F[G] | Xg(c) = 0 for all g e Q],

CN(F)

= {c

G

F[G] | x 9 (c) = 0 for all g e JV}.

Both of these codes have dimension \{q-\-1). If p = q, then these codes are the classical quadratic residue cyclic codes. For a £ Fg, put

-^2a^-a^geF[G}.

e(a) = q

gee

The GQR codes coincide with ideals generated in F[G] by the idempotents \~] e(a) and 2_, a€Q

e a

( )-

a€N

Every cyclic code C of length n = pm — 1 over Fp can be extended by the overall parity-check digit and embedded in the group algebra Fp[G], where G is the additive group of Fp™, as follows. Choose a primitive element a in Fp™, i.e., a generator of the multiplicative group of nonzero elements of Fpm. Let e be the identity of the group G. Map a codeword c = (co, c\,..., c„_i) to the element n —1

c >-> -

Y^ i=0

n—1

c e

* +Yl

c ai e

*

J F

( P[G])

i=0

of the group algebra _FP [G]. The defining set T of C is the set of all integers A; such that 0 < k < n and ak is a root of the generator polynomial of C.

Codes as Ideals in Group Rings

137

Let p be a prime, q = pr. Let p be an integer with 0 < p < m(q — 1). The generalized Reed-Muller (or GRM) code of order p and length qm over Fq is the linear subspace G R M ^ (p, m) of the space L of all functions / : F"1 —> Fq spanned by all polynomials in the coordinate variables x\,..., xm with degree at most p, i.e., (x[1---x^

\ii + --- +

im
By wtg(fc) we denote the number of nonzero digits in the g-ary notation for k. The following theorem shows that GRM codes over Fp coincide with powers of the Jacobson radical of a group algebra of an elementary abelian p-group. Theorem 9.2 Let F be a field with char(F) = p, G = {g | gpm — e), 1 < j < Pm - 1, V = id((l - g)i) C F[G], j = hp771'1 + --- + bm, where 0 < bk < p. If bi = • • • = bm = p - 1, then V = id(l + g -\ h g^'1) m has dimension 1 and weight WJJ(V) = p • Let r be the first index with br^p-l. Then if ( 6 r + 1 (V) = { pr~1{-ir + 2 ) ' • •''hm) * ( 0 , • •''0)' ' l / ^ i r + l) t/r = m o r ( 6 P + i , . . . , 6 m ) = (0,...,0). If G = Zp and m = 1, then this class of codes contains a subclass of MDS-codes with optimal parameters (p, £,p — £ + 1) and efficient encoding and decoding algorithms. HK

Theorem 9.3 Let G be the direct product ofm copies of the cyclic group of order p, and let 0 < p < m(p — 1). Then the generalized Reed-Muller code GRMFp(p,m) is equivalent to the code given by the power J{Fp[G])t of the radical of the group algebra Fp[G\, where t = m(p — 1) — p. It has length pm, dimension \{k | 0 < k < pm, wtp(k) < m(p - 1) - t}\ and minimum weight (b + l)pa, where t = a(p — 1) + b and 0 < b < p — 1. As an extended cyclic code its defining set consists of all numbers k such that 0 < k < pm - 1 and wtp(k) < t. Let F be a field of characteristic p, and let G be an arbitrary group of order pm. A Jennings basis of G is a sequence
138

Examples of

Applications

We may assume that 1 = hi < • • • < hm. The power J(F[G])h of the radical J{F\G])h is spanned by all products (g\ — l ) ' 1 • • • (gm — l)tm such that 0 < U < p-1 and £ ™ i Uk > h. The product (gt - 1 ) ' 1 •••(gm\)tm has weight I~[fei(** + !)• Ward has proved that they form a visible basis of F[G] in the sense that every code spanned by a set B of these products has minimum weight equal to the minimum weight of the products in B. Let K be a finite extension of Fq, and let K[x,y]m denote the set of homogeneous polynomials over K of degree m in commuting variables x and y. Denote by Fq the projective line over Fq, i.e., Fq — Fq U {oo} and put f(z) = f(z(f>), for z € Fq, f £ K[x,y]m, where
-\

f (z,l) (1,0)

iizGFq, ifz = oo.

Take any a = ( a o , . . . , a n _ i ) , where ceo, • • •, otn-i are distinct elements of Fq, and v = (vo,... ,vn_i), where vo, • • •, u n - i £ K \ {0}. For 1 < k < n, the Cauchy code with location vector a, scaling vector v, and location set La = {c*o, • • •, a n - i } is defined by Ck{a,v,q,K)

— {(v0f(a0),...,vn-1f(an-1))

| / e ^[a;,?/]^-!}.

In the special case where the location set is Fq \ {0}, and v = 1, then Ck{a, v, q, K) is the Reed-Solomon code; if the location set is Fq and v = 1, then Ck(a,v,q,K) is an extended Reed-Solomon code; and if the location set is a subset of Fq, then Ck(a,v,q,K) is the generalized Reed-Solomon code GRSk((x,v). Assuming that the polynomial / is nonzero, the code Ck(ct,v,q,K) has dimension k and minimum distance n — k + 1. A bicyclic code (also called 2-D cyclic code) is an abelian code w.r.t. to the direct product of two cyclic groups of order, say, n', n". If /3 (resp. 7) denote two elements of respective order n' and n" in the algebraic closure of Fq then such a code can be defined by the data of a certain subset A of [n — 1] x [n — 1] in the following manner C := {c(x,y) I (j',f)

&A^

c ( ^ ' , 7 j " ) - 0}.

Here c(x, y) is a polynomial representation of a generic codeword of C. Its degree in x (resp. y) is < n' (resp. < n"). An important class of bicyclic codes is introduced by specifying

A :={(/,/') I (i' + l)(i" + l ) < d}.

Codes as Ideals in Group Rings

139

These are the so-called hyperbolic codes of designed distance d. For instance there is such a code over Fg of parameters [49,35,7]. These codes can be decoded efficiently by the Sakata Algorithm. See [Pless, Huffman, Brualdi (1998)], p. 1614, for details and references. If G is non abelian then the ring F[G] is not commutative and codes will be denned as being one-sided ideals. Representation theory is essential. See [Curtis and Reiner (1962)] for background and undefined terms. A representation T of degree n of a group G over a field F is a homomorphism from G into GL(n, F). The character x afforded by T is the map g \—> det(T(g)). Let Ti, i = 1, • • •, k be the irreducible representations of G over K; let rij be the degree of T and Xi the representation it affords. With each Ti we attach a two-sided ideal of F[G] generated by the idempotent

Thus F[G] decomposes as a direct sum of minimal two-sided ideals F[G] := 8 t i ( e 0 If F = L is the splitting field of G, then each summand decomposes in turn as a sum of one-sided ideals. Theorem 9.4 If T is an irreducible representation of G over L with attached ideal V, then V =

®^WU

where each Wi is the one-sided ideal generated by

1

' geG

y

Given three integers m, n, r, the metacyclic group G(m, n, r) is defined by the presentation (x,y\ xm = yn = 1 &yx = xry). Absolutely irreducible representations of such groups are well-understood [Curtis and Reiner (1962)], pp. 333-340, and allow for a straightforward application of the general theory. A binary [125,20,44] code is constructed in that way by Sakata.

Examples of

140

Applications

The binary and ternary Golay codes are described as ideals in group rings of non-abelian groups in [Landrock and Manz (1992)] and extended Golay codes in [Bernhardt et al. (1990)]. The investigation of coding properties of ideals in semigroup rings has been started in [Cazaran and Kelarev (1997)]. Twisted group ring and their cocycles have also been used in constructing error-correcting codes (see [Baliga and Horadam (1995)], [de Launey et al. (2000)], [Horadam (1996)] [Horadam (2000)], [Horadam and de Launey (1993)], [Horadam and de Launey (1995)], [Horadam and Perera (1997)] and [Horadam and Udaya (2000)] for details and references). 9.2

Codes as Ideals in Matrix Rings

Let F be a finite field, and let MQ (F) be a structural matrix ring. Various types of ideals in structural matrix rings have been explored very well in the literature. Prom the point of view of coding theory, the Hamming weights and information rates of ideals are of interest. Indeed, the Hamming weight WH(C), i.e., the minimum number of nonzero coordinates of elements of the code C in a given basis, is important, because it gives the number of errors a code can detect or correct; and the information rate shows the ratio of the number of message digits, which form the information to be transmitted, to the number of all digits. This section contains recent results on the Hamming weights and information rates of ideals in structural matrix rings. We consider only ideals with Hamming weight greater than one, because codes with weight one cannot detect even a single error. The in-degree and out-degree of a vertex v £ V are defined by indeg(v) = \{w G V | (w,v) £ E}\, outdeg(u) = \{w £ V | (v,w) £ E}\. A vertex of D is called a source (sink, isolated vertex) if indeg(w) = 0 and outdeg(u) > 0 (respectively, indeg(w) > 0, outdeg(v) = 0, or indeg(w) — outdeg(w) = 0). Denote by so(D) and si(D) the sets of all sources and sinks of D, respectively. For each vertex v £ V, put so(w)

= {«e

SO(JD) I (u,v)

£

E},

si(u) = { « £ si(£>) | (v,u) £ E},

Codes as Ideals in Matrix

141

Rings

V = V\(so(D) U si(L>)) = {v G V | indeg(u), outdeg(v) > 0}. In order to describe all pairs of the information rates and weights of all ideals, it suffices to find maximal information rates of all ideals with each value of the Hamming weight. Denote by wici{D) the maximum Hamming weight of ideals of the ring MD(F). Theorem 9.5 ([Kelarev and Sokratova (2001)]) Let D = (V,E) be a graph defining a structural matrix ring MD(F). The maximum Hamming weight of ideals of the ring Mry (F) is equal to wid{D)=mzx.{l,\En(so(D)x

si(D))\, | si(u)|, |so(u)| : v £ V}.

(9.1)

Proof. Let us show that MB (F) always has an ideal with Hamming weight given by (9.1). Consider all possible cases, which may occur in (9.1). Case 1: m a x { l , | E n (so(D) x si(D))\, | si(u)|, | so(w)| : v € V} = 1. In this case the assertion is trivial, since the Hamming weight of the whole ring ME>(F) is equal to 1. Case 2: max{l, \E n (so(Z?) x si(D))|, | si(w)|, | so(u)| : v e V} = \E n (so(D) x si(D))|. Denote by I the ideal generated in MD{F) by the element x —

/

e-w.

j

weEn(so(D)x

si(D))

It is easily seen that

MD(F)

Fe

Yl w€En(so(D)x

si(D))

^=

£ w€En(so(D)x

FewMD(F)=Q. si(D))

Therefore I = Fx, and so wH(I) = wH(x) = \E n (so(D) x si(D))|. Case 3: max{l, \Ef\ (so(D) x si(D))|, | si(u)|, | so(v)| : v G V} = |si(u)|, for some u £V. Denote by I the ideal generated in MD (F) by the element

V=

J2 e (UiU) . u£ si(u)

By the definition of si(tt), we get yM[)(F) = 0. Hence

I= F

J2 e(«,-)+ Yl ve si(«)

J2 Fe(^,v)-

(ui,u)€Bi>€ si(u)

Every edge of D occurs at most once in all sums of this expression. Therefore the Hamming weight of J is equal to | si(u)|.

Examples of

142

Applications

Case 4: max{l, \En(so(D) x si(D))|, | si(u)|, | so(u)| : u G V} = |so(w)|, for some v G V. Denote by J the ideal generated in MD(F) by the element

u£ so(v)

The definition of so(u) yields I=F

= 0. Therefore

MD(F)Z

Jl e(«,«)+ S u£so(u)

H

(v,vi)€E

Fe

(«,-i)-

u€so(v)

Every edge of D occurs at most once in all sums of this expression. It follows that the Hamming weight of I is equal to | so(v) |. Thus in all the cases MD{F) has an ideal with Hamming weight given by (9.1). Next, we take any ideal I of MD(F), and show that it has Hamming weight at most (9.1). Obviously, we can assume that 1^0. Choose a nonzero element

(u,v)eE

where x^u^ G F. The following cases are possible: Case 1: X(U)„) ^ 0 for some u, v G V. Since u GV, there exists u\ such that (ui,u) G E. Similarly, {v,v{) G E for some v\ G V. It follows that l e («i,ui) ~ e(ui,u)xe^VtVl-) G I. Therefore in this case the Hamming weight of I is equal to 1. Case 2: £(„,„) = 0 for all pairs u,v eV, but £(„,„) / 0 for some u &V. Hence v G si(u). Moreover, v' G si(u) for all v' G V with x^u,v') ¥" 0- Since u G V, there exists ui such that (ui,u) G E. It follows that

zG si(u)

Therefore the Hamming weight of I is at most | si(u)|. Case 3: x^u^ = 0 for all pairs u,v £V, but £(„,„) 7^ 0 for some v GV. We see that u G so(-u). Moreover, v! G so(w) for all u' G V with £(„',„) ^ 0. Given that v GV, there exists wi such that (v, vi) G £ . It follows that Xe

(.V,Vl)

e

^

^e(2^l)-

z£ so(u)

Therefore the Hamming weight of / is at most | so(v)|.

Codes as Ideals in Matrix

143

Rings

Case 4: X(Ui„) — 0 if u G V or v G V. Then

w£Er)(so(D)x

si(D))

and so the Hamming weight of x is at most \E n (so(.D) x si(Z)))|. Thus in each of these cases the Hamming weight of I does not exceed w id(D) given by (9.1). This completes the proof. • For positive integers n, h, denote by dh{n) the maximum integer d such that there exists a linear (n,d) code with Hamming weight h (see [Pless, Huffman, Brualdi (1998)] and [Poli and Huguet (1992)]). If there are no codes of this sort, then we put dh(n) = 0. T h e o r e m 9.6 ([Kelarev and Sokratova (2001)]) Let D = (V,E) be a graph defining a structural matrix ring MD(F). For any 1 < h < Wid(D), all ideals of the ring MD (F) with Hamming weight h have information rate at most 1

dh

U 80WI) + d^E

£ dh(\ si(tO|) + E

W\ p&V

n

(S°P) x siP))l)

v€V

(9.2) Proof. Consider any ideal I of the ring MD(F), which has Hamming weight h, where 1 < h < Wid{D). Every element x G MD{F) has a unique representation in the form x = ^ZweE xwew> where xw G F. The element xw is called the projection of x on w. Let J-w

=

~ \ 3 J t u \ X KZ 1 j .

For any S C E, denote by Is the projection of I on S, that is the set wes Let supp(J) be the set of edges w such that Iw ^ 0. If supp(7) Pi V ^ 0, then I contains an element x such that X(ui) G E, and get le( Uljt;i ) = e(UliU)a:e(„>t;i) G I. This contradicts our choice of h and shows that supp(7)

C

{(v,si(v))\v

GV}U{(so(v),v)\v

eV}

144

Examples of

Applications

U(£n(so(D) x si(D)).

(9.3)

Suppose to the contrary that the information rate of I exceeds (9.2). Then it follows from (9.3) that one of the following cases occurs. Case 1: |supp(7) n {(u,v) \ v £ si(u)}| > dh(\si(u)\), for some u £V. Putting S = {(u,v) | v £ si(u)}, we get dim(Js) > dh(| si(w)|). Therefore wH(Is) < h. Take any element z £ Is- There exists x £ I such that z = xs- Since u £ V, there exists u0 £ V such that (u0,u) £ E. Clearly, the Hamming weight of xs is equal to the Hamming weight of e ^ ^ s s = e(„0)U)X G I. Therefore WH{I) < WH{IS) < h. This contradiction shows that the first case is impossible. Case 2: | supp(J) n {(u,v) \ u £ so(v)}| > dh(so(v)), for some v £ V. Putting S = {(u,v) | u £ so(u)}, we get dim(is') > dh(\ so(w)|). Hence WJJ(IS) < h. For each z £ Is, there exists x £ I such that z — xs- Since v £ V, there exists vi £ V such that (y, vi) £ E. The Hamming weight of xs is equal to the Hamming weight of xse(VtVl) = xe^^^ G I. Hence WH{I) < WH(IS) < h. This contradiction shows that the second case is impossible, either. Case 3: |supp(J) n (so(£>) x si(J5))| > dh((so(D) x si(£>))). Then it follows that dim(J n J2vev -^e(so(t<),v)) > dh(\so(v)\); whence U>H(I) < e U>H(I (~l YlvPV F (so(v),v)) < h. This contradiction completes the proof. • A tournament G is a graph such that, for all distinct u,v £ G, either (u,v) £ E{G) or (v,u) £ E(G), but not both. The following proposition shows that for many graphs D the bound (9.2) is exact. Proposition 9.1 ([Kelarev and Sokratova (2001)]) Let D = (V,E) be a graph defining a structural matrix ring Mr>{F), and such that every connected component of the subgraph induced in D by the set {u£V\ is a tournament, in D by the set

si(u) ^ 0}

(9.4)

and every connected component of the subgraph induced {v£V\

so(v) + 0}

(9.5)

Codes as Ideals in Matrix

Rings

145

is a tournament. Then, for every 1 < h < uiid(D), the structural matrix ring MD{F) has an ideal with Hamming weight h and information rate given by (9.2). Proof. Let C be a connected component of the subgraph induced in D by the set (9.4). Every tournament satisfying condition (3.10) is acyclic, i.e., it has no directed cycles. It is well known that every acyclic graph can be topologically ordered. This means that we may reorder the vertices {u\,U2,... ,Uk} of the tournament C so that it has an edge (u,, Uj) if and only if i > j . Put Si = si(uj). It follows from condition (3.10) that S1CS2C---CSk. By induction we can define subspaces Li of £ „ e s i ( U i ) F^(uitv) s u c n that w(Li) = h, dim(Lj) = dh(£ve si{Ui) Fe(uuv)) and L1 C L2 C • • • C Lk. Straightforward verification shows that the union of these subspaces is an ideal of MD(F). Now, let C be a connected component of the subgraph induced in D by the set (9.5). Relabel the vertices {vi, v2,..., vk} of the tournament C in the opposite direction so that it has an edge (vi,Vj) if and only if i < j . This time we put St = SO(VJ). Again it follows that S1CS2C---CSk. By induction we can define subspaces L, of Eufsi(«)^ e (»i,«) such that w(Li) = h, dim(Li) = dh{Y,U€si(vi)Fe^i^)) and L x C L 2 C • • • C Lk. Then it is easily seen that the union of these subspaces is an ideal of MD{F). Denote by S the sum of these ideals obtained above for all connected components C of the subgraph of D induced by the set (9.4), together with the sum of ideals given above by all connected components of the subgraph induced in D by the set (9.5). Then the sum

S+

J2

Fe

™

tue(£D(so(D)xsi(D))

has the required information rate given in (9.2).

•

The following example shows that the exact values of information rates intricately depend on the structure of the graph D, and for some graphs may be less than the bound (9.2).

146

Examples of

Applications

Example 9.1 The graph D = (V, E) with the set V = {1,2,3,4,5,6,7} of vertices and adjacency matrix 0 0 0 0 1 0 1

0 0 0 0 1 1 1

0 0 0 0 1 1 1

0 0 0 0 0 1 1

0 0 0 0 0 0 1

0 0 0 0 0 0 1

0 0 0 0 0 0 1

satisfies condition (3.10), and so it defines a structural matrix ring MD{F). Let h = 3, and let F = GF(2). The largest linear subspace with Hamming weight 2 in Fe^^+Fe^^+Fe^z) is generated by e ^ + e ^ + e ^ ) , and so it has dimension equal to c^d si(6)|). It contains all vectors of Hamming weight 3. Similarly, the largest linear subspace with Hamming weight 3 in Fe (6 , 2 ) + ^e ( 6 ) 3 ) + Fe( 6)4 ) is generated by e(6,2) + e(6,3) + e (6>4 ), and so it has dimension equal to c^d si(6)|), too. However, if we consider the ideal generated by these spaces, it has smaller Hamming weight. Indeed, e

(7,5)( e (5,l) + e(5,2) + e (5,3)) + e(7,6) ( e (6,2) + e (6,3) + e (6,4)) =

e

(7,l) + e (7,4)-

It follows that all ideals of Mp (F) with Hamming weight 3 have information rates strictly less than the upper bound given by (9.2).

9.3

Color Lie Superalgebras

One of the deepest results permeating physics is the spin-statistics theorem (see [Streater and Wightman (1964)]), according to which the space-time properties (spin) of elementary particles are correlated with their quantum statistical description. The two classes of particle statistics (Bose-Einstein and Fermi-Dirac, respectively) can be accommodated naturally in a larger algebraic scheme incorporating the notion of grading to reflect various sign factors in defining relations. At the level of nonassociative algebras, the structure and representation theory of Z2-graded Lie superalgebras have been extensively studied as symmetry algebras of physical systems (for a few examples of applications we refer to [Avan and Talon (1991)], [Coquereaux et al. (1992)], [Eyre (1997)], [Gould et al. (1993)], [Gould et al. (1991)]). In recent years

Combinatorial

Applications

147

the study of two-dimensional systems has led to the realization that richer algebraic schemes such as the so-called quantum algebras may be relevant (the spin-statistics theorem is also weaker in the two-dimensional case). Let F be a field with characteristic ^ 2 , 3 , and let G be an additive abelian group. Consider a bilinear alternating form eG x G - t K*, i.e., a function satisfying e(g + h,k) =

e(g,k)e(h,k),

e(g,h + k) =

s(g,h)e(g,k), s(h,g)~1,

e(g,h) =

for g,h,k £ G. A G-graded (nonassociative) algebra L = @g£GLg is called a color Lie superalgebra if, for all a £ Lg, b £ Lh, c £ L, [a,b] =

-e(g,h)[b,a],

[[a,b],c] = [a,[b,c]]

-e(g,h)[b,[a,c]].

If G = 0, then we get an ordinary Lie algebra; if G = Z2, e(0,0) = £(0,1) = £(1,0) = 1, £(1,1) = —1, then L is a Lie superalgebra. A general way of obtaining color Lie superalgebras is to introduce a commutator on a G-graded algebra R = ©eeG-R9 by setting [x,y]

=xy-e(x,y)yx.

An algebra graded by the group Z2 = {0,1} is called a superalgebra. A Lie superalgebra is a superalgebra L = L0+Li such that, for all a, b £ {0,1}, x £ La, y £ Lf,, z £ L, two identities hold:

[x,y] = llx,y},Z) = 9.4

-(-iyb,

[x,[y,z}}-(-iyb[y,[x,z}}.

Combinatorial Applications

It makes sense to use ring constructions in order to generate various other combinatorial objects with versatile applications, including applications in cryptography (see [Fellows and Koblitz (1994)], [Koblitz (1998)]). Here

148

Examples of

Applications

we only mention that cocycles have been used to obtain new generalized Hadamard matrices, and twisted group rings have played essential roles in the investigations of properties of these matrices (see, for example, [Horadam (2000)]). A natural and well-known way of defining a graph on the multiplicative semigroup of a ring is to use Cayley graphs. Let G be a semigroup, and let S be a nonempty subset of G. The Cayley graph Cay(G, S) of G relative to S is defined as the graph with vertex set G and edge set E{S) consisting of those ordered pairs (x, y) such that sx = y for some s G S (see, for example, [Babai (1995)], [Biggs (1993)], [Kelarev (2001)], [Kelarev (2003)], [Kelarev and Praeger (2003)], [Margolis and Meakin (1989)]). It would be interesting to investigate ring constructions defining families of Cayley graphs with nice combinatorial properties.

9.5

Applications in Logic

The role of Boolean rings in classical logic is well known. These rings, in some cases equipped with additional operations, and also certain other classes of rings and their constructions are used in other versions of logic, including modal logic, multivalued logic, fuzzy logic and quantum logic (see, for example, [Barwise (1977)], [Bole and Borowik (1992)], [Crossle et al. (1972)], [Dorninger et al. (1997b)]), [Enderton (1972)], [Hodges (1993)], [Hodges (1997)], [Hughes and Cresswell (1996)], [Kelarev and Stokes (1999)], [Kelarev and Stokes (1999)], [Kelarev and Stokes (2000)], [Kelarev and Stokes (2001)], [Kosko (1993)], [McNeill (1993)], [Muller and Lenski (1987)], [Nguyen (2000)], [Shoenfield (1967)], [Smorynski (1985)]). This short section has been included for completeness and in view of the importance of methods of logic for telecommunications, software engineering and data processing.

Chapter 10

Open Problems

Let us begin with the following general problem. Problem 10.1 Investigate properties of error-correcting codes considered as ideals in finite skew and twisted group and semigroup rings. When the author was a student, the following problem was mentioned to him by E.I. Zelmanov. Problem 10.2 Let D be a division ring, R = Dn a matrix ring. Describe semigroups S and decompositions into direct sums of additive subgroups R — (BsesR* s u c h that R is S-graded. Problem 10.3 Is it true that if the Jacobson radical is homogeneous in every S'-graded ring, then S is embeddable in a group G? This problem is recorded in [Kelarev (1992f)] and is related to Theorem 4.10. The answer is positive for commutative semigroups. Moreover, if 5 is a commutative semigroup and the Jacobson radical is homogeneous in all S'-graded rings then S is embeddable in a group G such that the Jacobson radical is G-homogeneous (see [Kelarev (1992f)]). Problem 10.4 Let R be an algebra over a field of characteristic zero. Suppose that R is graded by a (i) nilpotent group, (ii) solvable group. Is it true that the Jacobson radical of R is homogeneous? 149

150

Open Problems

Problem 10.5 Describe all nilpotent and solvable groups G such that, for each G-graded ring R, a certain power of J(R) is contained in J6I(R). Is it true that this property holds if and only if the orders of all finite subgroups of G are bounded from above? Problem 10.6 Is is true that the Jacobson radical is homogeneous in every ring graded by (i) a torsion-free group, (ii) a u.p.-group, (iii) a linearly ordered group? Problem 10.6(ii) was included in [Jespers (1993a)]. It is well known that, for every linear group G and each field F, the radical J(F[G\) is locally nilpotent (see [Karpilovsky (1991), §49]). Problem 10.7 ([Kelarev (1993d)]) Is it true that, for each linear group G and every strongly G-graded ring R, the Jacobson radical J(R) is locally nilpotent, provided that J{Re) is nilpotent, where e is the identity of G? It follows from [Karpilovsky (1991)], Proposition 30.41 and Lemma 30.16, that if G is a finitely generated nilpotent group and R = ©s£G-Rg is a strongly G-graded ring with nilpotent J(Re), then J{R) is nilpotent. On the other hand, the famous Zalesskii's theorem says that the Jacobson radical of any group algebra of a solvable group over a field is locally nilpotent (see [Karpilovsky (1991)]). This motivates the following Problem 10.8 Is it true that, for each solvable group G and every strongly G-graded ring R, the Jacobson radical J{R) is locally nilpotent, provided that J(Re) is nilpotent, where e is the identity of G? Problem 10.9 Describe all semigroups S such that every 5-graded ring R is quasiregular provided that all subrings RQ are quasiregular for all maximal subgroups G of S. Problem 10.10 Investigate left or right self-injective semigroup-graded rings. Describe all semigroups S such that each strongly 5-graded ring R is left self-injective provided that all Re are left self-injective for all idempotents e in S. Problem 10.11 Investigate Goldie, Krull, and Gelfand-Kirillov dimensions of semigroup-graded rings. Describe all semigroups S such that each

151

strongly 5-graded ring R has a finite Goldie (Krull, Gelfand-Kirillov) dimension provided that all Re have a finite Goldie (Krull, Gelfand-Kirillov) dimensions for all idempotents e in 5. Problem 10.12 Describe all semigroups 5 such that every 5-graded ring R is Brown-McCoy radical provided that all subrings Re (resp., RQ) are Brown-McCoy radical for all idempotents e (maximal subgroups G) of 5. Problem 10.13 Describe all semigroups 5 such that the support of every right Noetherian 5-graded ring R is finite. Does there exist a right Noetherian ring with infinite support graded by an infinite descending idempotent chain? Problem 10.14 Investigate semisimple and semiprime semigroup-graded rings. Describe all semigroups 5 such that every semisimple 5-graded ring R is right Artinian or Noetherian provided that all subrings Re (resp., RG) are right Artinian or Noetherian for all idempotents e (maximal subgroups G) of 5. Problem 10.15 For strongly semigroup-graded rings, investigate the relations between properties of the whole ring R and properties of subrings Re and RQ for idempotents e and subgroups G of 5. Describe all semigroups 5 such that (i) every strongly 5-graded ring R is left or right Artinian or Noetherian provided that all the Re are left or right Artinian or Noetherian for all idempotents e of 5; (ii) every strongly 5-graded algebra R over a field of characteristic zero is semiprime, semisimple, left or right hereditary or semihereditary provided that all the Re satisfy the same property for all idempotents e of 5; (iii) every strongly 5-graded ring R has a left Artinian classical quotient ring provided that all the Re have left Artinian quotient rings for all idempotents e of 5. Problem 10.16 For semigroup-graded P7-rings, investigate the relations between properties of the whole ring R and properties of subrings Re and -RG for idempotents e and subgroups G of 5. Is it true that if 5 is an arbitrary semigroup, R is an 5-graded P7-ring and the Jacobson radicals J{Re) are nil for all idempotents e of 5, then J(R) is nil?

152

Open Problems

P r o b l e m 10.17 For right Noetherian semigroup-graded rings, investigate the relations between properties of the whole ring R and properties of subrings Re and RG for idempotents e and subgroups G of 5. Describe semigroups 5 such that each right Noetherian 5-graded ring R with finite support is right fully bounded provided that all subrings Re are right fully bounded for all idempotents e of 5. P r o b l e m 10.18 Describe all semigroups 5 such that, for each 5-graded ring R, the Jacobson radical of R is equal to the largest ideal among ideals I of R with the property that / G ( / ) C J{RG) for all maximal subgroups

GoiS. P r o b l e m 10.19 Let 5 be a cancellative periodic partial groupoid such that all subgroups of 5 are finite and, moreover, their orders have a finite least common multiple. Suppose that R is an 5-graded ring inducing 5. Is it true that, for each r G J{R), there exists n > 0 such that all homogeneous components of nr belong to J{R)1 It is known that the answer is positive for finite groupoids. P r o b l e m 10.20 Is it true that, if 5 is a finite cancellative partial groupoid with idempotent e and R = ®s^sRs is an 5-graded ring inducing 5, then

J(Re) =

RenJ(R)l

P r o b l e m 10.21 Is it true that, if 5 is a finite cancellative partial groupoid and R is an 5-graded ring inducing 5, then a certain power of the Jacobson radical J{R) is contained in the largest homogeneous quasiregular ideal

oiRi P r o b l e m 10.22 Is it true that, if 5 is a finite partial groupoid and R = <&s€SR3 is an 5-graded ring inducing 5 and such that the radicals of all rings among the Rs are nilpotent, then the Jacobson radical of R is nilpotent? P r o b l e m 10.23 Is it true that, for every groupoid 5 and every 5-graded ring R satisfying a polynomial identity, if the radicals of subrings Re are nil for all idempotents e in 5, then the radical of R is nil? P r o b l e m 10.24 Let R be an algebra over a field of characteristic zero graded by an (infinite) cancellative partial groupoid. Is it true that the Jacobson radical of R is homogeneous?

Appendix A

Glossary of Notation

Symbol o (BgeoRg

Uiei Ri Ann r (X) BM(R) Bn B(R) B(S) C char(i?) ct-K-dim(fl) V, H, J, C, Tl E(G) G° G1 \G\ GR(pm,r) •"IE)

**'Xl

H(R) id(T) idr(T)

^X

Concept The circle composition A G-graded ring or algebra Direct product The right annihilator of X The Brown-McCoy radical of R Brandt semigroup The Baer radical ideal of R Baer radical of the semigroup S The field of complex numbers The characteristic of R The classical Krull dimension of R Green's equivalences Standard matrix unit The set of all idempotents of G Groupoid with zero adjoined Groupoid with identity adjoined The order of G Galois ring The W-class, T^class, £-class containing x The set of homogeneous elements of R Ideal generated by T Right ideal generated by T

153

.

Page 12 17 25 15 13 9 12 6 11 12 15 5 9 2 1 1 2 12 5 18 11 11

154

Symbol idj(T) I(X, R) J{R) K-dim R (M) C(R) M(G; I, A; P) M°(G;I,A;Q) MD (F) M(R; I, A; P) Mn{R), M/(fl) My ' (R) Mj (R) M\rci){R)

MJttb)(R) Mfh)(R)) Mj (R)) [P, U, V, Q] Q R Rg R[G] i?o [S] R/I R* S R[x\, • . . , xn] R\S; a, 5} R(x; a, S) R[S; a] RT [S] R[[x; a, 5]] R((x, a, 5)) fl#Z[G*]

Glossary of

Notation

Concept Left ideal generated by T Incidence ring The Jacobson radical of R Krull dimension The Levitzki radical of R Rees matrix semigroup over a group Rees matrix semigroup over a group with zero Structural matrix ring Rees matrix ring (Full) matrix ring The ring of row finite matrices The ring of column finite matrices The ring of row and column finite matrices The ring of row bounded matrices The ring of column bounded matrices The ring of matrices with finite supports . . . Morita context The field of rational numbers The field of real numbers Homogeneous component Group ring or semigroup ring Contracted semigroup ring The quotient ring of R modulo J Crossed product Polynomial ring Skew polynomial ring Skew Laurent polynomial ring Skew semigroup ring Twisted semigroup ring The ring of skew power series The ring of skew Laurent series Smash product

Page 11 35 12 20 13 6 6 34 33 31 31 31 32 32 32 32 33 11 11 19 26 26 12 27 27 28 28 28 29 30 30 34

155

Symbol s" S/I spec(i?) supp(r) S*\, S%*, Six Sym(n) S = UfcgjgSb (T),T+ T* (T) U(R) Un(R), Ui(R) V(R) Z Z„

Concept All elements above s Rees quotient semigroup The set of prime ideals of R The support of r Subsets of a Rees matrix semigroup Symmetric group Band of semigroups Groupoid generated by T Groupoid with identity generated by T Subring generated by T is the set The group of units of R The ring of upper triangular matrices The adjoint group of R The ring of integers The cyclic group of order n

Page 2 5 15 19 7

....

3 9 2 2 11 73 31 73 11 3

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Index

cocycle, 29 cocycle equation, 29 code 2-D cyclic code, 138 abelian code, 134 bicyclic code, 138 Cauchy code, 138 cyclic code, 133 generalized quadratic residue code, 135 Golay code, 140 group-algebra code, 134 hyperbolic code, 139 Reed-Muller code, 137 Reed-Solomon code, 138 Cohen and Montgomery's Theorem, 22 Cohen and Rowen's Theorem, 22 color Lie superalgebra, 147 conjugate, 2 coset, 3 crossed product, 27

adjoint monoid, 12 algebra, 11 contracted monoid algebra, 26 contracted semigroup algebra, 26 group algebra, 26 incidence algebra, 36 monoid algebra, 26 path algebra, 31 semigroup algebra, 26 algebras Dirichlet algebras, 133 Baer radical, 12 band, 4 band of semigroups, 9 left regular band, 4 left zero band, 4 rectangular band, 4 right regular band, 4 right zero band, 4 Beattie and Jespers' Theorem, 22 bilinear alternating form, 147 Birkhoff's Theorem, 16

degree in-degree, 35 out-degree, 35 derivation, 28 dimension Goldie, 15 uniform, 15

Cayley graph, 148 center, 3 chain, 4 characteristic, 12 circle composition, 12, 73 clique, 35 201

202

Index

direct product, 25 direct sum, 25

groupoid, 1 cancellative, 18 partial groupoid, 18 periodic, 2

element idempotent, 2, 12 nilpotent, 6 periodic element, 2 strongly nilpotent, 13 elements orthogonal, 12 epigroup, 5

height of a semilattice, 4 homogeneous components, 19 elements, 19 homogeneous sum, 18 ideal, 2, 11, 12 left or right ideal, 2 principal ideal, 2, 11 right ideal, 11, 12 ideal extension, 12 idempotent, 2, 12 orthogonal idempotents, 12 identity element, 1 incidence algebra, 36 incidence ring, 36 index, 3 irreducible polynomial, 11

factor principal factor, 5 factor of a semigroup, 5 field, 11 finitely generated, 2 generating set, 2 Golod ring, 13 graded ring X>-faithful, 21 faithful, 21 graph acyclic, 35 Green's equivalences, 5 group, 2 p-group, 3 abelian group, 2 abelian-by-finite, 4 adjoint group, 73 cyclic, 3 finite-by-abelian-by-finite, group of units, 73 nilpotent, 4 poly cyclic, 4 solvable, 4 symmetric group, 3 torsion, 3 group algebra, 26 group of units, 12 group ring, 26 skew group ring, 29 twisted group ring, 29

Jacobson radical, 12 Jennings basis, 137 Jespers and Okninski's Theorem, 82 Koethe problem, 13

4

Lagrange's Theorem, 3 length, 19 Lie superalgebra, 147 loop, 1 m.u.-semilattice, 102 Maschke's Theorem, 26 matrix ring, 31 generalized matrix ring, 32 generalized upper (lower) triangular matrix ring, 32 Rees matrix ring, 33 row and column finite, 32 row bounded, 32

203

Index row finite, 32 structural matrix ring, 35 module irreducible, 14 monoid, 1 free commutative monoid, 27 monoid algebra, 26 monoid ring, 26 skew monoid ring, 28 twisted monoid ring, 29 monomial ring, 26 Morita context, 33 Munn's Theorem, 45, 64 non-degenerate grading, 19 normal series, 4 normal subgroup, 3 order, 2 Ore extension, 28 p-element, 3 partial groupoid, 18 Passman's Theorem, 22 path, 31 path algebra, 31 Pi-algebra, 15 Pi-ring, 15 polynomial ring, 27 skew Laurent polynomial ring, 28 skew polynomial ring, 28 power series skew Laurent series, 30 skew Laurent series ring of derivative type, 30 skew power series, 30 skew power series ring of automorphism type, 30 skew power series ring of derivative type, 30 preorder, 36 prevariety, 110 primary component, 3 principal factor, 5

problem Koethe problem, 13 product crossed product, 27 direct product, 25 semidirect product, 25 smash product, 34 subdirect product, 25 projection, 19 quasigroup, 1 quotient group, 3 radical, 66 j4-radical, 67 Baer radical, 12 Brown-McCoy radical, 13 homogeneous radical, 19 invariant, 67 Jacobson radical, 12 left strong, 67 Levitzki radical, 13 normal, 67 prime radical, 12 principally left hereditary, 67 relation balanced, 36 representation generalized triangular matrix representation, 33 of a group, 139 ring G-graded, 17 contracted graded ring, 19 contracted semigroup ring, 26 division ring, 11 Galois ring, 12 generalized matrix ring, 32 generalized upper triangular matrix ring, 32 group ring, 26 groupoid-graded, 17 incidence ring, 36 locally nilpotent, 13

204

matrix ring, 31 monoid ring, 26 monomial ring, 26 Morita context, 33 nil, 12 nilpotent, 12 polynomial ring, 27 prime, 14 primitive, 14 principal ideal ring, 11 quasiregular, 12 quotient ring, 12 Rees matrix ring, 33 right T-nilpotent, 13 right Artinian, 13 right Goldie, 15 right Noetherian, 13 right perfect, 13 semigroup ring, 26 semigroup-graded, 17 semilocal, 13 semiprimary, 13 semiprime, 13, 14 semiprimitive, 13 semisimple, 45 semisimple Artinian, 13 skew group ring, 28, 29 skew Laurent polynomial ring, 28 skew Laurent series ring, 30 skew Laurent series ring of automorphism type, 30 skew Laurent series ring of derivative type, 30 skew monoid ring, 28 skew polynomial ring, 28 skew polynomial ring of derivative type, 28 skew power series ring, 30 skew power series ring of derivative type, 30 skew semigroup ring, 28 skew semigroup ring of automorphism type, 29 strongly graded, 19

Index structural matrix ring, 35 twisted group ring, 29 twisted monoid ring, 29 twisted semigroup ring, 29 upper triangular matrix ring, 31 root set, 135 sandwich matrix, 6 semidirect product, 25 semigroup, 1, 6 (cocycle) extension semigroup, 29 aperiodic, 6 Baer radical, 6 Brandt semigroup, 9 cancellative, 4, 7 combinatorial, 6 completely regular, 6 completely simple, 6 epigroup, 5 inverse, 6 left simple, 4 linear semigroup, 9 nil, 6 nilpotent, 6 periodic semigroup, 2 Rees matrix semigroup, 6 Rees quotient, 5 right (left) group, 7 right Ore, 10 t.u.p.-semigroup, 10 torsion-free, 6 u.p.-semigroup, 10 with zero multiplication, 6 semigroup ring, 26 contracted semigroup ring, 26 skew semigroup ring, 28 twisted semigroup ring, 29 semilattice, 4 convex subset, 105 multidiamond semilattice, 105 semilattice of semigroups, 9 skew derivation, 28 skew Laurent series, 30 source, 35

Index subalgebra, 12 subdirect product, 25 subgroup, 2 subgroupoid, 2 subring, 11 subsemigroup, 2 sum direct sum, 25 superalgebra, 23, 147 homogeneous, 129 Lie superalgebra, 23 support, 19 Sylow's Theorem, 3 union 0-direct union, 10 unit standard matrix unit, 9 unity, 1 variety, 16, 109 product, 109 semisimple, 109 Wedderburn's theorem, 20 zero, 1