Simple Noetherian Rings (Cambridge Tracts in Mathematics)

CAMBRIDGE TRACTS IN MATHEMATICS GENERAL EDITORS H. BASS, J. F. C. KINGMAN, F. SMITHIES J. A. TODD & C. T. C. WALL 69...

Author: John Cozzens | Carl Faith

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CAMBRIDGE TRACTS IN MATHEMATICS GENERAL EDITORS

H. BASS, J. F. C. KINGMAN, F. SMITHIES J. A. TODD & C. T. C. WALL

69. Simple Noetherian rings

JOHN COZZENS Department of Mathematica Rider College

AND

CARL FAITH Department of Mathematics Rutgers University

Simple Noetherian rings

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON • NEW YORK • MELBOURNE

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521207348 ©Cambridge University Press 1975 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1975 This digitally printed version 2008

A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Cozzens, John, 1942Simple noetherian rings. (Cambridge tracts in mathematics; 69) Bibliography: p. 129 Includes index. 1. Noetherian rings. I. Faith, Carl Clifton, 1927- joint author. II. Title. III. Series. 512'.2 75-11535 QA251.5.C69 ISBN 978-0-521-20734-8 hardback ISBN 978-0-521-09299-9 paperback

This book is for Midge and Mickey; Barbara and Kathy; Heidi and Cindy

Contents

Preface List of symbols

page ix x1

Conventions

xiii

Introduction

1

1

8

2

The correspondence theorem for projective modules Structure of Noetherian simple rings 3 Noetherian simple domains 4 Orders in simple Artin rings

23

5

~~~

42 64 ~

6 PCI-rings 7 Open problems

98 113

Afterword

118

Appendix

121

Notes

125

References

129

Index

133

Preface

This book is intended for the general reader of mathematics, and the authors have eliminated unnecessary mathematical machinery whenever possible, or postponed using it as long as possible. This was done, however, only after the authors saw that it could be done without any loss of clarity, or generality, in the statements of theorems. The book is as elementary and self-contained as practicable; and the little background required in homological and categorical algebra is listed in two short appendices. Otherwise, full definitions are given, and short, elementary full proofs are supplied for such tool theorems as the Morita theorem 1.12 (p. 15), the correspondence theorem 1.13 (p. 17), the Wedderburn-Artin theorem 2.23 (p. 32), the Goldie-LesieurCroisot theorem 4.4 (p. 65), and many others. The authors are indebted to Professor Hyman Bass, one of the general editors of this series, for suggesting that such a study of simple Noetherian rings would be of interest. We are immensely grateful to Professor Bass and to the Cambridge University Press for making it possible for this study to appear in the Cambridge Tracts in Mathematics. J.H.C.

C.F.

List of symbolst

iff

v 3 E

¢ c

s;; ~

<=>

0

N 7L. Q

u (")

+ 0

X

--+

f:A--+B 1-+ 4

X

n u t

Name

Example

double implication (or equivalence of propositions) universal quantifier existential quantifier membership nonmembership proper containment containment implication double implication equals empty set natural num hers integers rational num hers union intersection plus minus composition Cartesian product mapping mapping mapping embedding cross product coproduct

A is true iff B is true VaEA 3aEA

aEA a¢A AcB As;;B A~B

A<=>B A=B l, 2, ... 0, ± l, ± 2, ... afb, a, b E ?L., b =1= 0

AuB AnB a+ b a-b jog

AxB A--+B a~-+b

A4B XIAI DtAI

UtAt

Much of the material in this table is taken from Grundlehren volume 190 and is reproduced by permission of Springer-Verlag, Heidelberg_ [ xi ]

Simple Noetherian rings

Xll

equivalence or isomorphism (in the appropriate category) equivalence (as functors) nat ,..,

,..,

natural isomorphism (or equivalence) greater than less than greater than or equal to less than or equal to functor wedge perpendicular ("perp") summation

A::::::: B, mod-A : : : : mod-B

a> b a
a !5: b

ArvoB AVB AJ. and J.A L At tel

direct sum

A(f)B

L

tel

®At

tensor product A®B end of proof (occasionally, no proof) similarity (Morita A-B equivalence) Q ,.., A~B equivalence (as orders) ,..,I A..!..B left equivalence (as orders) ..!.. right equivalence (as orders) A ..!.. B mod-A = the category of (unital) right A-modules A-mod = the category of left A-modules Ab = the category of abelian groups (mod-Z:) L.H.S. (R.H.S.) = left (right) hand side {x I x has property P} = the set of all x with property P {xt I i E I} = a set of elements indexed by a set I

®

D

Conventions

The following conventions are the standard ones adopted by most modern texts on homological algebra or ring theory. Their inclusion is primarily for the reader's convenience. For the most part, by a ring A, we mean an associative ring with unit, usually denoted l. We let ring-1 denote a ring in which an identity element is not assumed. Thus, a ring-1 may have l. All ring homomorphisms will be assumed to preserve the respective units. We let mod-A (A-mod) denote the category of all unital right A-modules (left A-modules), the morphisms being right (left) A-linear homomorphisms. The fact that M is a right (left) A-module will be denoted variously by ME mod-A (ME A-mod) and MA (AM). f: M A-+ N A (f: AM -+ AN) will indicate that f is a morphism (A-linear homomorphism) between right A-modules (left A-modules) M and N. Iff: M A-+ N A (f: AM-+ AN), we shall write f on the left (right) of its argument. Thus f(m) ((m)f) will denote the image off on mE M A (mE AM). If M, NEmod-A and homA(MA, NA) ={!If: MA-+NA}, we set End MA = homA(MA, MA). Similarly, End AM= homA(AM, AM). Of course, End MA (End AM) is called the endomprphism ring of MA (AM). Whenever ME mod-A and ME B-mod and b(ma) = (bm)a, Vm EM, a E A, bE B, we call M a (B, A)-bimodule and denote this variously by ME B-mod-A or 8 MA" The morphisms in the category B-mod-A are homomorphisms f: M-+ N which satisfy f(bma) = bf(m)a

Vm EM, a E A, bE B.

Let ME mod-A and B =End MA" Since we have agreed to write homomorphisms on the left of their arguments, M is naturally a left B-module, in fact a (B, A)-bimodule where [ xiii ]

Simple Noetherian rings

xiv

b·m = b(m), VbEB, mEM. Similarly, if MEA-mod and B = End AM, ME A-mod-B. If ME B-mod and A =End 8 M then B =End MA is called the biendomorphism ring (bicommutator) of M and is frequently denoted by B = Biend 8 M. Clearly, B maps homomorphically into B (as rings) via left multiplication. This map is obviously injective whenever 8 M is faithful. If the map is a surjection 8 M is called balanced. In this case, center A : : : : center B via the canonical map of B into Biend 8 M. A basis is another term for a generating set of a module. However, free basis means that a basis {xt I i E I} is linearly independent in the sense that any (finite) linear combination .L xtrt is zero iff r1 = 0 for every i. (Finite here means that rt = 0 tel

for all but a finite number of i E I.) In this case, the sum .L xtR of the submodules generated by the set {xt I i E I} is tel

direct, notationally,

.L E8 xtR·

iel

As indicated in the list of symbols, M = A E8 B denotes a direct sum of modules A and B, and then A (also B) is said to be a direct summand of M, or more briefly, a summand of M. (Thus, every mE M can be expressed in one and only one way, m = a+ b, as a sum of elements a E A and bE B.) In order to save space, in defining 'right-handed' concepts, we shall assume the left-handed concept defined by right-left symmetry. (Similarly for left-handed concepts.) Example: right Noetherian. A mapping f: X -+ Y of sets X and Y is said to be injective, or an injection provided that f(x) = f(y) => x = y, Vx, y EX. It can be shown that a mapping f: X -+ Y is injective iff f is left cancelable in the category of set morphisms in the sense that for any two mappings g : U -+ X and k : U -+ X, then fg = fk iff g = k. Dually, a mapping f: X-+ Y is surjective, or a surjection, iff Y = im f. This happens iff f is right cancelable. The terms map and mapping are synonyms for morphism in the specified category. Thus, a homomorphism of groups is also called a map. We use monomorphism, or the abbreviation, mono, in the sense that the morphism is left cancelable in the

Conventions

XV

category; dually for epimorphism, or epi. Thus, an injection is mono in the category SETS of sets, a surjection is epi, and conversely. Similarly, in Ab (the abbreviation for mod-Z:, the category of abelian groups), a morphism f: X-+ Y is mono iff kerf= 0 iff f is injective, and dually, f is epi iff cok f = 0 iff f is surjective. Here, of course, kerf denotes the kernel off, and cokf denotes the cokernel (which is isomorphic to Yfimf). The term 'dually' referred to above is meant in the sense that one concept is obtained from the other by reversing arrows and directions in the definitions. For a more explicit statement of duality, consult Faith [73a, p. 60]. A sequence X -+ Y -+ Z of modules is exact if the image of X-+ Y is the kernel of Y-+ Z. This terminology also applies to long sequences of modules and maps, namely ... -+ X 1 -+ Xt+ 1 -+ Xt+ 2 -+ ... -+

Xn-+ Xn+l -+ ...

is exact, provided that each XI-+ xl+l-+ xl+2 is exact for every i, with possibly i ranging over Z:. An exact sequence

0-+X-+ Y-+Z-+0 is said to be short exact. Thus, 0 -+ X -+ Y is exact iff X -+ Y is mono, and X-+ Y-+ 0 is exact iff X-+ Y is epi. Thus there is always a canonical exact sequence

0-+X-+ Y-+ YfX-+0 for any submodule X of Y. Other requirements from homological algebra will be administered ad hoc, or, in a few cases, found in the Appendix, §l and §2. The term natural isomorphism is an old one which has been subsumed under the concept of natural equivalence of functors defined in the Appendix, §2, but often we use it in its old informal sense to mean that the mapping indicated by ~t is the canonical one, and is an isomorphism. A module M of mod-A is Noetherian if it satisfies the equivalent conditions: (N 1 ) a.c.c.: Any ascending sequence

M 1 s M 2 s ... s Mn s ...

xvi

Simple Noetherian rings

of submodules is ultimately stationary, that is, there exists an integer N > 0 such that MN = MN+k Vk :<':: l. (N 2 ) M has the maximum condition for submodules, that is, any nonempty setS of submodules contains a maximal element M(S). (Thus, M(S) ¢ M' for any M' ES.) (N 3 ) Every sub module of M has a finite basis, that is, is finitely generated. Dual to the concept of Noetherian is the concept of Artinian, namely, a right A-module M is Artinian provided that M satisfies the equivalent conditions: (A 1 ) d.c.c. Any descending sequence

of submodules is ultimately stationary. (A 2 ) M satisfies the minimum condition for submodules. The ring itself is right Noetherian (Artinian) provided that A is a Noetherian (Artinian) module in mod-A. This happens iff every finitely generated module in mod-A is Noetherian (Artinian), a fact which shows that this is a Morita invariant property (in the sense defined on p. 15.) Furthermore, a theorem of Hopkins and Levitzki states that any right Artinian ring A is right Noetherian. (Since this result for simple rings is an immediate consequence of the W edderburn--Artin theorem 2.23, p. 32, we omit the proof.) The Hilbert Basis Theorem states that if A is a right Noetherian ring, then so is the polynomial ring A[X] over A. This readily generalizes to finitely many variables. (See, for example, Faith [73a], p. 341.) A module N is simple if N has no nontrivial submodules. ForME mod-A, the socle of M, soc M, is the sum of all simple submodules of M. If soc M = M, we call M semisimple. 'Dually', a submodule N of M is a maximal submodule of M or simply maximal, if MjN is simple, and the radical of M, rad M, is the intersection of all maximal submodules of M. An ideal P of A is called prime (semiprime) if for all ideals I and J of A, IJ s;; P => I s;; P or J s;; P (J2 s;; P => I s;; P). Here and hereafter, by an ideal of A, we mean a two-sided ideal

Conventions

XVll

of the ring A. A ring A is prime (semiprime) if 0 is a prime (semiprime) ideal of A. A ring A is simple provided that there are no ideals except the trivial ones, A and 0. Clearly, every simple ring is prime (0 is a maximal ideal!). Simplicity of a ring A may be characterized by the property that every right module M =1= 0 is faithful, a Morita invariant property which shows that any full n x n matrix ring An over a simple ring A is simple. The term domain is an abbreviation for an integral domain, that is, a ring without zero divisors =1= 0. Finally, a field, defined as a ring in which every element x =1= 0 has a (two-sided) inverse x- 1 , may be characterized by the property that the only right ideals are the trivial ones. Thus, a field is simple.

Introduction

In this book we have collected the known theorems on the structure of simple right Noetherian rings, and more generally for simple rings containing a uniform right ideal U. By the Goldie [58, 60]t and Lesieur-Croisot [59] theorems, any prime right Noetherian ring R has a simple Artinian ring Q(R) of right quotients. By the Artin [27]-Wedderburn [08] theorem, Q(R) is isomorphic to a full n x n matrix ring Dn over a (not necessarily commutative) field D. Moreover, the endomorphism ring B of any uniform right ideal U of R is a right Ore [31] domain with right quotient field Q(B) : : : : D. The Goldie and Lesieur-Croisot theorem is historically the first representation of a right Noetherian non-Artinian prime ring in a matrix ring of finite rank with entries in a field. (The Chevalley-Jacobson density theorem for a primitive ring R yields a representation by infinite matrices over a field unless R is Artinian.) Significant simplifications in, and additions to, the Goldie and Lesieur-Croisot theory occur if R is assumed to be a simple ring. Henceforth R will denote a simple ring with identity element, and uniform right ideal U as described. We now outline the structure theory for R, which includes that for a simple right Noetherian ring. 1

Endomorphism ring theorem 2.17t

Every simple ring R with uniform right ideal U is the endomorphism ring, R : : : : End 8 U canonically, of a torsion free t

t

Goldie [58] refers to a. work by A. W. Goldie published in 1958; co-authors are referred to in the text in hyphenated form, in conformity with the widespread practice in mathematics. The notation 2.17 refers to a. theorem in Chapter 2 numbered 17. In general, for integers a. and b, the notation a..b will denote a. statement numbered bin Chapter a.. [ l ]

Simple Noetherian rings

2

module ( = U) of finite rank over a right Ore domain B (Faith [64]). 2

Projective module theorem 2.17

U is a finitely generated projective left B-module, where B = End U R (Hart [67]).

3

Uniqueness of orders theorem 2.22

Assume that R::::; End 8 .U' for a finitely generated projective module U' over a right Ore domain B'. Then there is an embedding B'-+ D such that B' and Bare equivalent orders of D. (Equivalence of orders is defined in the sense of Jacobson [43] in Chapter 2 (p. 26).) In the next theorem, trace 8 • U' = .L (U')f, that is, /eHomB•(U'.B')

the B'-submodule generated by all elements (u)f, with u f E Hom 8 .(U', B'). 4

E

U',

Least ideal theorem 2.6

In any representation R::::; End 8 .U' (as in 3), T' =trace 8 .U' is a (nonzero) least ideal of B'. Thus, T' = B' iff B' is a simple ring. 5

Idempotent right ideal theorem 2.21

Moreover, R is right Noetherian iff B' (as in 4) satisfies the ascending chain condition (a.c.c.) on idempotent right ideals contained in T'. In this case B satisfies the a.c.c. on principal right ideals. 6

Single ideal theorem 2. 7

There is a module U' and domain B' (as in 3) such that R ::::; End 8 • U' and such that B' contains at most one nontrivial ideal T'.

Introduction 7

3

Same centers theorem 2. 7

The center C of B is a field canonically isomorphic to the center of R, and U' and B' (in 6) can be chosen such that U' = U and B' = C + T, where T =trace 8 U. In other words, U is finitely generated projective over B' = C + T, and R : : : : End 8 .U canonically. Moreover, center B' = C and trace 8 .U = T.

8

Hereditary ring theorem 2.25

Every representation of R as an endomorphism ring R::::::: End B'U' with B' = End u~ (as in 3) has the property that B' is a simple ring iff R is right hereditary. This shows that the single ideal theorem is of some use: the representation of R over a domain with a single nontrivial ideal does occur (as in 6) when R is a nonhereditary simple ring. The results 3 to 8 are proved by Faith [72a] and are consequences of the correspondence theorem for projective modules taken up in Chapter l. Any simple Noetherian ring R is a maximal order in Q(R), in a sense defined in Chapter 2 (p. 27), and the question arises when is B = End U R a maximal order in D. When R is a twosided order we have:

9

Maxi·rnal order theorem 4.14

The domain B = End U R is a maximal order in D iff U is a reflexive right R-module. This happens, for example, when U is a maximal uniform (=basic) right ideal. By a theorem of Goldie, any uniform right ideal Y of R is contained in a basic right ideal. This, together with the previous theorems shows that R can be represented as the endomorphism ring of a finitely generated projective over a maximal order in a field.

Simple Noetherian rings

4 10

Reflexive ideal theorem 4.20

If R is two-sided Noetherian with right global dimension R :::;; 2, then every uniform right ideal of R is reflexive iff R is right hereditary. Theorems 9 and 10 are proved by Cozzens [75], and taken together, Theorems 1 to 10 reduce many structural questions on a simple right Noetherian ring R to corresponding questions on the structure of a maximal right order B (of a field D) having at most one nontrivial ideal. For example, the question of when U can be chosen such that B = End U R is itself a simple ring can be formulated entirely within D, namely

ll

Simple endomorphism rings theorem 2.20

Assume that R has the representation R = End 8 , U' as in 3. Then, there exists a simple right Ore domain B 0 and a finitely generated projective left B 0 -module U 0 such that R : : : : End 80 U 0 iff there exists a right ideal I of B' such that

(I: I) = {dE D I di s;; I} is a simple subring of D. (Note that since End I 8 , ::::::: (I: I), this is equivalent to requiring that some right ideal of B' has simple endomorphism ring.) (Faith [72a].) The question of the existence of such a representation of R (as in ll) is one of the main structural problems on a simple right Noetherian ring R, and in this case (and only then) R is similar ( = Morita equivalent) to a right Ore domain A, that is, then (and only then) there is a category equivalence mod-R : : : : mod-A, for some domain A. A positive result in this direction is the following:

12

Global dimension 2 theorem 2.40

A simple Noetherian ring R of right global dimension :::;; 2 is similar to a (simple) domain A. (Faith [72a], Hart-Robson [70], Michler [69]). The proof requires ideas behind Bass's global dimension 2 theorem (4.16).

Introduction

5

This holds if R is a principal right ideal ring, since then r.gl.dim R ~ 1, but in this case much more is true: 13

Principal right ideal theorem 4. 7

Any prime principal right ideal ring R is isomorphic to a full n x n matrix ring An over a right Noetherian hereditary

domain A. (Goldie [62].) Curiously, A need not be a principal right ideal domain for this to happen (Swan [62], see also 3.6, p. 49). (When R is simple then of course A is.) 14

Ore domains and the Faithr-Utumi theorem 4.6

A refinement of the Goldie-Lesieur-Croisot theorem is the theorem ofFaith-Utumi (proved in Chapter 4) which states that if R is any right order in Dn, then R contains a right order F n• where F is a right order in D. A change of matrix units in Dn may be necessary for this (unless R is also a left order) and, moreover, F does not in general contain an identity element. [If it did, then R would contain the full set of matrix units of Dn, and hence R would itself be a full n x n matrix ring Hn, where H = R (") D. This is, in general, not the case.] One can show that F can be chosen to contain no nontrivial ideals when R is simple. 15

Classification of simple domains

Inasmuch as Ore domains play a role in the structure of simple Noetherian rings analogous to the role of fields in the structure of simple Artinian rings, their classification, especially the simple ones, is fundamental to the structure theory of simple Noetherian rings. The rest of the introduction is devoted to this. Restricting our attention for the moment to simple integral domains, we can distinguish three main types: (D 1 ) For any simple domain (or ring) R of characteristic 0, and finitely many commuting outer derivations 81 , . . • , 8n, the ring ~R of differential polynomials is a simple

Simple Noetherian rings

6

(D2 )

(D 3 )

right Noetherian Ore domain. (Ore [33], Littlewood [33], Amitsur [57].) We call these Amitsur-Littlewood (simple Ore) domains. For any field F, the localization RM at a maximal ideal M, of the skew polynomial ring R = F[x] with respect to a p-derivation ()is a simple principal left ideal (pli-) domain, not a field. (Jacobson [43].) We call these domains simple localizations. For any field F, if F[x] is a primitive ring, e.g., ifF is a transcendental algebra over the center C, then A = F Q9 cC(x) is a simple pli and pri (principal right ideal)domain.

D 3 is an observation derivable from Jacobson's theorem [64a] characterizing when F[x] is a primitive ring. (See Theorem 3.20 and 3.21, pp. 61-2.) For this reason, we entitle this class Jacobson domains. (Actually, D 3 is a subclass of D 2 .) 16

V-domains

In this the concluding section of the Introduction, we summarize some results on a special class of simple Ore domains for which there is a rather satisfactory structure theory paradoxically resembling (commutative) Dedekind domains. The question of the existence of integral domains, not fields, having the property that every simple right module is injective was first raised by Faith [67a, Problem 17, p. 130]. The history of this problem is as follows: Kaplansky showed that commutative regular rings could be characterized by the injectivity of the simple modules. We use the terminology right K-ring (after Kaplansky) for a ring with every simple right module injective. Villamayor characterized a right K-ring R by the property that every right ideal is the intersection of maximal right ideals, or equivalently, every right module M has the property that the intersection of its maximal submodules, namely rad M, is equal to 0. Such rings are called right V-rings, and we refer interchangeably to right K-rings and V-rings. The first examples of right V -domains, not fields, given by Cozzens [70], were differential polynomial rings K[x, d] with

Introduction

7

respect to a derivation d over a (Kolchin) universal differential field K. Analogous examples of these right V-domains were localizations of twisted polynomial rings K[x, a] with respect to an automorphism a of a field K over which certain designated algebraic equations have solutions. These are taken up in Chapter 5. More generally, simple Noetherian V-domains, not fields, were shown to exist in all homological dimensions by CozzensJohnson [72]. Osofsky has shown the existence of V-domains having infinitely many nonisomorphic simple modules (see 6.20). However, the problem of exhibiting examples with just finitely many nonisomorphic simple modules remains open. All of the above examples which are pri-domains also have the property that each cyclic module, not isomorphic to the base ring, is injective. Such rings are called PCI-rings (proper cyclics injective). Chapter 6 is devoted to the study of these rings: they are either semisimple Artinian, or else semihereditary simple right Ore domains in which any finitely generated right ideal I can be generated by two elements, one of which is an arbitrary nonzero element of I. Other resemblances to Dedekind domains are noted.

I. The correspondence theorem for projective modules

In this chapter, we prove the correspondence theorem for projective modules, referred to in the Introduction, on which most of the known structure theory for simple Noetherian rings depends. If M, N E mod-A and B = End MA, then the trace of N in M is the A -submodule of M generated by all elements {J(x) I x E N,fE homA(N, M)}. This module is denoted trace MN and is a (B, A)-bimodule. In particular, trace AM is an ideal of A for any module M. l.l PROPOSITION AND DEFINITION An object U of mod-A is a generator if the equivalent conditions hold: (l) The group-valued functor homA(U, -):mod-A"-" Ab (the category of abelian groups) is faithful. Thus, given any nonzero k : X A -+ Y A there exists a homomorphism h : U A -+ X A such that

kh =I= 0.

(2) trace MU = M for all ME mod-A. (3) trace AU = A. (4) un ::::; A E8 X for some integer n > 0 and X E mod-A. (5) un-+ A-+ 0 is exact for some n > 0. Proof For any index set/, U(l) will always denote the direct sum (coproduct) of I copies of U. (l) => (2): Define

M, ( ••• ,u1 , ... )~---+f(u) VfEhomA(U,M). U
TT{

NotethatimTT = traceMU.IfimTT =1= M,setC = MfimTTand let v : M -+ C be the canonical projection. By assumption, there exists an f E hom A( U, M) such that vf =1= 0. However, im f s;; im TT, contradicting vf =1= 0. Thus, im TT = M. (2) => (1): Let g: M-+ N be nonzero. If gf = 0 for all f E homA(U, M), then clearly gTT = 0 implying that g = 0 by surjectivity of TT. Thus, gf =I= 0 for some f E homA(U, M). [ 8 ]

Correspondence theorem for projective modules

9

(2) => (3) is clear; the equivalence of (3), (4) and (5) is immediate by 1.2 which follows. (3) => (1): Since A is free, there exists an epi A (I>--+ M for some set I and hence, an epi um--+ M. By the same arguments used in the proof (2) => (1), (1) is clearly satisfied. D 1.2 PROPOSITION AND DEFINITION An object UE B-mod is projective if the following equivalent conditions hold: (1) The group valued functor hom 8 (U, -): B-mod "-" Ab is exact. (2) (Dual basis lemma) There exist sets {x1 I i E I} of elements of U and {!; I i E I} of elements of hom 8 ( U, B) such that for each x E U, (x)f1 = 0 for almost all i E I and

x =

,2 (x)f1x1

Vx

E

U.

tel

(3) U ·is isomorphic to a (direct) summand of a free B-module. (4) Every exact sequence M--+ U--+ 0 in B-mod splits. Proof The equivalence of (1), (3) and (4) is elementary, and

will be left for the reader. We shall contend with the equivalence of (2) and (3). (4) => (2): Let {x1 I i E I} be any generating set for 8 U, {e1 I i E I} a basis for Bm andf: Bm--+ U the B-map defined by e1 1-+ x 1• Since f splits there exists a map g : U --+ Bm such that gf = 1u. For any x E U,

where b; = 0 for almost every i E I. Define a family {!1 I i where eachf1 : 8 U--+ 8 B, by x ~--+ b1 for all x E U. Clearly,

E

I},

X= _2b;X; =

,2 (x)f1x1

VxE U.

(2) => (3): Let f: Bm--+ U be defined as before and define g : U --+ Bm by

(u)g =

,2 (u)f1e1

VuE U.

tel

Clearly, g is a B-map and gf = 1u.

D

Simple Noetherian rings

10

Remarks (a) The proof shows card I can be chosen equal to the cardinal of any basis of U, and, moreover, the equation for x in (2) shows that {x1 I i E I} is a basis of U. (b) The set {!1 I i E I} is a basis of the 'dual' module Hom 8 (U, B) if 8 U is finitely generated. (c) If A is any simple ring, then any right ideal U =1= 0 is a generator of mod-A, for T = trace AU is an ideal containing AU, hence T = A. 1.3 PROPOSITION

If U is projective in B-mod, then

T = trace

Bu

n

=

K.

KSB

and KU=U

That is, Tis the intersection of all right ideals K of B such that KU = U. In particular, U = TU. Moreover, T 2 = T. Proof By (2) of 1.2, U = TU since each (u)f1 E T. Thus, the L.H.S. 2 R.H.S. Conversely, if U = IU, for some (right) ideal of B, then T = trace U = trace(IU) = IT s;; I. Thus, R.H.S. 2 L.H.S. The fact that T 2 = T follows immediately from the dual basis lemma. D

Let U be projective

1.4 CuT-DowN PROPOSITION Bm~U-+0

exact in B-mod, and let T = trace 8 U. If B 0 is any subring of B containing T, then U is projective in B 0 -mod and p induces an exact sequence B 0 m-+ U-+ 0. Furthermore, the inclusion map End 8 U -+ End morphism. If U is faithful over B, then center B 0 = B 0

(")

80

U is an iso-

center B.

Proof Let T s;; B 0
Correspondence theorem for projective modules

11

B 0 m and therefore, B 0 m--+ U--+ 0 is split exact. Thus, U is projective in B 0 -mod. Now let t E T, u E U, bE B and let fE End 80 U. Since bt E T £ B 0 , (btu)f = bt(u)f = b(tu)f.

Since U = TU is generated by the set {tu It E T, u E U}, it follows that f E End 8 U. This proves that End 8 U = End 80 U. Finally, if x E center B 0 , bE Bandt E T, then (xb)t = x(bt) = (bt)x = b(tx) = b(xt) = (bx)t.

Hence, (xb - bx)T = 0 which implies that (xb - bx)U = 0 since TU = U. Since U is faithful, xb = bx. Thus, x E center B.

D 1.5

PROPOSITION

(Schanuel's Lemma)

Let

be two r~olutions forMA with P 1 projective in mod-A, i = l, 2. Then K 1 E8 P 2 : : : : K2 E8 P 1. Proof Let P be the fiber product of ?T1 and ?T 2, i.e., P = {(pl, P2) E P 1 E8 P 2 I1T1(P1) = ?T 2(p 2)}. It is an easy matter to verify that the following diagram is commutative with exact rows and columns:

0

0

! ! ! 1 !0

! ! ! !0

K2=K2 o~K 1 ~P

II

~P 2 ~0 n2

o~K 1 ~P ~M~o

Simple Noetherian rings

12

1.6 LEMMA (l) LetUEB-modandA = End 8 U.Themap 4>: hom 8 (U, B)--+ homA(U, A) defined by

[v](4>(f)(u)) = ((v)f)·u Vu, v

E

U,f E hom 8 (U, B)

is a bimodule homomorphism. (2) By symmetry, if B = End U A• the map hom 8 (U, B) defined by ((u)ifl(f))[v] = u}(v)

Vu, v

E

ifJ: homA(U, A)--+

U,f E homA(U, A)

is a bimodule homomorphism. In this case, 4> is an isomorphism with inverse 1/J. Proof The proofs of (l) and the first part of (2) are routine. Temporarily denote 4>(f) by [f, ] and 1/J(g) by ( , g). Then [v]([f, ](u)) = (v)f-u Vu,vE U,fEhom 8 (U, B). LetfEhom 8 (U, B). Then t/J [ f ' f 1-+

"'

f

] 1-+ ( ' [ ' ])

and ((u)( ,[f, ]))[v] = u·[f,v] = (u)f-v Vu, v E U,f E hom 8 (U, B),

since [f, v]

ifJ o 4>

= l

E A and the right A action on U is evalution. Thus, on hom 8 (U, B). Similarly, 4> o ifJ = l. D

l. 7 PROPOSITION Let V E B 0 -mod be faithful and B = Biend 80 V. If U 0 = hom 80 (V, B 0 ) and U = hom 8 (V, B), the canonical embedding B 0 4 B induces an inclusion U 0 s;; U. Moreover, if T 0 = trace 80 V, then BT0 s;; B 0 • Proof Let A = End 80 V. Then B = End VA. Let ( , ) : hom 80 (V, B 0 )--+ homA(V, A) be the (A, B 0 )-homomorphism defined in 1.6. For b E B, v, v' E V and f E U 0 , we have

(bv)f·v' = (bv)(f, v') = b((v)(f, v')) = b((v)f·v') = (b(v)f)·v'

Correspondence theorem for projective modules

13

since V E B-mod-A. Since 8 V is clearly faithful, (bv)f = b(v)f, Vb E B, v E V and f E U 0 • Thus, U 0 s;; U as asserted. D 1.8 CoROLLARY Let U be a left B-module. Then, 8 U is a generator iff 8 U is balanced, and finitely generated projective over A= End 8 U. Proof =>: That 8 U is balanced follows from the Proposition since T = B. Next

un : : : :

B

EB X

X

for some

E

B-mod

by projectivity of 8 B. Apply hom 8 ( - , U) and get the mod-A isomorphism An : : : : U

EB hom 8 (X,

U).

Thus, U A is finitely generated projective. The converse is obtained by reversing the steps. D 1.9 CoROLLARY

(l) If U is a generator in mod-A, then

U* = homA(U, A)::::::: hom 8 (U, B) is a generator in A -mod, and finitely generated projective in mod-B. Moreover there is a canonical isomorphism A : : : : End U* 8 . (2) If U is finitely generated and projective in B-mod, then U is a generator in mod-End 8 U. Proof Let u--+ U** { u ~--+ u)

<' >

<,

denote the canonical (B, A)-homomorphism of U into its Abidual U** = homA(homA(U, A), A). Thus,f< , u) = (f, u) = f(u), Vf E U*, u tor in mod-A, then by l.l, l =

n

.L 1=1

E

U. If U is a genera-

f 1(u1) for certain f 1 E

homA(U, A), u 1 E U. Consequently, n

_2 f1<

, ul) = lA

1=1

which shows that AU* is indeed a generator. Then Corollary l. implies the rest. D

Simple Noetherian rings

14

1.10 PROPOSITION AND DEFINITION An object U of mod-A is a progenerator if U satisfies any of the following equivalent properties: (1) U is a generator in mod-A and finitely generated and projective in mod-A. (2) U is a generator in mod-A and in End-U A-mod. (3) U is finitely generated, balanced and projective both in modA and in End U A-mod. (4) U is a generator in End U A-mod, finitely generated and projective in End UA-mod and A ::::; Biend U A canonically. Proof An easy application of 1.8 and 1.9(2). D l.ll PROPOSITION AND DEFINITION A bimodule 8 U A is a (B,A)-progenerator provided the equivalent conditions hold: (1) U is a progenerator in mod-A and B ::::; End U Acanonically. (2) UisaprogeneratorinB-modandA::::; End 8 Ucanonically. (3) homA(U, -):mod-A "VV mod-B defines a category equivalence. Proof (1) and (2) are clearly equivalent by 1.10. (3) ~ (1): Any equivalence is clearly faithful and exact, hence U A is a progenerator by 1.1 and 1.2. Set T equal to the inverse of S = homA(U, -).Then To S :::::: 1mod-A and So T :::::: 1mod-B· Now, S(M) ~t hom 8 (B, S(M))

~t homA(T(B), TS(M)) n~t homA(T(B), M)

for all ME mod-A. In particular, S(A) = homA(U, A) ::::; homA(T(B), A) implying that UA::::; T(B). Hence,

B::::; ST(B) ::::; S(U). Thus, B ::::; End U A(1) ~ (3): Let S = homA(U, -) and T = hom 8 (U*, -) where U* = homA(U, A) ::::; hom 8 (U, B) by 1.7.

TS(M) = T(homA(U, M)) = hom 8 (U*, homA(U, M)) ~t homA(U* ® 8 U, M) nat

~

h omA( A , M )

nat

~

M.

Correspondence theorem for projective modules

15

(The verifications that U* 0 8 U ::::; A and U 0 AU* ::::; B will be deferred until after the proof of 1.12.) Similarly, 8T(N) n~t N for all N E mod-B. Thus, 8 is an equivalence with inverse T. D 1.12 PROPOSITION AND DEFINITION Two rings A and B are similar or Morita equivalent, A ,.., B, if they satisfy the equivalent conditions: (8 1 ) mod-A ::::; mod-B, (8 2 ) A-mod::::; B-mod, (83 ) There is a (B,A)-progenerator V. If A ,.., B, then the lattice of right ideals of B is isomorphic to the lattice of A submodules of V under the map I~IV.

Moreover, the lattice of ideals of A is isomorphic to the lattice of ideals of B. Proof (81 ) and (83 ) are equivalent by l.ll and (82 ) and (8 3 ) are equivalent by symmetry. The remaining statements follow immediately from the correspondence theorem 1.13 and will be omitted. 0 If Pis any property of a ring A or mod-A and whenever A ,.., B, B has property P, Pis said to be a Morita invariant (property) of A. For example, simplicity, r.gl.dim R = n, Artinian, Noetherian, semiprime Goldie are all Morita invariants while field, domain, and pli-domain are not. Of interest to us will be the weaker condition: if A = End 8 V, with 8 V finitely generated and projective, and A has property P, then so does B. To that end, suppose A = End 8 V, where 8 V is finitely generated and projective. What are the relationships between the various module lattices which naturally arise1 This question is answered completely in the correspondence theorem 1.13. In order to facilitate the proof of 1.13 we shall first make some definitions and observations. Let V E B-mod and A = End 8 V. Set U = hom 8 (V, B). Then V and U are canonical (B, A) and (A, B)-bimodules respectively. Define two maps ( , ) and [ , ] as follows:

[ ' ]: v

X

u -+ B,

Simple Noetherian rings

16 where (v,f)

~

[v,f] = (v)f, Vv ( ' ): u

E

V,f E U.

X

v --+A,

where (g, v') is that element of A =End 8 V taking v to (v)g·v', that is, [v](g, v') = (v)g·v'

Vv, v'

E

V, g E U.

One readily checks that [ , ] and ( , ) are maps (balanced bilinear in the sense that, mappings V--+ B and U --+ B induced by [ phisms, and [va, u] = [v, au], Vv E V, u E U,

balanced bilinear for example, the , ] are homomora E A) such that

V·(u, v') = [v, u]·v'

and (u,v)·u' = u·[v,u']

Vu,u'E U,v,v'E V.

Since [ , ] (resp. ( , )) is a balanced bilinear map from V x U--+ B (resp. U x V--+ A), [ , ] (resp. ( , )) induces a map, also denoted [ , ] (resp. ( , )), [ , ] : v ®Au--+ B, ( ' ) : u ® B v--+ A.

Clearly, [ , ] is a (B, B) homomorphism (resp. ( , ) is an (A, A) homomorphism) and im[ , ] = trace 8 V. Moreover, [ , ] (resp. ( , )) is epi if and only if it is an isomorphism. For suppose [ , ] is epi and x = .L v; ® u; with x E ker[ , ], that is,

.L [v;,

I

u~]

= 0. Since l

E

B can be written in the form

I

l =

_2 [vi, ui], i

x = =

= =

f

[vi, ui](f v; ® u;)

,2 [v;, ui]v; ® u; l.i ,2 vi(ui, v;) ® u; l.i ,2 vi ® (ui, v;)u; l.i

= fvi

® ui(f[v;,u;])

= 0.

Correspondence theorem for projective modules

17

Thus, [ , ] is an isomorphism. The proof of the parenthetical statement is analogous. In fact, by 1.2(2), ( , ) is an isomorphism if and only if 8 V is finitely generated and projective. For subsets V1 £ V, U 1 £ U, set [V1,

Ud = {[v1,ud I V1E v1,U1E U1}·

If v1 E mod-A and u1 E A-mod-B, then clearly, [V1, Ud is a right ideal of B. Similar definitions are made for (U', V') etc. Set T = trace 8 V = [V, U]. The important lattices which anse are

lat(B8 ) lat(VA) lat( 8 VA) (lat(B8 ))T

= the lattice = the lattice =the lattice = the lattice form I'

The lattice operations v and

of right ideals of B of A-submodules of V of (B, A)-submodules of V of right ideals of B of the = IT, IE lat(B8 ). 1\

are defined on (lat B 8 )T by

and I A J = (I n J)T. We also have (lat 8 B 8 )T consisting of the set of all ideals of B of the form I' = IT, IE lat 8 B 8 • (The operations v and 1\ are defined as in (lat B 8 )T.) Others are T(lat 8 B 8 )T, and T(lat 8 VA) consisting of the set of all (B, A)-submodules of V of the form W' = T W, W E lat 8 VA. Here the operations v and 1\ are defined by

w

1

v

w

2

W1 A W2

=

w

1

+

w

2>

= T(W1 n W2).

1.13 THE CORRESPONDENCE THEOREM FOR PROJECTIVE

Let A = End 8 V, 8 V finitely generated projective and T = trace 8 V. Then, (l) The mapping

MoDULES.

P{

lat VA--+ (lat B 8 )T W r-+ [W, U]

Simple Noetherian rings

18

is a lattice isomorphism with inverse defined by K r-+ KV,for any right ideal K = KT of B. (2) The mapping

lat 8 VA--+ (lat 8 B 8 )T { Wr-+ [W, U] induced by ( l) is a lattice isomorphism. (3) The mapping

1/1: lat AAA --+ T(lat 8 VA) J r-+ VJ is a laUice isomorphism with inverse Wr-+ (U, W).

is an isomorphism of multiplicative semigroups. Proof (l) Since T = [V, U] and A = (U, V), [W, U]T = [W, U][V, U] = [[W, U]· V, U]

=

[W·(U, V), U]

= [W, U]

'v'W E lat VA-

Thus, pis actually a mapping into (lat B 8 )T. Next, [W1 + W2, U] = [W 1, U] + [W2, U] and [W1 n W2, U] = ([Wl, U] (') [W2, U])T, vwl, w2 E lat VA which together imply that p is a lattice homomorphism. The first statement is obvious. As for the second, since 8 V is projective, ([W 1, U] n [W2, U])· V = [W1, U]· V n [W 2, U]· V = wl n w2·

Thus, [W1 n W2, U]

as asserted.

= ([W1, U] n [W2, U])[V, U] = ([W 1 , U] n [W2 , U])T

Correspondence theorem for projective modules

19

Finally, the map is clearly an order homomorphism, as is the map

IT

f-+

lTV= IV

which is easily seen to be its inverse. (2) Clear. (3) ifJ is clearly a map into T(lat 8 VA) since VI = (TV)I = T(VI). Similarly, the map T(lat 8 VA)--+latAAA defined by TW f.-+ (U, TW) is a mapping into lat AAAObserve that

If-+ VI

f-+

(U, VI) = I

and

TW

f-+

(U, TW)

f-+

V · (U, TW) = [V, U]TW = T 2 W = TW.

Thus, the maps are mutually inverse to one another. That they are lattice homomorphisms follows from the fact that U 8 is projective, and hence (U, W1 n W2 ) = (U, W1 ) n (U, W2 ). (4) Clear with the observation that

[VI, U][VJ, U] = [[VI, U]VJ, U] = [Vl·(U, VJ), U] = [VIJ, U]

1.14 CoROLLARY Let V be finitely generated faithful and projective over a ring B, let A = End 8 V, and let T = trace 8 V. For example, V can be an arbitrary generator in mod-A, for any ring A, and B = End VA. (l) Let P be any property of multiplicative semigroups which is preserved under homomorphisms. Then (ideals of A) has property P if and only if T(ideals of B)T has property P. (2) A is semiprime if and only if TQ = 0 for every nilpotent ideal Q of B. (3) A is prime if and only if TQ = 0 for every annihilator ideal Q of B. (4) If B is semiprime, then B is prime if and only if A is prime. ( 5) A has a finite ideal lattice if and only if the set of ideals of the

Simple Noetherian rings

20

form TIT for an ideal I of B is finite. In particular, A is a simple ring if and only if T K = 0 for every ideal K =I= T contained in T. (6) If B is semiprime, then A is simple if and only if Tis the least ideal of B. In this case a right ideal I of B satisfies I = IT if and only if I is an idempotent right ideal contained in T. Proof The corollary is a consequence of the semigroup isomorphism ideals of A _. T(ideals of B) T stated in (4) of 1.13. Thus, if Pis a statement that every ideal of a certain set P( R) of a semigroup R is zero, then, in particular, P(A) = 0 implies that P(T(ideals of B)T) = 0. Since TU = U, then H T = 0 for a right or left ideal H of B if and only if H = 0. Thus, THT = 0 if and only if TH = 0. The other parts are consequences. If B is semiprime, then T I = 0, for some right ideal I of B implies that (IT) 2 = 0, and IT = 0, so I = 0. Thus, in this case, B is prime if and only if A is prime. Furthermore, then A is simple if and only if Tis the least ideal of B. In this case, if I = I 2 is a right ideal contained in T, then I 2 IT 2 J2 =I. So I= IT. Conversely, I = IT implies

I2 = I (TIT) = IT = I since TIT must equal T when I

=1=

0.

D

1.15 DEFINITION A right B-module M is a rational extension of a submodule N provided that given x, y EM, y =1= 0, there exists b E B such that xb E N and yb =1= 0. M A 2 N A is an essential extension if Vm E M, m =1= 0 ~ mA n N =I= 0. Remarks (a) Any rational extension is an essential extension, and conversely, if the singular submodule is zero. If ME mod-B, the singular submodule of M, Z(M) is defined by Z(M) = {mE M I mi = 0 for some essential right ideal I of B}.

Correspondence theorem for projective modules

Biend h 8 , where h 8 = injective hull of B 8 • Then B is the maximal rational extension of B in the sense that for any rational extension B' of B, there exists an injective ring homomorphism h: B'--+ B such that hl 8 = 18 . B is called the Johnson-Utumi maximal quotient ring of B. (See FindlayLambek [58], Johnson [51], Johnson-Wong [59], or Faith [67a], Chap. 8.) We do not require this remark, however. (b) Let

B

21

=

1.16 PROPOSITION Let A = End 80 V, where Vis a faithful, projective left B 0 -module, let B = Biend 80 V, U 0 = hom 80 (V, B 0 ) and U = hom 8 (V, B). Then, U = U 0 B; and 8 V is projective. Moreover, if T 0 = trace 80 V and T = trace 8 V, then T 0 is a left ideal of B satisfying

T 0 2 = T 0 = TT 0 = BT0

and T 0 B = T 0 T = T = T2• Thus, there is a lattice isomorphism (lat B 8 )T--+ (lat B 0 8 )T0

(l) { lf--+ IT 0

o

which induces an isomorphism (2) T(lat 8 B 8 )T--+ T 0 (lat

80

B 08 0 )T0

of multiplicative semigroups. Finally, every right ideal I of B is a rational extension of IT 0 in mod-B0 • In particular, B is a rational extension of the subring Bo. Proof That U = U 0 B, and that 8 V is projective, follow immediately from the dual basis lemma 1.2(2) and 1.7. Next, T 0 V = V = TV, T 0 2 = T 0 and T 2 = T by 1.3. Thus, T = [V, U] = [T 0 V, U] = T 0 [V, U] = T 0 T. Similarly, T 0 = TT 0 • Clearly, the map (l) is a lattice isomorphism. Likewise, the induced map (2) is bijective. The identity

[T 0 (TIT)T 0 ][T0 (TJT)T 0 ] = T 0 (TIT)(TJT)T 0 shows that (2) is an isomorphism of multiplicative semigroups.

22

Simple Noetherian rings

Finally, to show that I is a rational extension of IT 0 in mod-B 0 , let a, bE I, b =1= 0. Then, b V =1= 0 ~ bT0 =I= 0. Thus, bt0 =1= 0 for some t0 E T 0 and clearly, at 0 E IT 0 . Thus, I is indeed a rational extension as claimed. In particular, B a rational extension of BT 0 = T 0 implies B is a rational extension of Bo. D 1.17 DEFINITION An object ME mod-A is called finite dimensional, or is said to satisfy the a.c.c. Ei), if each independent family of submodules of M is finite. If M is finite dimensional, we write dim M < oo. It can be shown that when dim M < oo, the maximal number of independent submodules of M is always the same natural number (cf. Goldie [60]). This number is called the (Goldie) dimension of M. Any Noetherian module is finite dimensional. 1.18 PROPOSITION Let A = End 8 V with 8 V finitely generated projective and faithful, and U = hom 8 (V, B). Then (1) dim VA = dim B 8 , (2) dim AA = dim U 8 • Proof (1) Let V1 E8 V2 EE) •.. Ei) V1 be a direct sum of A-submodules of V. Then [V 1 , U] + ... + [V~o U] is a direct sum of right ideals of B. For, clearly, each [V1, U] is a right ideal of B. If L. b1 = 0, b1 E [V~o U] then L. b1V = 0 ~ b1 V = 0, Vi ~ b1 = 0, Vi since 8 V is faithful. Thus, dim VA ~ dim B 8 • Similarly, dim B 8 ~ dim VA. The proof of (2) is analogous. D

2. Structure of Noetherian simple rings

We begin this chapter with a characterization (2.6) of when a simple ring A has the structure A : : : : End 8 V, where V is a balanced, finitely generated projective over an integral domain B. This happens iff A has a right ideal W which is lateral in the sense that the endomorphism ring of W is an integral domain. (See 2.6.) Moreover, in this case, we have A : : : : End 80 V, where B 0 is a domain with at most one nontrivial ideal T, and B 0 = T + C, where C = center B 0 ::::::: center A. The structure theory for a simple ring A with uniform right ideal W is exactly the same only in this case Band B 0 are right Ore domains (2.17). Furthermore, we have the (Goldie) dimension of V* over B equals that of A over A dim 8 V* = dim AA. Moreover, A is right Noetherian iff B satisfies the a.c.c. on idempotent right ideals contained in T. (See 2.21.) A simple ring A : : : : End 8 V (as above) is similar to a right Ore domain iff some right ideal of B has a simple endomorphism ring (2.20). Two different representations

A : : : : End

8,

V;,

~ = l,

2,

as endomorphism rings of finitely generated projectives occur only if B 1 and B 2 have isomorphic right quotient fields Q = Q(B 1 ) ::::::: Q(B 2 ), and, identifying Q with Q(B2 ), it is true that B 1 and B 2 must be equivalent right orders in Q. Thus, for each isomorphism class [A] of simple rings with uniform ideals, we can associate the class (B) of equivalent orders in a field Q. (See 2.22.) For two orders B 1 and B 2 , there exists a similarity (as rings) B 1 ,.., B 2 only if B 1 £ B 2 (as orders). Furthermore, when B 1 is simple, then equivalence of orders B 1 £ B 2 implies similarity as rings VB 2 iff B 1 is right hereditary (2.26). [ 23 ]

Simple Noetherian rings

24

We investigate the structure of simple hereditary rings, and show interalia that each right ideal is generated by two elements. (These results hold more generally for hereditary Noetherian prime rings, but we restrict ourselves to the case of simple rings.) We conclude Chapter 2 by showing that every simple right Goldie ring of right global dimension not exceeding 2 is similar to a right Ore domain. The proof borrows an idea from Bass's characterization of Noetherian rings of global dimension ~ 2 as those Noetherian rings for which the duals of finitely generated modules are projective. 2.1 DEFINITION Let I be a right ideal of R. The idealizer of I, Idealizer (I) = {a E R I ai s;; I}. Clearly, Idealizer (I) is the largest subring of R in which I is an ideal. 2.2

PROPOSITION

If I is a right ideal of R,

Idealizer (I) E d JI I :::::; n (R )R.

Proof The map defined by a + I r-+ a1 where a1(r + I) = ar + I, r E R is clearly a surjective ring homomorphism with kernel I. D Recall that if X s;; R and I = X .L = {r E R I xr = 0, Vx E X}, I is called a right annulet of R or a right annihilator of R. 2.3 PROPOSITION AND DEFINITION A right ideal V of a ring A is lateral provided that V =1= 0 and End VA is a domain. A principal right ideal V = vA is lateral iff Idealizer (v.L)jv.L

is a domain. Proof The canonical isomorphism vA :::::; Afv.L together with 2.2 suffices. D 2.4 PROPOSITION In order that a right ideal V =/= 0 of a ring A be lateral, any of the following conditions are sufficient: (l) V = vA for some v E A, with v 3 =1= 0, such that for any x E A, x.L 2 v.L ~ x.L = v.L or vx = 0.

Structure of Noetherian simple rings

25

(2) (Koh) V = vA for some v E A such that v.L is a maximal right annulet. (3) V = vA, where vis contained in a minimal left annulet of A. Proof (1) By 2.3, we must show for a, b in Idealizer (v.L), that abE v.L and b ~ v.L ~a E v.L. But one verifies: (va).L

2

v.L

+

bA => v.L.

Hence v(va) = v 2 a = 0. Since v 3 =1= 0, then v 2 .L = v.L, so a E v.L as required. (2) In the proof of (1), we have that (va).L :::> v.L, so (va).L = A, hence va = 0, and a E v.L. (3) Then v.L is a maximal right annulet. D 2.5 LEMMA Let 8 V be a faithful finitely generated projective module and A = End 8 V. Then VA is isomorphic to a right ideal of A iff B 8 embeds in V 8 *. D 2.6 LEAST IDEAL THEOREMt A ring A is a simple ring with a lateral right ideal iff A is isomorphic to the endomorphism ring of a balanced, finitely generated projective left module V over a domain B such that T = trace 8 V is the least ideal of B. Proof (<=): By 1.14(6), A =End 8 V is simple iff T = trace 8 V is the least ideal of B. Hence, A is simple. Since 8 V is faithful and projective, we can assume that VA is a right ideal of A by 2.5. Since 8 Vis balanced, End VA = B is a domain and hence, Vis a lateral right ideal of A. Conversely, if V is a lateral right ideal, then B = End VA is a domain. Since VA is a right ideal of A and A is simple, VA is balanced and 8 V is finitely generated and projective, i.e., A = End 8 V as required. D By Proposition 1.4, if A = End 8 V, B = End VA and 8 V is finitely generated projective and faithful, then A = End 80 V, where B 0 = C + T, C = center B::::; center A, and 80 V is t

In the Introduction, we have stated the Least Ideal Theorem a.s it applies to a. simple Goldie ring, and this theorem specializes to that one by taking V to be any uniform right ideal when A is right Goldie. See 2.34(2).

26

Simple Noetherian rings

finitely generated projective and faithful. Moreover, T 0 = trace 80 V = trace 8 V = T and center B 0 = center B. For T 0 = T by the proof of the second part of 1.4, and if c E center B 0 , bE B, (be - cb)T = 0 because BT = T and ct = tc, Vt E T. Hence be = cb, Vb E B implying that c E center B. The opposite inclusion is obvious. Thus, we have, 2.7 SINGLE IDEAL THEOREMt A simple ring A with a lateral right ideal V is isomorphic to the endomorphism ring of a finitely generated projective left B-module where B is a domain with at most three ideals. In fact, B can be chosen so that B = C + T, T =trace 8 V and C =center B::::::: center A is a field. D Simple rings with uniform right ideals 2.8 DEFINITION A right order R of a ringS is a subring of S such that S = {ab - 1 I a, b E R, b a unit of S}. 2.9 DEFINITION Q is a right quotient ring (quoring) of R if R is a subring-1 of Q such that: (Q1) R has regular elements. (Q2) Every regular element of R is a unit of Q. (Q3) If q E Q, then q = xc- 1 , with x, regular c E R. 2.10 DEFINITIONS Let R 1 and R 2 be right orders in a quotient ring Q. Then (1) they are equivalent, R 1 i R 2 , if there exist regular elements a 1 , b1 , a 2 , b2 ofQ such that a 1 R 1 b1 £ R 2 , a 2 R~ 2 s;; R 1 ; (2) they are right equivalent, R ..!:.. R 2 , if there exist regular elements a 1 , a 2 of Q such that a 1 R 1 £ R 2 , a 2 R 2 £ R 1 ; (3) they are left equivalent, R 1 ..!.. R 2 , if there exist regular elements b1 , b2 of Q such that R 1 b1 s;; R 2 , R 2 b2 s;; R 1 • These are equivalence relations. Also, R 1 ..!:.. R 2 ~ R 1 2- R 2 I Q and R 1 ,.., R 2 ~ R 1 ,.., R 2 • t

In the Introduction, we have stated the Single Ideal Theorem a.s it applies to a. simple Goldie ring. 2. 7 also includes the Same Centers Theorem described in the Introduction.

Structure of Noetherian simple rings

27

2.11 LEMMA Let R be a right order in a quotient ring Q, and letS be a subring-1 of Q. (1) If there exist regular elements a, b, c, d of Q such that aRb s;; S, cSd s;; R, then Sis a right order in Q, and R 2- S. (2) If there exist regular elements a, c of Q such that aR s;; S, cS s;; R, then Sis a right order in Q, and R ..!:.. S. Similarly for left equivalence. Proof (1) If q E Q, then a- 1qa = xc- 1 for some x, c E R, and then q = (axb)(acb)-I, so that S is a right order of Q similar to R. Proof of (2) is obtained by taking b = l.

D

2.12 DEFINITIONS A maximal 2, right order is a right order maximal amongst the right orders which are equivalent to it. Maximal ~ and maximal ..!.. right orders are defined likewise. 2.13 DEFINITION A ring R is said to be right Goldie provided that R satisfies the a.c.c. on right annihilators, the (a.c.c.).L and dim R < oo. A right Goldie domain of dimension 1 is called a right Ore domain. It is elementary to show that any right Goldie domain has dimension 1. 2.14 THEOREM The following conditions on a ring R are equivalent: (1) R is semiprime right Goldie; (2) R is semiprime, Z(R) = 0 and dim R < oo; (3) R has a semisimple right quotient ring S. We shall postpone the proof of 2.14 until Chapter 4. There, the orders in simple Artin rings will be discussed at great length. The following concept plays a fundamental role both in the

28

Simple Noetherian rings

study of orders in semisimple rings and the representation theory for simple rings. 2.15 DEFINITION ME mod-A is called uniform if M satisfies any of the following equivalent conditions: (1) Vm, m' EM, mAn m'A =I= 0; (2) .Ji = injective hull M is indecomposable; (3) dim MA = l. 2.16 EXAMPLES (1) Simple modules. (2) If A is any right Ore domain, A is uniform in mod-A. By Proposition 1.18(1), if A= End 8 V where 8 V is finitely generated projective and faithful, dim B 8 = dim VA- Thus, B is right Ore iff VA is uniform. In particular, if A is simple and U A is a right ideal of A, K = End U A is right Ore iff U A is uniform. Moreover, whenever U A is uniform, any right ideal J of A with endomorphism ring a domain is contained in U AFor, if K' = End J A is a domain, since A = End K'J, dim K' K' = dim J A 5 dim A A < oo. Thus, J A is uniform. Since VJ =I= 0, vJ =1= 0 for some v E V. However, JQ is clearly a minimal right ideal of the quotient ring of A. In fact, JQ = injective hull of JA. Thus vj =1= 0, Vj EJ, and hence the map J--+ V defined by j ~ vj is a monomorphism. 2.17 ENDOMORPHISM RING AND PROJECTivE MoDULE THEOREMf The following conditions are equivale1tt: (1) A is a simple ring with uniform right ideal V. (2) A is isomorphic to the endomorphism ring of a finitely generated projective left B 0 module V over a right Ore domain B 0 having center C a field such that either B 0 is simple or B 0 = C + T 0 , T 0 = trace 80 V, is a ring possessing precisely 3 ideals.

0 2.18 THEOREM (Hart [67]) If A is a simple ring, then A has a uniform right ideal iff dim AA < oo. Moreover, in this case, A t

2.17 includes the Endomorphism Ring Theorem and the Projective Module Theorem described in the Introduction. This theorem also contains a statement of the Single Ideal Theorem.

29

Structure of Noetherian simple rings

is necessarily a Goldie ring and there exists a right Ore domain K and an integer n > 0 such that A : : : : eKne where e is an idempotent belonging to Kn. Proof <= : Clear. ~= By the preceding Proposition, A : : : : End 8 V, 8 V finitely generated projective and Bright Ore. By Proposition 1.18(2), dim AA = dim V 8 * < 00. If U is a uniform right ideal of A, then U A is a generator and hence, un : : : : A E8 B for some integer n > 0. Let e E End un be the projection idempotent e : un --+ A --+ 0. If K = End u A then A : : : : eKne. D 2.19 LEMMA Let A =End 8 V, B =End VA be right Ore with quotient field D, and let 8 V be finitely generated projective and faithful. If W A s;; VA and I = [W, U] where U = hom 8 (V, B), (I: I) = {dEDI dis;; I}::::::: End I 8 andEnd I 8 f End W A where g is defined by

g(f)(xv) =f(x)·V VfEEndi8 ,

XEI,VE V.

Proof Clearly, each element dE (I: I) induces, by left multiplication, an element d1 of End I 8 , and hence, a ring homomorphism of (I : I) into End I 8 • To exhibit its inverse,/ E End I 8 extends to anj:l--+1, and/(1) = d1 satisfies d1i = /(i) = f(i), T/ i E I, so d1 E (I: I). The desired inverse is defined by f r-+ d1 . The proof of the second isomorphism is similar. D 2.20 SIMPLE ENDOMORPHISM RINGS THEOREM Let A be a simple ring with a uniform right ideal V and let D denote the right quotient field of the domain B = End VA- The conditions listed below are equivalent: (1) A is isomorphic to the endomorphism ring of a finitely generated projective left R-module U where R is a simple right Ore domain. (2) A is isomorphic to the endomorphism ring of a finitely generated projective leftS-module U, with Sa right Ore domain such that the endomorphism ring of some right ideal of S is a simple ring. (3) Some right ideal of B has simple endomorphism ring. (4) The endomorphism ring of some A -submodule of V is simple.

30

Simple Noetherian rings

(5) The endomorphism ring of some uniform right ideal of A is simple. (6) There is a right ideal I of B and a simple subring G of D such that GI s;; I and Dis the right quotient field of G. (7) There is a progenerator Pin mod-A such that L = End P A is a simple domain, that is, A ,.., L. Proof (1) ~ (2): Trivial. (2) ~ (5): A = End 8 U. Therefore, by Lemma 2.5 U A is isomorphic to a uniform right ideal of A. By the remarks preceding 2.17, U A s;; VA- If I is a right ideal of S such that End I 8 is simple, then IUA is a uniform right ideal of A and End 18 ::::::: End IUA by Proposition 2.19. (5) ~ (4): Trivial. (4) ~ (3): Suppose Y A s;; VA has a simple endomorphism ring. There exists a right ideal I of B such that IV = Y (I= [Y, V*]) and by Proposition 2.19, End YA::::::: End lB. (3) ~ (6): If End IBis simple, G = (I: /) = {dE D I dl s;; I} : : : : End I B and Dis clearly the right quotient field of G. (6) ~ (7): Suppose G s;; (I:/) = End lB. Clearly, P = IVA is uniform and

If J is any nonzero ideal of (I : /), J n G =1= 0 since D is an essential extension of G. Therefore, 1 E J which implies that J = (I : /), that is (I : /) is simple. Since A is simple, P A is a generator in mod-A and therefore, finitely generated projective over L = End P A- Simplicity of L implies that P is an ( L, A) progenerator. (7) ~ (1): Trivial. D

2.21 IDEMPOTENT RIGHT IDEAL THEOREM A ring A is a simple right Noetherian ring iff A is isomorphic to End Bo V, Bo V finitely generated projective, B 0 a right Ore domain having center C a field and such that either B 0 is simple or B 0 = C + T 0 , T 0 = trace Bo V, is a ring possessing precisely 3 ideals and satisfying the a.c.c. on idempotent right ideals contained in T 0 • Moreover, when this is so, B = End VA satisfies the a.c.c. on principal right ideals.

Structure of Noetherian simple rings

31

Proof ~: Clearly, A contains a uniform right ideal V and hence the first part of the necessity follows from 2.17. By 1.14(6) and 1.13(1), B 0 satisfies the a.c.c. on idempotent right ideals contained in T 0 • <=: First, A is simple by 1.14(6). Next, the a.c.c. on idempotent right ideals contained in T 0 implies that VA is a Noetherian module by 1.14(6) and 1.13(1): hence, so is AA since A E8 B for some n > 0. If x 1 B s;; x 2 B s;; ••• is an ascending chain of principal right ideals of B, then

vn : : : :

is an ascending chain in VA which must terminate since VA is Noetherian. Thus, for some k > 0 xk V = xk + 1 V

Vj ~ 0.

Now B is a right Ore domain having right quotient field D say. Thus, d = xk+ 1 - 1xk ED which satisfies dV = V and dT = T since [xk V, U] = xkT = [xk+l V, U] = xk+lT and TU = U. Since d induces an epimorphism of the Noetherian module VA> dis a unit of B ~ xkB = xk+ 1 B. D Two questions naturally arise pertaining to the above representation of A: (1) Is the right Ore domain B necessarily Noetherian 1 (2) If A : : : : End 8 , V', where 8 , V' is finitely generated projective and B' right Ore, what is the relationship between B and B'1 The answer to (1) is 'no' as will be shown in Chapter 4. The next Proposition gives us some indication of the rela.tionship between B and B'. 2.22 UNIQUENESS OF ORDERS THEOREM Let B 1 be a right Ore domain with right quotient field D 1, and U 1 finitely generated projective in Bcmod, and A 1 = End 81 Ut> i = 1, 2. If there is a ring isomorphism f: A 1 --+ A 2 , then there is an isomorphism g: D 1 --+ D 2 such that g(B 1 ) and B 2 are equi'tJalent right orders in D 2 •

32

Simple Noetherian rings

Proof By 2.5, U1 is canonically isomorphic to a uniform right ideal of A 1• Therefore,f(U 1 ) = V2 is a uniform right ideal of A 2 • In fact, V 2 is isomorphic to an A 2 -submodule W 2 of U 2 • Moreover, W 2 = I U2 where I = [ W 2 , U2 *] is a right ideal of B 2 and End W 2 A ::::::: End I 82 ::::::: {dE D 2 1 di £ I}= B. Clearly, B 2 is a right order of D 2 and in fact, equivalent to B 2 • For, if x E I, x =I= 0, then Bx £ B 2 and xB 2 £ B. Now, R = Biend 8 w2 =End w2A is a right Ore domain 2 2 containing B 2 by 1.16 and since RT £ B 2 and B 2 £ R, where T = trace 82 W 2 , R and B 2 have the same right quotient field D 2 and are equivalent right orders. Similarly, S = Biend U 1 Al and B 1 are equivalent right orders of D 1 • Let g: D 1 --+ D 2 be the unique extension of the isomorphism S--+ B (defined by f). Since S is equivalent to B 1 and B is equivalent to B 2 , g(S) = B is equivalent to g(B 1 ) and hence, g(B 1 ) is equivalent to B 2 • D Having established necessary and sufficient conditions for a simple ring A to be similar to a right Ore domain B, it is instructive to analyze the structure of the ring A for low global dimensions.

Zero global dimension By definition, the right global dimension of a ring A is zero iff every right A-module is projective (injective), and this is equivalent to stating that every right A-module is semisimple in the sense that every module is a direct sum of simple modules. By the Wedderburn-Arlin Theorem, these rings are finite products of matrix rings over fields. (See, for example Faith [73a, pp. 369-70, 8.8 and 8.11 ]. ) We next derive theWedderburnArtin theorem for simple rings using our methods. 2.23 THEOREM (Wedderburn-Artin) A ring A is simple Artinian iff for some n > 0, A : : : : Bn, B a field. Proof <= : Clear. ~: By 2.21, A : : : : End 8 V, 8 V finitely generated projective, B a right Ore domain satisfying (2) of 2.17 and the a.c.c. on idempotent right ideals contained in T = trace 8 V.

Structure of Noetherian simple rings

33

By 1.13(1) and 1.14(6), VA is a Noetherian module. Thus, for some n > 0, there exists an epimorphism A An--+ VA--+ 0 which implies that VA is an Artinian module ~ B satisfies the descending chain condition (d.c.c.) on idempotent right ideals contained in T ~ Soc B 8 =1= 0 ~ B is a field, i.e., Soc AA =1= 0 ~ Soc B 8 =1= 0. Thus, B is a field, 8 Vis a finite dimensional vector space over B, hence 8 V ::::; Bn for some n > 0 ~A ::::; Bn. D By the symmetrical properties of matrix rings one sees that A has zero right global dimension iff A has zero left global dimension, since A is right semisimple iff A is left semisimple. Right global dimension of A = 1 A ring A is called right hereditary if the right global dimension of A equals 1; equivalently, each right ideal of A is projective. 2.24 LEMMA Any finite dimensional right hereditary ring R is right Noetherian. Proof Let I be any right ideal of R. Clearly, by finite dimensionality of R, I contains a finitely generated essential right ideal J. Since I is projective, I is a summand of R<X> for some X, and by finite generation of J, J is contained in a finite free summand of R<X>, say R. Since homR(IfJ, R) = 0, any homomorphism/: I--+ R satisfyingfiJ = 0 must already be 0. Thus, I s;; R and is a summand. D Thus, a simple right hereditary ring with a uniform right ideal is similar to a right Ore domain which is necessarily Noetherian. (For r.gl.dim = 2, see 2.40.) The following proposition provides a characterization of this class of rings in terms of the uniform right ideals. 2.25 HEREDITARY RING THEOREM Let A be a simple ring with a uniform right ideal U. Then the following conditions are equivalent: ( 1) The endomorphism ring of every nonzero uniform right ideal V of A is simple. (2) Every uniform right ideal V of A is finitely generated projective.

34

Simple Noetherian rings

(3) A is right hereditary and right Noetherian. Proof (l) ~ (2): Let B = End VA. Since VA is a generator, 8 V is finitely generated projective and a generator since B is simple. Thus, VA is finitely generated projective since A = End 8 V. The converse is similar. (l) ~ (3): Let U be any uniform right ideal of A. Then UA is finitely generated projective by (l) ~ (2). Thus, every Asubmodule of U is finitely generated projective implying that uA is Noetherian. Since uA is a generator, un : : : : A E8 X for some X E mod-A implying that AA is likewise Noetherian. By an easy induction on l, every A-submodule of U 1 is finitely generated projective. Thus, every right ideal of A is (finitely generated) projective or A is right hereditary. (3) ~ (1): If A is right hereditary and Noetherian, and W is any uniform right ideal of A, W A is an A-progenerator implying that A ,.., B = End W A- Thus, B is simple. D

In general we have observed that there is no precise relationship between an order A and an equivalent order B. However, when A is a simple, Noetherian, hereditary domain, there is a far more fundamental connection. Namely, 2.26 PROPOSITION Let B be a right Ore domain with right quotient field D, and let A be a subring of D which is a right order in D. Consider tke conditions: (l) A ,.., B. (2) There is a right ideal I of B and a ring isomorphism A : : : : (I : I) = {dE D I di s;; I}. (3) A and B are equivalent right orders. Then (l) ~ (2) ~ (3). If B is a simple ring, then every ring satisfying (2) or (3) is simple only if B is right hereditary and right Noetherian. In this case, (l) ~ (3) for any maximal order A. Proof In order to establish (l) ~ (2) ~ (3), we first prove the following lemma: 2.27 LEMMA Let B 1, i = l, 2, be right Ore domains having the same right quotient field D. If B 1 ,.., B 2 , then B 1 and B 2 are

Structure of Noetherian simple rings

35

equivalent right orders and there is a right ideal I of B 2 together with a ring isomorphism

Bl :::::; (/: /). Proof By 1.10, B 1 = End P 82 and B 2 = End 81 P for some (B 1 , B 2 )-bimodule progenerator P. By 2.5 P is isomorphic to a right ideal I of B 2 and clearly, B 1 :::::; (I: /). For any x E /, B 1 x s;; I s;; B 2 and xB 2 s;;I s;; B 1 • Thus, B 1 and B 2 are equivalent right orders. D Now assume that B is simple. If I is any nonzero right ideal of B, I is a left ideal of the ring A = (I : /) and A is a right order of D equivalent to B. A ,.., B implies that A is simple. Thus, I is a generator in A-mod and mod-Band by 1.7 finitely generated projective in both. Thus, B is right hereditary and right Noetherian provided that every equivalent right order of Dis simple. Conversely, assuming that B is a simple, right hereditary, right Noetherian domain, let A be a maximal right order equivalent to B. Then, xAy s;; B for nonzero elements x, y E B. Thus, xy = bE B and y = x- 1 b. Set A' = xAx- 1 • Clearly, A'b s;; B and hence A'I s;; I for I = A'bB. LetS = (I:/) :::::; End 1 8 • Since 18 is a progenerator, S,.., Band hence, is simple. Since A is a maximal order, A = x- 1 Sx and thus, A is likewise simple. Since A ,.., S, A ,.., B. D In a different direction, we have the following result which has as an immediate corollary the fact that each right ideal of A in 2.25(3) can be generated by 2 elements. Thus emerges an intimate connection between these rings and the classical Dedekind domains. 2.28 THEOREM (Webber [70]) Let A be a simple, right hereditary, right Noetherian ring. Let I, J and K be any three nonzero right ideals of A, J essential in A. Then, there exists a right ideal L suCh that I E8 K :::::; J E8 L. Proof If I :::::; J, we are finished. Otherwise, choose j 0 E J which is regular and set J 0 = j 0 I. Since I :::::; J 0 , if we set L 0 = K, I E8 K :::::; J 0 E8 L 0 •

Simple Noetherian rings

36

Moreover, J 0 c J. Thus, {a E A I Ja s;; J 0 } must be zero since this set is a 2-sided ideal of A, and A is simple while J A = J :::> J 0 . Hence, 3j1 E J such that j 1L 0 ¢. J 0 • Set J 1 = J 0 + j 1 L 0 • We have an epimorphism

cp: Lo---+ jlLofJ o r. j1Lo = J 1/Jo· ker cp. Then L 1 E8 J 1 : : : : J 0 E8 L 0 by

Let L 1 = Schanuel's Lemma. The ascending chain J 0 c J 1 c ... must terminate in finitely many steps. D 2.29 CoROLLARY Each right ideal of a simple right hereditary right Noetherian ring A can be generated by 2 elements. A duality for torsion modules Let A be a right Goldie ring and ME mod-A. An element x E M is called a torsion element in case xc = 0 for some regular

element c EA. Using the fact that A is a right Goldie ring, the set t(M) of torsion elements of M can be seen to be a sub module of M, called its torsion submodule. The proof ofthis requires the theorem of Goldie and Lesieur-Croisot (given in Chapter 4) which states that A has a right quotient ring Q(A), or equivalently, for given elements a, bE A, and b regular, the Ore condition holds: 3a1 E A, regular b1 E A such that

ab 1 = ba1 • Given that t(M) exists, we say that M is a torsion module if t(M) = M; M is torsion-free if t(M) = 0.

Let A be a semiprime, two-sided hereditary Noetherian ring with quoring Q and I an essential right ideal of A. Then (l) I*::::::: I-l = {qEQ I qi s;; A}. (2) I-1-1-1 = I-1. (3) I-1-1 = I. (4) AJI is Artinian. Proof The proof of (l) is similar to that of 2.19 while the proof of (2) is trivial. Note that (l) and (2) are valid for any semiprime, two-sided Noetherian ring. Henceforth, we shall 2.30 LEMMA

37

Structure of Noetherian simple rings

freely identify the modules I* and I- 1 • Since I**:::::; I (proof?) and I- 1 - 1 s;;; I, I- 1 - 1 = I. To show that AJI is Artinian, observe that I contains a regular element c since it is essential. (See 4.4 for a proof of this fact.) If A 2 I 1 2 I 2 2 ... 2 I 2 cA is a descending chain containing I, Ac- 1 = (cA)- 1 2 I- 1 2 ... 2 In - 1 2 ... 2 I 1 -

1

2 A

is an ascending chain contained in Ac- 1 • Since A is left Noetherian, for some n, In - 1 = In+i - 1 , Vj ~ I. Thus, In = In+i• Vj ~ I by (2). 0 2.3I DEFINITION Let f(f and f(f' be two categories, T : f(f "-" f(f' and S : f(f' "-" f(f be two contravariant functors satisfying To S ::::: I'i', and So T ::::: I'i'. Then the pair (T, S) is called a duality between f(f and f(f' and the categories f(f and f(f' are said to be dual to one another. 2.32 PROPOSITION Let A be a semiprime, two-sided hereditary Noetherian ring and let ffA (resp. Aff) denote the full subcategory of mod- A (resp. A -mod) consisting of all finitely generated torsion modules. Then, the functor hom A(-, K) defines a duality between ffA and A.r where K = QfA, and Q is the two-sided quotient ring of A. Proof By 2.30, each cyclic and hence, each finitely generated torsion module is Artinian and thus, has finite length. If AJM is a simple right or left A-module, then it is trivial to check that

(AJM)#

= homA(AfM, K)

:::::; M- 1 /A

and is simple. Moreover (M- 1 /A)#:::::; AfM. Now, if M has finite length n, let N be a submodule of length n - I. If we apply # twice to the exact sequence 0--+ N--+ M--+ MJN--+ 0

we obtain the following commutative diagram by injectivity of K (on both sides): o~N~M~MJN~o

0

~

lN##--+ M##--+ l M lfN##--+ 0

Simple Noetherian rings

38

By induction on n, N##::::::: N and ditto for MJN since this module is necessarily simple. Thus, M## : : : : M completing the proof. D 2.33 CoROLLARY Under the hypothesis of Proposition 2.32, each object in ~ (resp. Aff) has a composition series of finite length. D Structure of projectives over simple hereditary rings Let M be an arbitrary finitely generated right A-module. Our goal is to show that M : : : : M 1 E8 M 2 where M 1 is projective (in fact M 1 ::::::: An E8 I for some n > 0 and right ideal I), and M 2 is the torsion submodule of M. Since we know the structure of M 2 , we shall have gained a good handle on the subcategory of all finitely generated A-modules. Recall that if I is any right ideal of A, there exists a right ideal K of A maximal with respect to the property that I n K = 0 (Zorn). Such a K is called a relative complement of I. The set of all K which are relative complements of some right ideal I of A are called the complement right ideals of A. 2.34 PROPOSITION (Goldie [60]) Let A be a semiprime right Goldie ring. Then (I) each right annulet of A is a complement right ideal; (2) if U is uniform and 0 =F u E U, au = 0 ~ aU = 0; (3) if I is any right annulet, I = a.L for some a E A; (4) each minimal right annulet is uniform. Proof (Goldie) (I) Let I be any right annulet, K a relative complement and I' 2 I a relative complement of K. Since I E8 K is essential, (x: I

E8 K)

=

{a E A

I xa E I E8 K}

is essential in A for all x EA. If x E I', x(x: I E8 K) s;; I' n I E8 K = I and hence .LJx(x : I E8 K) = 0. Since the singular ideal of A, Z(A) = 0, then .LJx = 0 ~ x E I. (2) We must show that .LU = {x E A I x.L n U =F 0}. Set X = R.H.S. Claim: XU s;; Z(A). Since Z(A) = 0, this will imply that X = .LU.

Structure of Noetherian simple rings

39

Let x EX, u E U and I a right ideal of A. If ul = 0, then xu! = 0 ~ (xu).L (") I =F 0. If ul =F 0, then x.L (") U (") ul =F 0 by uniformity of U ~ (xu).L (") I =F 0. Thus, (xu).L is essential ~ XUE Z(A). (3) Let I be any right annulet and U a uniform right ideal satisfying I (") U = 0. We claim there exists u E U such that u.L 2 I and u.L (") U = 0. For, consider .LJ (") U. If .LJ (") U = 0 ~ U.LJ s;; .LJ (") U = 0 ~ (.LJU) 2 = 0 ~ .LJU = 0 by semiprimeness. Hence, U s;; (.LJ).L = I, a contradiction. Thus, we can choose u E .LJ (") U, u =F 0 such that u.L 2 I (obvious) and u.L (") U = 0 by a similar argument. Let I be any right annulet and K a relative complement. Choose U 1 s;; K uniform. Since I (") U 1 = 0 there exists u 1 E U 1 such that u 1 .L 2 I and u/ n U 1 = 0. If u 1 .L n K = 0 we stop. Otherwise, choose U 2 s;; u 1 .L (") K uniform, u 2 E U 2 satisfying u 2 .L 2 I and u 2 .L (") U 2 = 0. If u/ (") u 2 .L (") K = 0 we stop, otherwise continue. Eventually, since the U1 are independent and dim A < oo, there exists an integer n such that u 1 .L (") ... (") un.L (") K = 0. Then by (1), I= u/ n ... n un.L and clearly, I = a.L for a = u 1 + ... + Un. (4) Clear. D 2.35 LEMMA Let A be a simple, right hereditary, right Noetherian ring. Then A is isomorphic to a direct sum of uniform right ideals of A. More generally, each projective right A-module is isomorphic to a direct sum of uniform right ideals of A. n

.L ® 11, 11,j = I, ... , n, a right i= 1 ideal of A, and PAis projective, P : : : : .L ® J 1 where each J 1 is a Proof

Recall that if A : : : :

tel

right ideal of A contained in some 11 (see Cartan-Eilenberg [56], p. 13, Theorem 5.3 or Lambek [66], p. 85). By 2.34, there exists a uniform right ideal U = a.L for some a EA. Thus, by projectivity of aA, A::::::: U(j)aA. By induction, A : : : :

m

.L ®

U k where each U k is a uniform right

k=1

ideal of A. Thus, each projective right A-module can be similarly decomposed. D

40

Simple Noetherian rings

In 2.36 and 2.37 which follow, A will denote a two-sided hereditary, Noetherian simple ring. 2.36 PROPOSITION Let M be an arbitrary finitely generated torsionfreeA-module. ThenMisprojective.Infact,M::::::: An ®I for some right ideal I of A. Moreover, any nonfinitely generated projective A -module is free. Proof Clearly, M 4 M E8 AQ which is finitely generated and torsion free over Q. Thus, M Q9 AQ 4 Qn and hence, elements of M can be thought of as n-tuples of elements of Q. Multiplying by a common (left) denominator embeds M in Am for some m > 0. By 2.35 we can assume M is isomorphic to a direct sum of uniform right ideals of A. If dim A = n, any direct sum of m ~ n uniform right ideals is clearly isomorphic to a right ideal p

of A which is essential iff m = n. Hence, M : : : :

.L E8 I 1 E8 I'

J=l

where each I 1 is essential and I' is some right ideal of A. By repeated applications of 2.28, M : : : : AP E8 I, I a right ideal of A. The final statement, which we leave for the reader, is an easy application of 2.28. D 2.37 THEOREM Let ME mod-A be finitely generated. Then M : : : : t(M) E8 M' where M' is torsion free. D All of the results presented thus far in this section hold for a more general class of rings, namely, the Dedekind prime rings. The reader should consult Eisenbud-Robson [70a,b] for a thorough exposition of the properties of this important class of rings.

Rings of global dimension 2 In this section we prove that simple right Goldie rings of right global dimension ~ 2 are all similar to right Ore domains. Hence, in this case at least the underlying domains are also simple (and right Noetherian! cf. 2.25). 2.38 PROPOSITION If U is any right annulet of A where r.gl.dim A ~ 2, then U is projective.

Structure of Noetherian simple rings

41

Proof Since U = a\ AJU 4 A via the map f defined by

+

I

U

~:--+a.

Consider the following diagram:

A

/'

A ~

U 0 Since r.gl.dim A

/'

~

y

~

AJU

/'

0

~

cokf 0

2, U A is projective.

~

0

D

2.39 LEMMA Any projective uniform right ideal U of A is finitely generated. Proof Just parrot the proof of 2.24 using the obvious fact that any cyclic submodule of U is an essential submodule. D 2.40 GLOBAL DIMENSION 2 THEOREM Let A be a simple right Goldie ring of r.gl.dim A ~ 2. Then A is similar to a right Ore domain. Proof Let U A be a minimal right annulet of A. By 2.38 and 2.39, U A is finitely generated projective and necessarily, a generator by simplicity of A. Thus, A ""' B = End U A by 1.12.

D

3. Noetherian simple domains

The main purpose of this chapter is to construct the simple Ore domains, referred to as D 1 , D 2 , and D 3 in the Introduction. Without going into great detail, these rings are: (D 1 ) the ring !!)R of differential polynomials in commuting derivations a1 , ••• , an over a simple Ore domain R; (D2 ) the skew polynomial ring A = k[x, p, a] with respect to a p-derivation a(and suitable localizations of A); and (D3 ), the ring D Q9 cC(x), where D is a transcendental central division algebra over C, and C(x) is the field of rational functions over C in a variable x. These provide us with the models needed in order to venture further in the structure theory of simple rings in Chapter 4 and the following chapters. Differential polynomial rings A derivation of a ring R is an additive homomorphism

a: R --+ R such that

a(ab) = a(a)b

+

aa(b)

Va, b E R. For noncommutative rings, we have, for each x not m the center of R, an inner derivation Dx defined by Dx(a) = ax - xa Va E R. Also, any derivation a of R extends to a derivation A of the (ordinary) polynomial ring R[x] by setting A(x) = I. Thus,

Let a be a derivation of R, and let !!)R be the set of all polynomials in the symbol a. Then!!) = !!)R is a ring with respect to [ 42 ]

Noetherian simple domains

43

the operations of addition of polynomials, and multiplication induced by the relation

Sa= a8

+

8(a)

VaE R.

!!) is called the ring of differential polynomials in 8 over R or the ring of linear differential operators (l.d.o.) with coefficients in R. If a =

n

.L 1=0

a 181 E !!), an # 0, we define the degree of a, deg a, to

ben if n 2: 0; - oo otherwise. If R is a domain, then one readily obtains the formula deg(ab) = deg a

+

deg b Va, bE!!).

PROPOSITION If R is left (right) Noetherian, so is!!). Proof The argument used in the proof of the classical Hilbert Basis Theorem is applicable, mutatis mutandis. D

3.1

When R is a field, even more can be said: let a, bE!/). Then there exist q, r E !!), 0 :::; deg r < deg b, such that

a= qb Similarly, there exist q', r'

E

+ r.

!!), 0 :::; deg r' < deg b, such that

a= bq'

+ r'.

To demonstrate the above, we can clearly assume that deg a 2: deg b. Suppose that the leading coefficient of a equals an and that of b equals bm. Clearly, deg(a- anan-mbm -lb) < n. The proof is now completed by induction. As we shall see later, !!) need not be a pli-domain (principal left ideal domain) whenever R is. The importance of !!) in the study of simple rings begins to emerge with the next proposition. 3.2 THEOREM Let R be an ordinary differential ring which is also a Q-algebra. (R is called a Ritt algebra.) Then the following are equivalent: (a) !!) is simple.

Simple Noetherian rings

44

(b) a is outer and R has no a-invariant ideals other than 0 and R. An ideal I of R is called a-invariant if a(I) £;; I. Proof (b) ~ (a): Suppose 0 c I c !!) is a non-zero ideal of !!).

Let t =

n

.L 1=0

r 1a1 be an element of I of least non-negative

degree. We shall show that n = 0. Let

t = rnan

+ p 1(a),

where deg p 1( a) < n. The set L of all r

E

R such that either

r = 0 or r is the leading coefficient of an element in I of degree n is clearly a a-invariant ideal of R. Thus, L = R, which implies that there exists an element of the form t = an + p 1(a) in I.

(Note that the leading coefficient oft equals I.) Next, for all r E R, tr - rt E I and has degree less than n. Therefore, tr = rt, Vr E R, implying that t belongs to the centralizer of R in!!). Moreover, if

t = an

+

rn-lan-l

+ ... +

ro, tr = rt Vr E R

(I)

implies that

by equating the coefficients of both sides of (I). Thus, if n =F 0, ais inner, contradicting (b). Thus, n = 0, implying that I = !!). (a) ~ (b) Suppose!!) is simple and ais defined to be the inner derivation determined by a, Da, i.e., a(r) = ar - ra, Vr E R. Then it is trivial to show that !!}(a + a) is a two-sided ideal of !!) and clearly, =F !!) . If I is a nontrivial a-invariant ideal of R, then the set

1=

{~ r a I r

is clearly a nontrivial ideal

1

of!!).

1

1E

I}

Thus, (a)

~

(b).

D

3.2a THEOREM Suppose R is a commutative domain with the characteristic of R, char R, = p 0. Then !!)R is simple iff R is a field and dimcR is infinite, where C = constants a = {r E R I a(r) = 0}. Proof ~: First, a(rP) = 0 for all (nonunits) r E R. Assuming that R is not a field, there exists a nonunit r0 E R satisfying

*

Noetherian simple domains

45

8(r0P) = 0. Thus, R has a nontrivial 8-invariant ideal, namely Rr0PR, which contradicts the simplicity of !!)R. Hence, R must be a field whenever !!)R is simple and char R = p. If dimcR is finite, then, by Jacobson [64b, p. I90, Exercises 3 and 7] !!)R is not simple. <=: By the remarks following 3.I !!) = !!)R is a pli (pri)domain. Let I be a nonzero ideal of!!). Then, I = !!)f for some

f

E !/).

Suppose f =

n

.L 1=0

a 181 where we may clearly assume that

an = I. For each a E R, fa = {3f for some f3 E R since I is an ideal. By comparing constant terms, we obtain n

L a 8 (a) = 1

1

f3a 0 •

1=0

Moreover, by comparing terms of degree n, we obtain ana = a = f3an = {3. Now using the easily established fact that f8 = Sf and the observation that 8 can never be inner, we find that each coefficient off is either 0 or belongs to the center of R, thus, n

,2 a181(a)

=

a0 a

1=0

or n

,2 a181(a)

=

0 Va

E

R.

I= 1

However, if n ~ I, this readily implies that dimcR is finite, a contradiction. D By parrotting the argument that r .gl.dim R[ X] I, we obtain

~

+

I.

I

~

r.gl.dim!!)

~

r.gl.dim R

r .gl.dim R

+

Thus, when R is semisimple, !!) is hereditary. Examples where r.gl.dim!!) < r.gl.dim R + I abound as we shall now show.

An example Let R = k[X] be the ordinary polynomial ring in one indeterminate, k a field with characteristic equal to 0, and 8 be defined as the formal derivative of R. Then, clearly, the hypotheses of

Simple Noetherian rings

46

Proposition 3.2 are satisfied. Thus, !!)R is simple. This particular ring, denoted also by A 1 , has been the source of many important examples and counterexamples. (See Rinehart [62] and Webber [70].) Abbreviate !!)R to !!). We shall first show that r.gl.dim!!) = r.gl.dim R. 3.3 LEMMA Let P be a nonzero prime ideal of R and I a right ideal of!!) satisfying P!!)

~

I c !!).

Then, I = P!!). In particular,.!!)jP!!) is simple. Proof Suppose I =F P!!). Choose a =

n

.L 1=0

a 181 E I with a 1 E R,

an¢; P and n minimal with respect to this property. That this choice can actually be made is clear. Since aa - aa has leading coefficient n8(a)an T/ a E P, by minimality of n, n 8(a)an E P implying that n8(a) E P. Since char R = 0, we have 8(P) ~ P, a contradiction. Hence, n = 0 which is again a contradiction. D 3.4 THEOREM Let I be a right ideal of!!). Then there exists a chain of right ideals between I and I** such that each factor is, up to isomorphism, of the form !!) f P!!), P a prime ideal of R. Proof Since I*** = I* (2.30), it suffices to show that if J is a right ideal of!!), J ::::> I, Jfi is cyclic, and J* = I*, a chain of right ideals exists between J and I whose factors are all of the stated form. Since J* = I*, I' = IS- 1 = J' = JS- 1 where S is the set of all nonzero elements of k[X] and IS- 1 = {is- 1 1 iEI,sES}. To show this, suppose on the contrary that I' c J'. Now, !!)S- 1 = !!)k<X> and since !!)k<X> is a pri-domain and thus, necessarily, I'* ::::> J'*, we can find an element q in the quotient field of!!) satisfying ql' c !!)S- 1 , qJ' ¢ !!)S- 1 • In particular, qi c !!)S- 1 , qJ ¢ !!)S- 1 • By multiplying q by a suitable element of S, we can further assume that qi c !!), qJ ¢ !!). However, by 2.30, this implies that I* ::::> J*, a contradiction.

Noetherian simple domains

47

By assumption, J = a2) + I for some a E !!) and since J' = I', it is quite easy to show that Ann~( a

+ I)

I

= {dE!!) ad E I}

contains a nonzero ideal T of R. Clearly, Jfi is an epimorphic image of !!)jT!!). Since RJT is Artinian, there exists a chain of ideals T 1 between Rand T such that TtfTt+ 1 is simple, in fact, isomorphic to RJP1 where P 1 is a maximal ideal of R. Let Jtfi denote the image of T 1!!)jT!!) under the surjection

!!)jT!!) ---+Jfi---+ 0. Tt!!)fTt+l!!)::::::: TtfTt+l ® R!!)::::::: RfPt ® R!!)::::::: !!)fPt!!)· Since !!)jP1!!) is simple, JtfJ1 + 1 is either zero or isomorphic to !!)I Pt!!)· D 3.5 COROLLARY r.gl.dim!!) = r.gl.dim R. Proof It suffices to show that pdi~ < r.gl.dim R for all right ideals I of!!). Clearly, pdT!!)~ :::; pdTR for all ideals T of R since !!)R is free. By repeated applications of 3.4 we have pdi~ :::; max{pdi~ **, r .gl.dim R - I}. Thus, it suffices to show that pdi~**

:::; r.gl.dim R - I.

Since 0 ---+ K ---+ !!)n ~ I* ---+ 0 is exact for some n > 0, so are 0---+ I** ---+ !!)n* ---+ C = coker f* ---+ 0 and 0 ---+ C* ---+ !!)n** ---+I*** ---+ C' = coker f** ---+ 0. However, by a diagram chase, one obtains that K::::::: 0* and hence, !!)n*JI** 4 C** which is clearly a submodule of a free !!}-module. Thus, pdi~** = 0 or :::; r.gl.dim!!) - 2 which impliespdi~** :::; r.gl.dim R- I sincer.gl.dim!!):::; r.gl.dim R +l.

Remark In the particular case we are considering, since r.gl.dim !!) :::; 2, projectivity of I** follows immediately from 4.I6.

D

Simple Noetherian rings

48

Next, we shall show that each right ideal I of!!) can be expressed in the form I =

m

.L

d1!!) where, for each 0 :::; k :::; m - I, there

1=0

k

exists an irreducible Pk

E

k+l

R such that dk + 1 pk E

I

.L

d1!!) and

1=0

k

~~ dl!!) ~~ dl!!)::::::: !!)jpk!!).

By 2.33, Ifd 0 !!) is Artinian whenever d 0 is any element of I. Using the fact that if J = d 1 !!) + d 0 !!), d 1 N s; d 0 !!) for some ideal of R (see the proof of Theorem 3.4), the result follows immediately. Let I be any right ideal of!!). Then !!) ®I : : : : !!)

® !!).

(2)

(Compare this with Proposition 2.27.) Note that while !!) is not a pri-domain (for example, I = x!/) + 8!!) is not principal), it satisfies a property much stronger than 2.28. To show (2) is satisfied, note that since

Schanuel's lemma yields k+l

k

!!)

®

.2 dl!!) : : : : pk!!) ® .2 dl!!)

1=0

1=0

k+l

: : : : !!)

®

.2 d;!!).

1=0

Hence, by decreasing induction, !!) ® I : : : : !!) ®!!). Another important consequence of this is the following: since!!)® I : : : : !!) ®!!) for all right ideals I of!!), applying 2.36 we obtain that each projective module of !!) is either free or isomorphic to a right ideal of!!). Thus, Mn(!/)) is principal n 2: 2 (Chevalley [36], Th. 7, p. 20). However, since !!) is not, we obtain an example showing that being a pri (resp. pli)-ring is not preserved under similarity. We now summarize what we have obtained:

Noetherian simple domains

49

3.6 THEOREM Let R = k[X] where k is afield and char k = 0, 8 the ordinary formal derivative of R, and P}, the ring of differential polynomials with coefficients in R. Then P} has the following properties: (a) P} is a simple two-sided Noetherian domain; (b) r.gl.dim P} = I; (c) P} is not a pri (pli)-domain; (d) Mn(P}) is a pri (pli)-ring T/ n ~ 2; (e) P} tfJ I ::::; P} tfJ P}, VIE lat P}~; (f) each finitely generated projective P}-module P ::::; p)n - l ® I, where I is isomorphic to a right ideal of P} and n ~ I. Generalities on differential modules

Returning to the general case, let R be an ordinary differential ring, assumed commutative, and P} = P}R· Any M E P}-mod is usually called a differential module. Note that such an M is necessarily an (R, R)-bimodule and a (P}, R)-bimodule as well. The action of () on an element mE M is written 8(m). If M, N E P}-mod, M ® RN and homR(M, N) are in P}-mod, that is, are differential modules. To see this, formE M, n EN, define r(m ® n) = rm ® n

Vr

E

R

and 8(m Q9 n) = 8(m) ® n + m ® 8(n). That M ® RN E P}-mod is easy to check by linearity. For fE homR(M, N), define rf(m) = r(f(m))

Vr E R, mE M

and 8 o f(m) = 8(f(m)) - f(8(m)), Vm EM. It is also easy to check that 8 of E homR(M, N) and homR(M, N) E P}-mod. As an immediate consequence of the fact that M ® RN and homR(M, N) are differential modules, we obtain 3.7 PROPOSITION hom~(M

For all M, N, P E P}-mod,

® RN, P) ::::;

hom~(M,

homR(N, P)).

Proof Let cp: homR(M ® RN, P)--+ homR(M, homR(N, P)) be the canonical isomorphism. (Cf. Cartan-Eilenberg [56].) One

Simple Noetherian rings

50

simply checks that if f E hom~(M ® RN, P) then cp(f) E hom~(M, homR(N, P)). This is routine and left for the reader.

D The next proposition will find use in the next chapter. Recall that M E mod-R is reflexive (torsionless) if the canonical map of M into M** is an isomorphism (monomorphism). 3.8 PROPOSITION Let ME ~-mod be reflexive in mod-R. Then M ~t M** as ~-modules. Proof Since M ~t M** as R-modules by hypothesis, all that remains is to check that the natural isomorphism above is an isomorphism of ~-modules. This is left for the reader. D It is possible to generalize the notion of the ring of differential polynomials in an obvious manner. For, suppose R is a partial differential ring, that is to say, R is a ring with n commuting derivations, a1 , ••. , an defined on R. Define ~ to be all polynomials in the a1 with coefficients in R. A typical element of~ has the form

Equality and addition are defined as in commuting polynomial rings, with multiplication being induced by the relations

ala

=

aat

+

at(a)

Va E R, i

= I, ... '

n.

~ is also called the ring of linear differential operators with coefficients in R. This ring can be inductively defined as the differential polynomial ring with respect to the derivation an over the differential polynomial ring with respect to the derivations a1 , ••• , an_ 1 over R. Thus, by 3.1, ~is Noetherian whenever R is.

Twisted group rings Let R be a ring and G a group acting on R. Recall that the twisted group ring La(R) over R (with a group G acting on R) can be defined as follows: (La(R),

+)

= (R(G),

+ ),

Noetherian simple domains

51

where R(G) is the ordinary group ring; multiplication in La(R) is induced by the relations aa = a(a)a

Va

E

G, a

E

R.

(Here a(a) represents the group action of a E G on a E R.) It is well known that for G =
(the infinite cyclic group generated by p), La(R) = Lp(R) is a simple pli (pri)-domain whenever R is a field and pis an automorphism of R having infinite period (Jacobson [43], p. 38 and Jacobson [64a], p. 2ll). Moreover, in this case, each left (right) ideal of Lp(R) can be generated by an element a E LP(R) having the form a = n

.L 1=0

a 1p1• Since a fixed R and G will be assumed during the

ensuing discussion, we shall replace the symbol La(R) with L. Let MEL-mod, and let R be commutative. Note that if we denote the L action of a on mE M by a(m), we obtain for each a E G, an additive homomorphism of M satisfying a(am) = a(a)a(m) for all a E R. Using these observations, it is quite easy to define an L action on homR(M, N) and M ® RN whenever M, N E L-mod. homR(M, N): for f E homR(M, N), .L aqa E L, q

define

,2 aqaf(m) q

M ® RN: for

=

,2 aqa(f(a- 1 (m)))

Vm EM.

q

.LI m1 ® n 1 E M ®

RN,

.L aqa E L, define q

,2 aqa ,2 m1 ® n 1 = ,2 aq(a(m1) ® a(n1)). q

I

q,l

One readily verifies that the above definitions actually do endow homR(M, N) and M ® RN with the structure of an £module. 3.9 DEFINITION For MEL-mod, the set I(M) = {mE M I a(m) = m, a E G} will be called the G-invariant subset of M. Other notation, MG. Remark In general, I(M) is not an L-submodule of M, merely an l(L) submodule of M. The proofs of the following propositions are, for the most part, routine, and will therefore be omitted.

Simple Noetherian rings

52 3.10 PROPOSITION

Let M, N E L-mod. Then

D R is naturally a left £-module, namely, for all a

=

.L aqa E L, q

rE R,

We shall always denote this £-module by LR. 3.ll CoROLLARY

For MEL-mod, I(M) = homL(R, M).

D 3.12 PROPOSITION isomorphism

For all M, N, PEL-mod, the natural

homR(M ® RN, P) : : : : homR(M, homR(N, P)) is L-linear.

D

3.13 CoROLLARY homL(M ® RN, P) : : : : homL(M, homR(N, P)).

D

3.14 PROPOSITION ForME L-mod, M reflexive in Mod-R, the natural isomorphism M ~ homR(homR(M, R), R) = M** is an isomorphism of £-modules.

D

Skew polynomial ringst Let p be an endomorphism of a field k. A map () : k --+ k is called a p-derivation of kif 8(a + b) = 8(a) + 8(b) and 8(ab) = p(a)8(b) + 8(a)b hold for all a, bE k. Using k, p and ()one can define the ring R = k[t; p, 8] of skew polynomials in t, consisting of all polynomials

t

This section (Cozzens [72]) is reproduced from Journal of Algebra, Volume 23, October 1972, by permiBBion of Academic Press, New York.

53

Noetherian simple domains

where a 1 E k for each i, with the usual addition, multiplication being defined by ta = p(a)t

+

a( a) T/ a E k.

It is not difficult to verify that R = k[t; p, a] is a pli-domain and is principal on the right if and only if p is surjective. Since k[t; p, a] is never simple if a = 0 and k[t; p, a] is usually simple if p = id, one naturally asks whether k[t; p, a] can be simple if p is not surjective and a is a nontrivial p-derivation. If k is commutative, the answer is no. For by a lemma due to Cohn ([6Ia], Lemma page 537), if k is commutative, then any p-derivation a of k is inner [i.e., for some x E k, a( a) = p(a)x xa for all a E k] unless p is an inner automorphism of k. If a is an inner p-derivation of k determined by x E k, then one easily checks that the principal left ideal of R = k[t; p, a], R(t + x), is two-sided and nontrivial. Thus, one is naturally forced to consider noncommutative k. Theorem 3.I5 provides complicated sufficient conditions which lead to simple pH-domains. Finding a triple (k, p, a) satisfying the hypothesis of 3.I5 is where the fun begins. A remarkable construction due to Cohn [6Ia] and an embedding theorem due to Jategaonkar [69] ultimately produce (k, p, a). 3.I5 THEOREM Let E be afield of characteristic 0, p: E--+ E a ring monomorphism and a a p-derivation of E satisfying (a) pa + ap = 0. (b) If x E E satisfies xpn(a) = pn-i(a)x for all a E E where n ~ j ~ I are both arbitrary, then x = 0. (c) There does not exist an x E E such that for some mE N and for all a E E an[pm(a)]

+

xpm(a) = pm+n(a)x

n = I, 2.

(d) There does not exist an x E E satisfying p(x) = x and a(x) = 0 such that for some n E N and for all a E E,

an(a)

+

xa

= pn(a)x.

Then R = E[t; p, a] is a simple pli-domain. Proof Let I be an ideal of R such that 0 c I c R. Then

Simple Noetherian rings

54 I = Rf for some f tradiction. m

.L

Let f =

1=0

R, where f =F 0. The proof will be by con-

E

a 1t 1, where we can assume that am = I.

Case I. m = 2n. Given a EE,

fa

=

{3f for some f3 E E,

(3)

since I is an ideal. Expanding Eq. (3) and equating coefficients we obtain p2n(a) = {3, 2 a2n-1P n-l(a) = f3a2n-l• a2n-18p2n-2(a)

+

a2n-2P2n-2(a) = f3a2n-2• 2n

,2 a181(a)

=

p2n(a)a0.

(4)

1=0

By successive applications of (b) and (c), we obtain a 2n-l = a 2 n- 2 = ... = a 1 = 0. Now

i.e., (t 2n

+ ao)t

t2n+l

+ {3')/

ft

=

(t

=

(t

+ {3')(t 2n +

+ aot

= t2n+l

for some {3'

E

E,

a 0 ). Hence

+ p(ao)t +

8(ao)

+ f3't2n + f3'ao.

Thus {3' = 0 and, consequently,

a0

=

By (4) 82 n(a) + aoa = Thus f = t 2 n. Since

t2na

=

p(a0) and 8(a0 ) = 0. p2n(a)ao for all a E E contradicting (d).

i i=O

(~)p2n-2i[82i(a)]t2n-2i, J

p 2n- 2(8 2(a)) = 0 for all a E E. Therefore, 82(a) = 0 for all a E E, contradicting (c). Case 2. m = 2n + I.

55

Noetherian simple domains Proceeding as above, we obtain p2n+l(a) =

8(p 2n(a))

+

{3,

a2nP 2n(a) = f3a2n•

(5)

2n+l

,2

a181(a) = {3a0 •

1=0

Clearly, Eq. (5) contradicts (c). Thus, no suchf can exist, implying that I = 0 or I = R.

D

Definition of E, p and 8 Let k be any commutative field of characteristic 0 and denote by A, the free associative algebra over k on the free generating set B = {x0 , x 1 , ••• }. Since A is the free associative algebra on B, there exists a unique endomorphism p of A such that p(x0) = -x0

and

p(x1) = xt+ 1

Vi

~

I.

Clearly pis injective. We define

[x1, x 0] = p(x1)x0 - x 0x 1 Vi ~ I. Let k = k(t1n, i E N U {0}, n E N) be the rational function field in the indeterminates t1n over k. Let a be the k-endomorphism of k defined by a(t1n) = tt,n+l for all i EN U {0} and n EN. Let D = k[ Y, a] be the twisted polynomial ring in Y with coefficients in k. Let K be the left quotient field of D. If we set

x= 1

x:

t11 Y

then A ---+K is an embedding (seeJategaonkar [69]). Identify x1 with 1 E K. We shall extend p to K as follows: Define p(ton) = - t0 n Vn EN, p(ttn) = tt+l,n Vi ~ l, and p(Y) = y Clearly, pa = ap and

x

p(tol Y) = - tol Y p(t1, 1 Y) = tt+ 1 , 1 Y Vi ~ I.

Since pa = ap, p extends to a ring monomorphism of K and

Simple Noetherian rings

56

moreover, does what it should with respect to the generators of

A. Let E be the subfield of K generated over k by x 1 for all i ~ I and [x1, x 0 ] for all i ~ I. Define a p-derivation 8 of K as follows: 8(a) = p(a)x0

-

X0a

Va E K.

Clearly, p8 + 8p = 0 and E admits both p and 8. Also note that x 0 and x 0 2 ¢;E. (See Cohn [6la].) The following lemmas show that (E, p, 8) satisfies the hypothesis of Theorem 3.15. Remark Tcq = {a ETc I a( a) = a} = k and

kP

= k(tol2, to22, • • •' ton2• • • • ).

3.16 LEMMA IfaEK andatn 1 Y = tn- 1 , 1 Yawheren- j ~ 0, for all n ~ N > 0, N some suitably large integer, then j = 0

and aE k. Proof Set tn 1 Y = tn Y and assume that a E K is of the form

Now,

and

where Therefore, tn _1 Y a =

(~ tna( a

1)

Y

1

by definition of multiplication inK. If

then, in particular,

)

I(~ c Y 1

1 )

57

Noetherian simple domains Hence,

Thus,

thus, am1a(bm)/a(am1)bm = am(tn-N~\tn)

Vn ~ N.

If m # m 1 or j > 0, let n --+ oo to get the desired contradiction. Hence, m = m 1 andj = 0. Now C0 = a(b 0 ), d 0 C0 = e0 b0 and d0 tna(a0 ) = e0 a 0 tn. Therefore, a(a0 )ja0 = e0 /d 0 =

C0 /b 0

= a(b 0 )/b0 ,

implying that a(a0 jb 0 ) = a 0 jb 0 • Thus, where

a 0 = ab 0

a E k.

(8)

and

Now c1 = ta(b 1 )/a(t)

and

a(b 0 ) =

C0 •

By substituting these expressions in (8) we obtain d0 ta(b 1 )/a(t)

+ d 1 a2 (b

0)

= eobl

+

ela(b 0 ).

Since a0 = ab 0 , we obtain d0 ta(ab 1 )

+ d 1 a(t)a2 (a

0)

= e0 a(t)b 1

+ e1a(t)a(a

(10)

ab 1 ).

(II)

0 ).

Comparing (9) and (10), we obtain t(a(a1 )

-

a(ab 1 )) = (a(b 0 )/b0 )a(t)(a1

-

Equation (II) holds for all n ~ N (remember, t = tn)· If a 1 - ab 1 # 0, letting n--+ oo produces a contradiction.

Simple Noetherian rings

58

Assume a 1 = ab1 for all i :::; m - I by induction. By a process similar to the above (by examining the coefficient of ym+k- 1 in (6) and (7)), we obtain am+k-1(t)ek-1ak-1(abm)

+

am+k-1(t)ekuk(abm-1)

- dkak(t)ak+ 1(abm_ 1) = am+k-1(t)ek-1ak-1(am)

+

am+k-1(t)ekuk(am-1)

- dkak(t)ak+ 1(am_ 1).

Thus, am = abm and hence, a= aE k.

3.17 LEMMA aEk.

D

If aE E satisfies p(a) =a and 8(a) = 0, then

Proof We can assume once again that a=

(i a P)/(I b Y 1

1

1=0

8(a)

1 ),

= 0 implies that t01 a(a) = at01 • Lett =

and at=

(~ a1a1 (t)Y1 )/(~ b

(~ c1 Y 1)t

=I.

bm

I =0

1

=

t 01 ,

Yl

t(~ a(b1)Y1) ,

where c1 = ta(b1)fa1(t) for all 0 :::; i :::; m. Thus,

If

then, in particular, dkak(cm) = ek and dkak(ta(an)) = ek(ak(anan(t)))

59

Noetherian simple domains since

d(~ ta(a1)Y1)

=

e(~ a1a1(t)Y1)

by definition of equality inK. Therefore, ta(an)fanan(t) = tjam(t)

implying that (12) Now,

since p(a) = a. Therefore if

(~f1 P) (~ p(b )P) 1

=

(~ g Y (~ b Y 1

1 )

1

1 ),

Aak(p(bm)) = gkbm

or A = gk since bm = I. Thus, p(an) = an by definition of equality in K. By (12) and the definition of p, m = n and a(an) = an. Thus, an= a E k. Now, dk-1ak-1(cm)

+ dkak(cm-1)

= ek-1ak-1(bm)

+ ekak(bm-d

and dk_ 1ak- 1(ta(am))

+ dkak(ta(am-d) = ek-1ak-1(amam(t))

+

ekak(am-1am-1(t)).

Therefore, ekak+m-1(t)[ak(abm-1 - am-1)] = dkak(t)[ak+1(abm-1 ~ am-1)], ek = dkak(cm)·

Hence Cmam- 1(t)[abm-1 - am-1J = ta(abm-1 - am-1)

or assuming that abm _ 1

-

am_ 1 # 0,

tam- 1(t)jam(t)t = a(abm-1 - am-1)/(abm-1 - am-1).

(13)

Simple Noetherian rings

60 fk-1ak- 1(p(bm))

+ Aak(p(bm-1)) =

Yk-1ak- 1(bm)

+ gkak(bm-1) (I4)

and fk-1ak- 1(p(am))

Yk-1ak- 1(am)

+ Aak(p(am-1)) =

+ gkak(am-1).

Therefore, fkak(p(abm-1 - am-1))

=

gkak(abm-1 - am-1)

and A = gk imply that p(abm-1 - am-1) = abm-1 - am-1 =

Thus,

fJ E JcP.

fJ.

By (I), a({J)a[am- 1(t)] = {Jam- 1(t). Therefore, {Jam- 1(t) = yEkq = k.

Thus am_ 1 = abm_ 1 since p(fJ) = fJ and p(y) = y. Assume by induction that a; = ab; for all I :::; j :::; m.

and k

k

1=0

1=0

,2 d 1a 1(ta(an_ 1)) = ,2 e a (an_ an- (t)). 1

1

1

1

Consequently, dk[ak+ 1(a0

-

=

ab 0 )]

by induction. As before, p(a0 = ab 0 • Therefore, a = a E k.

a0

ek[ak(a0

-

ab 0 ) = (a0

-

ab 0 )] -

ab 0 ) and hence

D

3.I8 LEMMA (E, p, 8) satisfies the hypothesis of Theorem 3.I5. Proof (a) p8 + 8p = 0 is trivially satisfied by construction. (b) If there exists an x E E such that Va E E, xpn(a) = pn-i(a)x

n 2: j 2: I

then in particular, xpn(x1) = XXn+l = xtn+ty = pn-i(x1)x = tn-i+t, 1 Yx,

Vi > 0, contradicting 3.I6.

Noetherian simple domains

61

(c) Note that 8(a) = p(a)x0 - x 0 a and 82 (a) = p2 (a)x0 2 Xo a, Va E E by definition of a and 82 • If

-

2

8(pm(a)] = pm+l(a)x - xpm(x)

where x

E

E

E

E,

E, then pm+ 1 (a)(x 0

Va

Va

x)

-

=

(x0

-

x)pm(a)

E contradicting (b).

A similar argument works for the n = 2 case. (d) If there exists an x E E satisfying p(x) = x and 8(x) such that for some n EN and for all a E E, an(a)

then by 3.17, x

E

+

xa = pn(a)x,

k. Thus, an(a) = x[pn(a) - a]

which just cannot happen.

D

Jacobson domains We complete this chapter with yet another class of simple domains which we call Jacobson domains. If Dis any field with center C, and if the polynomial ring D[x] is primitive, then D Q9 cC'(x) is a simple pli- and pri-domain, not a field. This holds, for example, whenever Dis transcendental over C, that is, whenever there exists dE D which is not algebraic over C. The first proposition is taken from Jacobson [43]. 3.19 PROPOSITION If Dis afield with center C, then every ideal I of the polynomial ring D[x] is generated by a central polynomial p(x) E C[x]. Proof Since R = D[x] has a right and left division algorithm

R is a principal (right and left) ideal domain. Thus I = p(x)R, for some monic p(x) E R of least degree. Then, Vd E D, r(x) = dp(x) - p(x)d is a polynomial in I of lower degree. It follows that r(x) = 0, and so p(x) commutes with every d E D, hence p(x) lies in the center C[x] of R. D 3.20 THEOREM (Jacobson [64a]) If Dis afield, then D[x] is not primitive if and only if every nonzero f(x) E D[x] is a factor

Simple Noetherian rings

62

of a nonzero central polynomial, that is, iff Vf(x) =F 0 there exist g(x) E D[x] and 0 =F p(x) EC[x] withf(x)g(x) = p(x). Proof R = D[ x] is primitive iff there exists a simple faithful right module V, or equivalently some maximal right ideal m(x)R contains no nonzero ideal. Using the last proposition we see that R is not primitive iff every maximal right ideal m(x)R contains a nonzero central polynomial p(x) that is, iff m(x) is a factor of a nonzero central polynomial. However, this is equivalent to stating that every nonzero f(x) is a factor of a nonzero central polynomial, since f(x) a product of irreducible polynomials m 1 (x), ... , m1(x), and every irreducible polynomial m(x) generates a maximal right ideal. D

In the next theorem, C(x) is the field of rational functions over C. 3.21 THEOREM Let D be a field with center C. Then A = D ® cC(x) is a simple pli- and pri-domain embedded in the right quotient field Q of D[ x ]. Moreover A = Q iff D[x] is not primitive. In particular, whenever D is transcendental over C, then A is not a field. Proof We know that Q exists since D[x] is a right Ore domain. (Any polynomial ring over an Ore domain is an Ore domain.) Furthermore A embeds in Q under the mapping

f(x) ® p(x)- 1 r-+f(x)p(x)- 1

defined for all f(x) D[x], then

E

D[x], p(x)

E

C[x]. If A = Q, and if f(x)

E

f(x)-1 = g(x)p(x)-1

for suitable g(x)

E

D[x] and p(x) EC[x], so f(x)g(x) = p(x)

is a factor of a central polynomial. Conversely, if the latter holds for any f(x) E D[x], then A contains f(x)- 1 = g(x)p(x)- 1 for any f(x) =F 0, so that A 2 Q, and hence, A = Q. We complete the proof by showing that D[x] is primitive whenever Dis transcendental. Lett ED, and suppose that t(x

+

l)h(x) = p(x)

Noetherian simple domains

63

is a factor of a central polynomial p(x) # 0. Write p(x) = a 0 h(x) = d 0

+ a 1x + ... + anxn, + d 1x + ... + dmxm,

where a 1 E 0, d; ED, Vi,j. Then, m = n - I, and comparing coefficients, we see that ao =do, d0 t d1 t dn-2t

+ d1 + d2

= a 1, = a 2,

+ dn-1

= an-1• dn-1t = an.

Then, solving recursively for dt> i = 0, ... , n - I, yields dn_ 1 as a polynomial of degree n - I in t with coefficients in C; d 1 = -a0 t - a 1 , d 2 = -a0 t 2 - a 1t - a 2 , ••• , and Thus, a 0 tn

+ a 1tn- 1 + ... + an_ 1t + an

= 0

so that t is a zero of a nonzero central polynomial, that is, t is algebraic. Using 3.20, we see that whenever D[x] is not primitive, then D must be algebraic. D

4. Orders in simple Artin rings

In this chapter we study rings which are right orders in simple Artin rings and explicitly determine these via the Goldie and Faith-Utumi Theorems. We also present some further results concerning representations of semiprime maximal orders. Although the rings considered are more general than simple Goldie rings, the conclusions and the techniques of proof are invaluable in the structure theory of simple Goldie rings. Recall that a right ideal I of R is called nil if for all x E I, xn = 0 for some n > 0. The identity (ra)n+ 1 = r(ar)na for elements a, r E R shows that Ra is a nil left ideal iff aR is a nil right ideal. Thus, R has no nil right ideals =F 0 if and only if R has no nil left ideals =F 0. LetS be a ring satisfying the maximum condition for right annulets. If S has a nonzero nil right or left ideal A, then S contains a nonzero nilpotent ideal. Proof (Utumi [63]) By the preceding remark, we can assume A is a nil left ideal =F 0. Let 0 =F a E A be such that a.L is maximal in {x.L I 0 =F x E A}. If u E Sis such that ua =F 0, then (ua)n = 0 and (ua)n- 1 =F 0 for some n > I. Since [(ua)n- 1 ].L 2 a.L, and since uaEA, it follows that [(ua)n- 1].L = a.L. Since ua E [(ua)n- 1 ] \ then aua = 0. Thus, aSa = 0, and then (a) 3 =0, where (a) is the ideal of S generated by a. D 4.1 LEMMA

4.2 LEMMA Let R be any ring which satisfies the (a.c.c.).L and (a.c.c.)Ei). If a E R there exists n > 0 such that anR + an.L is an essential right ideal. Proof There is an n 2: I such that an.L = an+ u. Then an R (") an.L = 0. Let I be a right ideal and suppose that

+ an.L) = nJ + ... is

I (") (anR

0.

Then the sum I + an] + a direct and, because R has (a.c.c.)Ei), we conclude that I = 0. D 2

[ 64 ]

Orders in simple Artin rings

65

As an immediate consequence of 4.2 we have that if x satisfies x.L = 0, then xR is an essential right ideal of R.

E

R

4.3 DEFINITION A ring R is said to satisfy the right Ore condition if given a, bE R, b regular, 3a1 , b1 E R, b1 regular, such that ab 1 = ba1 • It is easy but somewhat tedious to show that R has a right quotient ring if and only if R satisfies the right Ore condition. (See, for example, Jacobson [64a], or Faith [73a], Chapter 9.) Recall that Z(R) = {x E R I x.L is an essential right ideal of R} is a right ideal of R, called the right singular ideal of R. It is easy to show that Z(R) is in fact, a two-sided ideal of R. The next theorem provides necessary and sufficient conditions for R to have a classical right quotient ring which is a (simple) semisimple Artin ring. 4.4 THEOREM (Goldie [58, 60], Lesieur-Croisot [59]) A ring R has a semisimple right quotient ring if and only if R is semiprime and satisfies (a.c.c.)® and (a.c.c.).L. Proof (Goldie [69]) We first assume that R is a semiprime ring satisfying (a.c.c.)® and (a.c.c.).L, and prove that an essential right ideal E contains a regular element. E is not a nil ideal by 4.1, so it has an element a 1 # 0 with a/ = a 1 2 .L. Either a 1 .L (") E = 0 or a 1 .L (") E # 0. In the latter case, choose a 2 E a 1 .L (") Ewith a 2 # 0 and a 2 .L = a 2 2 .L. If a 1 .L (") a 2 .L (") E # 0, then the process continues. At the general stage we have a direct sum

a1 R

E8 ... E8 akR E8 (a/ n ... n ak.L n

E),

where ak E a 1 .L (") ••• (") ak _ 1 .L (") E, ak # 0 and ak .L = ak 2 .L. The process has to stop, because R has (a.c.c.)Ei). Let this happen at the kth stage. Then

a/ n ... n ak.L n E

= 0

=a/ n ... n ak.L

and hence (a12

+ ... +

ak 2).L

=a/ n ... n ak.L

= 0.

Let Z be the singular ideal of R and z E Z. Then zn R ® zn.L is

66

Simple Noetherian rings

an essential right ideal for some n > 0 and zn.L is also essential. Hence zn R = 0. Thus, Z is a nil ideal and hence is zero. Set c = a 12 + ... + ak 2 E E; as c.L = 0 we deduce that cR is essential by 4.2. Hence .Lc s;; Z, so that .Lc = 0 and c is a regular element. This establishes the existence of regular elements in R. Suppose that a, d E R with d regular and E = {x E R

I axE dR}.

Then dR is essential and hence so is E, so that E contains a regular element d 1 • Thus, the right Ore condition is satisfied and R has a right quotient ring Q. Next suppose that F is an essential right ideal in Q, then F (") R is essential in R. Now F (") R has a regular element, which is a unit in Q, and hence F = Q. Let J be a right ideal and K a right ideal of Q such that J (") K = 0 and J E8 K is essential (use Zorn's lemma); then J E8 K = Q. Thus, the module QQ is semisimple and Q is a semisimple ring. Conversely, let R have a semisimple right quotient ring Q. Then a right ideal E of R is essential if and only if EQ = Q. To see this, suppose I is a nonzero right ideal ofQ, then I(") R # 0 and I (") R (") E # 0, taking E to be essential in R. Hence I(") EQ # 0, which means that EQ is essential in Q; then EQ = Q as QQ is a direct sum of simple modules. On the other hand, when EQ = Q is given and I is a nonzero right ideal of R then IQ (") EQ # 0, trivially, and hence I (") E # 0, so that E is an essential right ideal of R. These conditions are equivalent to saying that E has a regular element, because whenever IE EQ, I = ee-l, e E E, c E Rand regular. On the other hand, when E has a regular element, then EQ = Q and E is essential in R. We now conclude that R is a semiprime ring, because if N is a nilpotent ideal of R, .LN is essential as a right ideal and hence, has a regular element. Thus N = 0. Let S be a direct sum of nonzero right ideals {Ia I a E A} of R which is essential as a right ideal. S has a regular element c, expressible as a finite sum

67

Orders in simple Artin rings

Now cR is essential and lies in Ia 1 + ... + I«n; it follows that A has only the indices a 1 , ••• , an, and R has (a.c.c.)Ei). Finally, the maximum condition holds for right annihilators, because R is a subring of the Noetherian ring Q. D 4.5 CoROLLARY A prime ring R has a simple Artinian right quotient ring if and only if R satisfies (a.c.c.)® and (a.c.c.).L.

D A set M = {e11 I I :::; i, j :::; n} of elements of R is called a set of n x n matrix units of R if

and n

,2 e11 =I 1=1

for all i,j, k and l. (8 1k is the Kronecker delta.) Faith-Utumi theorem

IfF is a right order in a ring D, then then x n matrix ring F n is a right order in Dn, and so is any ring R such that F n s;; R s;; Dn. The theorem below states that if D is a semisimple ring, every order R of Dn is obtained this way. 4.6 FAITH-UTUMI THEOREM[65] LetRbeanysubring-I of a matrix ring S = Dn, where D = centralizer 8 M of a set M = {e11} 1~1 = 1 of matrix units. Let

I aM

s;;

R}

B = BM = {bE R I Mb s;;

R}.

A = AM = {a E R

and Then n

BA = (BA n D)n =

_2

(BA n D)e11

t.J= 1

is a subring-I of R, AB is an ideal of R, and (BA) 2 s;; AB. Furthermore, F = BA (") D and U = B (") D = A (") D are

68

Simple Noetherian rings

ideals of P = R n D such that F 2 U 2 • Thus, R 2 BA = Fn 2 Un2• If Dn is a right quoring of R, then M can be so chosen so that A contains a unit a of S, B contains a unit b of S, and BA = Fn is a right order of Dn equivalent to R and to P n· If R is also a left order of Dn, then any choice of matrix units M has these properties. When D is a semisimplet ring and Dn is a right quoring of R, then D is the right quoring of F, and Dn = {af- 1 I a E F n• regular f E F}. Proof Since AP s;; A, P B s;; B, and P s;; D, then P F s;; F, and F P s;; F, so that F is an ideal of P. Since B is a right ideal of R, then U = B (") D is a right ideal of P = R (") D, and U = A (") D is a left ideal of P. Furthermore, U 2 F = BAnD 2 (B n D)(A n D) = U 2 • Now, every element t E Dn has a unique representation n

t =

.L

tlieli, where

1,1=1

n

tli =

2: e ktek 1

1

E

D.

k=1

Thus, if t = ba E BA, bE B, a E A, then tii E BA (") D = F since M B s;; B and AM s;; A. This proves that BA = F n· Moreover, (BA) 2 = BABA s;; AB, since A is a left, and B is a right ideal. Now assume that Dn is a right quoring of R. First, let M' be any set of n x n matrix units of Dn, with centralizer D'. Clearly, there is a common denominator a in R of the elements in M', that is, a regular element a E R such that M'a s;; R. Then,

M = a- 1 M'a is a set of matrix units of S, and D = a- 1 D'a = centralizer 8 M

is a ring isomorphic to D'. Furthermore, a is a unit of S contained in A = {x E R I xM s;; R}.

t

This holds more generally for a.ny semiloca.l quoring Dn (Faith [71]). The original theorem of Fa.ith-Utumi [65] was stated for a. field D. (Some of 4.6 is taken from Faith [65].)

Orders in simple Artin rings

69

Moreover B = {y

E

R

I My s;; R}

also contains a unit b of S. We now show that any set M of matrix units of S, for which A and B just defined each contains a unit of S, has the stated properties. Since every set M of matrix units satisfies this requirement if R is also a left order in S, then every set of matrix units has the stated properties in this case. Since AB is an ideal of R, and since AB contains a regular element t = abE R, then Dn is a right quoring of AB, and every y E Dn has the form y = xt(ct)- 1 , for elements x, regular c E R. Since bSb - 1 = S, where S = Dn, every element yES has the form y = by1 b- 1 , fCir some y 1 = xr- 1 ES, where x, regular rE R. Thus,

with bxa E BA and braE BA. This proves that Dn is a right quoring of BA. Now bRas;; BA, and BA s;; R, so that BA is a right order of Dn equivalent to R. Moreover, since F is an ideal of P, then Fn is an ideal of Pn containing a regular element t. Thus, tPn s;; Fn, so Fn is equivalent to Pn. Assume that Dn is a semisimple right quoring of F n· Since semiprime right Goldie rings are Morita invariant, F is semiprime right Goldie and, hence, F has a semisimple right quoring Q. (IfF is a subring-1 of D, then the subring F' generated by F is semiprime right Goldie, with semisimple right quoring Q, and then Q is the right quoring of F.) Since every regular element f E F is regular in F n (identifying f with diag f), f - 1 E Dn> and in fact, f- 1 ED. Q can be embedded in D by a mapping which induces lp. Then, F n is a right order in Qn, and

However, Qn is semisimple, and hence every right regular element of Qn is a unit, so Qn is a right quoring of F n· This proves Qn = Dn, so that Q = Dis the right quoring of F. D

Simple Noetherian rings

70 Principal ideal rings

If R is a right order in Dn, the theorem of Faith and Utumi states that there exists an order F in D and an inclusion F n s;; R s;; Dn. Even though Fn is an order in Dn, the ring structure of R does not depend entirely on F n· Under a stronger hypothesis, however, it does.

4.7 PRINCIPAL RIGHT IDEAL THEOREM (Goldie [62]) If R is a prime ring and a principal right ideal ring, then there exist a right Ore domain F, an integer n > 0, and an isomorphism R::::::: Fn. Proof (Faith-Utumi [65]) Using the notation of 4.6, we can write B = cR, for some c E R. Since B contains a regular element, cis a regular element. Since e11 B s;; B, that is, e11cR s;; cR, then c- 1e11c = f 11 E F (i,j = I, ... , n). Now M = {111 I i, j = I, ... , n} is a complete set of matrix units in S, with associated field H = c - 1 De. Since M s;; R, R = n

.L f 11 K,

where K = H (") R. Since R is an order in Hn, then

1.1=1

K is an order in H; that is, K is an Ore domain.

D

F need not be a principal ideal ring (cf. Swan [62]). Nevertheless, F is right Noetherian and hereditary, since these properties are Morita invariant.

Supplementary remarks (I) Let R be a right order in the ringS = Dn, n > I. Iff is any idempotent of S such thatfS is a minimal right ideal, thenf = ac- 1 , with a, c E R. Thus e = c- 1fc is idempotent and 0 # ce E Se (") R. Since Se (") R # 0, it follows from primeness of R that (eS (") R)(Se (") R) # 0, so that eSe (") R # 0. Then the proof of 4.6 shows that F = eSe (") R is a right Ore domain, and D : : : : eSe is its right quotient field; we thus obtain that S : : : : Dn, where D contains a right Ore domain K, which is contained in R. This illustrates the precise nature of 4.6 which states much more. (2) Next we show that, in general, not every set M of matrix units has the property without the hypothesis that R is also a left order in S.

Orders in simple Artin rings

71

Let K be a right Ore domain with right quotient field D that is not a left Ore domain. Let x, y be nonzero elements of K such that Kx (") Ky = 0, and let R = (Kx Kx

Ky)· Ky

!) with a, c Kx, and !) is an arbitrary

[R is the set of all 2 x 2 matrices (;

E

b, dE Ky.] Since K is right Ore, if A = (;

element of D 2 , there exists q E K, q # 0, such that aq, bq, cq, dq E K, and then

(ac db) (qx0 qy0)

B =

(qx0

Thus, A = BC- 1 , with B, C =

E

0 ) qy

R. E

R. Hence, R is a

right order in D 2 • As in the theorem, identify D with the subring of D 2 consisting of all scalar matrices ( ~

~)

with k E D. Now assume

for the moment that R contains a subring F 2 , where F is a domain £D. The contradiction is immediately evident since the form of R, R = (Kx Kx

Ky), Ky

where Kx (") Ky = 0, precludes the possibility of its containing a nonzero scalar matrix

(~ ~)

with dE K.

D

(3) A ring R, as in 4.6, need not contain nontrivial idempotents. Perhaps the simplest example is as follows: LetS = 0 2 be the ring of all 2 x 2 matrices over the rational number field Q,

and let R be the subring consisting of all matrices (;

!) ,

where b, c are even integers, and a, d are integers which are either both even or both odd. Then R = (2Z:) 2 + l is not an

72

Simple Noetherian rings

integral domain. However, R has classical quotient ring S but does not contain idempotents # 0, I. Maximal orders Throughout this section, all rings will be at least right orders in semisimple rings, i.e., semiprime right Goldie rings. The term order will be reserved for a semiprime two-sided Goldie ring. Recall that a right order R is maximal (left-equivalent, rightequivalent) provided that R s;; R' s;; Q and R ,.., R' (R ..!.. R', R ..!:.. R') imply that R = R'. 4.8 PROPOSITION (a) Let R, S be right orders with R s;; S, R ,.., S. There exist right orders T, T' such that

R s;; T s;; S, R s;; T' s;; S,

R ..!.. T

..!:..

S,

R ..!:.. T' ..!.. S.

(b) R is maximal iff R is a maximal left- and right-equivalent order. Proof Consult Faith [73a], p. 413-4. D

4.9 PROPOSITION A simple Goldie ring R is a maximal order inS. Proof SupposeS 2 R and Sa s;; R for some regular element a E R. Clearly, SaR = R by simplicity of R. Since 8 2 = S, SR = S(SaR) = S 2 aR = SaR = R. However, SR = S. Thus, S = R and hence R is a maximal left-equivalent order. Similarly, R is a maximal right-equivalent order and hence, by 4.8, a maximal equivalent order in Q. D

The essential argument used here was first used by Faith [65] who showed, interalia, that center R coincides with center Q. 4.10 PROPOSITION Let R be a maximal right order, MR a finite dimensional torsionless generator and k = End M R· If k' 2 k and k' :.., k, then R = End k'k' ® kM. Proof Set M' = k' ® kM and R' = End k'M'. Since k'M' is finitely generated and projective, M~. is a generator and hence, balanced, i.e., if k" = End M~., R' = End k.M'. Moreover, if kM = kne, k'M' = k'ne and R' = ek~e imply-

Orders in simple Artin rings

73

ing that R' is an order in the quotient ring of R. Thus, End M~, = End M~. Since ak' s; k, where a is a regular element of k, a induces a monomorphism M~-+ MR implying that M~ is finite dimensional, torsionless and hence, T = trace M~ =F 0. Since M~ is clearly faithful, T R' s; R by 1.6. However, T R is essential since M~ is faithful and torsionless. Thus, T contains a regular element of R implying that R' ..!:.. R and hence R' = R by maximality of R. D 4.ll PROPOSITIONt Let R be a maximal order, MR a finite dimensional torsionless generator, k = End MR, M* = homR(M, R) and M# = homk(M, k). Then (a) End M#k = End M*k = R (b) k' = End RM# = End RM* is an essential extension of k and hence, an equivalent order in the quotient ring of k. (c) dim RM* = dim M R (d) M* : : : : M***. Proof (a) Set R' = Biend RM# =End M#k and T = trace RM#. By 1.6, R'T s; R and since RM# is a generator, T = R. Thus, R' = R. (b) Since k' = Biend M#k, Tk' s; k where T = trace M#k. Hence, k' is an essential extension of k and, since Tis an essential right ideal of k, T contains a regular element of k and thus, k',.!:., k. (c) By 1.18(1), dim MR = dim kk and dim RM* = dim k'k'. Since both k and k' are orders and kk' is an essential extension of kk, dim kk = dim kk' = dim k'k' as claimed. (d) By (c), dim RM* = dim M**R = dim RM***. Since RM* is isomorphic to a direct summand of RM*** (see Jans [64], p. 67), we have RM* : : : : RM***. D

The next lemma allows us to assume in many instances that the module in question is a generator (Jategaonkar [71]). 4.12 LEMMA Let R be a maximal right order, M a finite dimensional, torsionless, faithful R-module and k = End M R· Then k is a maximal (right-equivalent, left-equivalent) order when-

t More generally, 4.11 holds for finite dimensional, torsionleBB, faithful M 8 • The proof is almost identical.

Simple Noetherian rings

74

ever End(M E8 Rh is a maximal (right-equivalent, left-equivalent) order. Proof Viewing the elements of M E8 R as column vectors, k- = End(M

E8 R)R =

( k M*

M) R

and its quotient ring JJ is Jj

= End((M E8 R) ® RQ)Q = ( End(M ®

RQ)Q homQ(M ® RQ, Q)

where Q = the quotient ring of R. Furthermore, D = End(M ® RQ)Q = the quotient ring of k (see Zelmanowitz [67]). Suppose Tc is maximal and k' 2 k with k' ""' k, say, ak'f3 s; k with a and f3 regular elements of k. Set k'

S

k'M )

-

= ( M*k' M*k'M + k.

Sis clearly an order in Jj containing Tc. Since

Tc(~ ~)TesTe(~ ~)Tc s; Tc (routine computation), S ""' Tc if the ideals

Tc(~ ~)Tc

Tc(~ ~)Tc

and

contain regular elements of Tc. To demonstrate

this, we shall consider only the first ideal. Since

Tc( ~ ~) Tc

contains all matrices of the form kak ( 0

0 ) [M*, aM]

such a regular element exists provided the ideal I = [M*, aM] is essential as a right ideal of R; equivalently, Ir = 0 ~ r = 0. If Ir = 0, aMr = 0 since M is torsionless. Thus, Mr = 0 (a is regular) ~ r = 0 since M is faithful. Consequently, I is essential as asserted. The proofs of the parenthetical statements are special cases of the above. D

Orders in simple Artin rings

75

When R is prime, Hart-Robson [70] have shown that eRne is maximal for all idempotents e E Rn and n 2: I. The argument used in the proof of 4.12 is essentially theirs. 4.13 THEOREM Let R be a maximal order and M a finite dimensional, torsionless, faithful R-module. Then End RM* = m is a maximal order and is the maximal right equivalent order containing k. Proof It suffices to show that whenever k' 2 k = End M R and k' ..!:.. k, k' s;; m and if k' 2 m with k' ..!.. m, k' s;; m. For if this condition is satisfied and k' 2 m with k' ""' m, k' ""' k and by 4.8 there exists an orders with k' 2 s 2 k, k' ..!.. s and s ..!:.. k. By assumption, s s;; m and since k' ..!.. s, k' ..!.. m. Thus, k' s;; m, and m is maximal as claimed. To show that the above condition is fulfilled, we can clearly assume that M is a generator by 4.12. If k' 2 k with k' ..!:.. k, set M' = k' ® kM. By 4.10, R =End k'M' and clearly, homk.(M', k')--+ homR(M', R) is a monomorphism. Since MR is a generator, R =End kM and M#::::::: M*. Since M~ is an essential extension of IJJR, homR(M', R) 4 homR(M, R). One readily checks that the composition of maps

is the map defined by

f--+ JIM (restriction to M) Vf E homk,(M', k'). Thus, any k-linear map f: M --+ k' is actually a map into k since f induces a k' -linear J: M' --+ k' such that /I M = f. Next, if T = trace kM, Tk' s;; k. For, if (u)f E T, a' E k', fa' E homk(M, k') = homk(M, k) by the above remarks. Hence, (u)fa' E k. Finally, M#Tk' = M#k' s;; M# which implies that k' s;; m. In order to complete the proof, suppose that k' 2 m with k' ..!.. m. Then by an obvious modification of 4.10, R = End(M# ® mk')k'· By modifying the above argument, we obtain homm(M#, k') = homm(M#, m) and hence, k'T# s;; m where

76

Simple Noetherian rings

T# = trace M#m· Since m = End M**R (see the proof of (a) (c) in 4.14) and T#M** = M**, k' s;; mas claimed. D

~

4.14 MAXIMAL ORDER THEOREM Let R be a maximal order and M a finite dimensional, torsionless, faithful R-module. Then (a) ~ (b)~ (c), where (a) M R is R-reflexive, (b) k = End M R is a maximal order, (c) k = End RM*, where

M* = homR(M, R). Moreover, if M is a generator, then (b) ~ (a). Proof We shall first assume that M is a generator and prove the stated equivalences for this case. Then we shall discuss how the general case can be reduced to the case of a generator.

(a)

~(c):

Since M = M**, k = End MR = homR(M, M**) = homR(M*, homR(M, R)) = EndRM*.

The third equality follows from one of the standard functorial isomorphisms (e.g., see Cartan-Eilenberg [56], ex. 4, p. 32). (c)~ (b): Immediate by 4.ll and 4.13. (b) ~ (a): By 4.ll, k' = End RM# = k. Since M#::::::: M* and homk.(M#, k')::::::: homR(M*, R) = M**, M## : : : : M** as (k, R)-bimodules. By finite projectivity of kM, M## : : : : M as (k, R)-bimodules. Hence, M is R-reflexive as asserted. If M is no longer assumed to be a generator, note that M is reflexive iff M = M E8 R is reflexive, and End M R is maximal only if End M R is maximal. Moreover, M is always a generator for arbitrary M. Thus, (a) ~ (b), and (b) ~ (c) follows immediately from 4.ll(b). If k = End RM*, k is maximal by 4.13. Consequently, (c) ~ (b). D Remark To see that (b)

~

(a) does not hold in general, let

R be any maximal order having a faithful ideal I which is not reflexive as a right ideal and set k = End I R; e.g., any ndimensional regular local ring with n ;;:: 2. These are maximal

Orders in simple Artin rings

77

orders by Auslander-Goldman [60] and must have nonreflexive right ideals by Mattis [68]. Since k 2 R and k,.., R, k = R. Thus, I R is not reflexive but k is maximal. In order to generalize the above to maximal right orders, care must be exercised to insure that all of the endomorphism rings in question are right orders in the appropriate semisimple ring. For example, if MR is a finite dimensional torsionless generator, there is no guarantee that k' = End RM* is a right order in the quotient ring of k. However, when R is prime and M R reflexive, we can conclude that k is maximal. Specifically, 4.15 PROPOSITION Let R be a prime maximal right order, M R a finite dimensional reflexive R-module and k = End M R· Then k is a maximal right order. Proof Since 4.12 is valid for right orders, it suffices to assume that M R is a generator. Moreover, the proof of 4.13 shows that if k' . !:.. k or k' ..!.. k, Tk' s;; k provided that R = End k'k' ®k M. This, of course, implies that k is maximal. Thus, it suffices to show thatifk'..!.. k, R = Endk,k' ®kM(4.10coversthecasek'..!:.. k). To that end note that for some n > 0, R' = End k'k' ®k M : : : : ek~e and R : : : : ekne. Since k' a s;; k, ek~eknekna.ekne £ ekne where a = ai n• In the n X n identity matrix. Since ekne is prime, the ideal eknekna.ekne is essential and hence, contains a regular element of ekne. Thus, ek~e . !:.. ekne ~ ek~e = ekne by maximality of R(ekne). D The question of whether one-sided versions of 4.13 and 4.14 hold for general semiprime R remains open. If k' ..!.. k always implies ek~e ..!.. ekne, then, of course, the above argument shows that finite dimensional, faithful, reflexive modules have maximal endomorphism rings. As an interesting aside, note that the proof of 4.15 shows that m = kT, the (left) quotient ring of k with respect to the smallest filter of all left ideals which contain T = trace kM. The case of global dimension at most 2

In order to show that reflexive generators, specifically in the case of maximal orders, are bonafide generalizations of finitely

78

Simple Noetherian rings

generated projective generators, we shall show (4.17) that if each finitely generated reflexive module is projective, then the global dimension of the ring cannot exceed 2 (the converse is well-known). Then, if U R is any finitely generated reflexive nonprojective (necessarily r.gl.dim R > 2), U R E8 RR is a nonprojective reflexive, generator. For simplicity, we shall assume that R is a semiprime two-sided Noetherian ring. To prove the above assertion we need the following characterization of twosided Noetherian rings of global dimension ~ 2. 4.16 PROPOSITION (Bass [60]) If R is left and right Noetherian, the following are equivalent: (a) r.gl.dim R ~ 2. (b) The dual of any finitely generated right R-module is projective 0 Let MR be any finitely generated R-module and t(M) its torsion submodule. We can assume that MR is a generator by theremarks preceding the conclusion of the proof of 4.14. Clearly, M* = (Mjt(M))* since (t(M))* = 0. Thus, we can further assume that M is a torsion free and hence, a torsionless, finitely generated R-module. Since M R is clearly finite dimensional, RM* : : : : RM*** and assuming reflexive modules are projective, M* is projective. Consequently, by 4.16, r.gl.dim R ~ 2. Thus, we have shown that 4.17 PROPOSITION The following conditions are equivalent for any semiprime two-sided Noetherian ring R. (a) r.gl.dim R ~ 2. (b) each finitely generated reflexive module is projective. 0 A canonical and important example of a reflexive module is given by 4.18 PROPOSITION Any maximal uniform right ideal of R (more generally, any right annulet) is reflexive. Proof By Bass [60] U = X* for some finitely generated left R-module X. Since we can assume that X*R is a generator, dim X*R =dim X***R and consequently, U =X* =X*** =

U**.

0

Orders in simple Artin rings

79

By parroting an argument used in Proposition 2.26 we readily obtain the necessity of the following characterization of simple Noetherian domains having global dimension less than or equal to 2. 4.I9 THEOREM (a) Let R be a simple two-sided Noetherian domain. Then r.gl.dim R :::; 2 iffVn ~ I, whenever B is a maximal order and B ~ Rn, B is simple. (b) If R is simple Noetherian with r.gl.dim R :::; 2, MRfinite dimensional and torsionless, then k' = End RM* is a simple Noetherian ring Morita equivalent to R. Proof (b) Since M R is finite dimensional and torsionless, M R 4 R for some n ~ I, implying that T = trace M R # 0. By simplicity of R, T = R and hence M R is a generator. Therefore, by 4.I6 RM* is a progenerator and hence, k' is Morita equivalent to R. (a) (Necessity): We shall present only then = I case. The general case is identical. Suppose xBy s; R. Set a = xy and B' = xBx- 1 • Then B' a s; R and hence I = B'aR is a right ideal of R satisfying B' s; End I R· Since End I R ,.., R, B' = End IR by maximality of B' (B' maximal iff B is). Moreover, B' = End RI* by 4.Il. Since r.gl.dim R :::; 2, RI* is a progenerator implying that B' and hence, B is simple. (Sufficiency): By 4.I7 it suffices to show that each finitely generated, reflexive R-module is projective. To that end suppose I R is finitely generated reflexive and dim I R = n. Then clearly R 4 I R and since dim RI* = n, I R 4 R. This readily implies that B = End IR,.., Rn = End RR. B is a maximal order since I R is a reflexive generator and hence, simple by hypothesis. Finally, by simplicity of B, I R is projective, completing the proof of 4.I9. D

As a final remark regarding uniform right ideals we have If R is a simple two-sided Noetherian ring satisfying r.gl.dim R :::; 2 and each uniform right ideal is reflexive, then R is hereditary. Proof We can assume R is a domain by 4.I9. Since each right ideal is now reflexive, each right ideal is projective by 4.I7. 4.20

REFLEXIVE

IDEAL

THEOREM

80

Simple Noetherian rings

Thus, even in such well-behaved rings, one cannot expect to find too many reflexive uniform right ideals.

The nonreflexive case The object of this section is to provide further insight into the structure of the endomorphism ring of a finite dimensional torsionless R-module M, in particular, a basic right ideal of R, and to sharpen Faith's representation theorem mentioned in the introduction. In 4.21 we shall assume that R is simple. While 4.21(b) is 4.13 in a less general setting, its proof is an interesting application of the correspondence theorem. 4.21 PROPOSITION Let R be a simple order, MR a finite dimensional torsionless R-module and k = End M R· Then (a) k is a maximal left equivalent order: (b) m = End RM* is the maximal equivalent order containing k. Proof (a) MR is necessarily a generator for mod-R. LetT = trace kM. If k' 2k with k' ..!.. k, say k'a s;; k, then k'aT = T = k' T since T is the least ideal of k. Thus, k' T M = k' M s;; M and hence k' s;; k. (b) If k' 2 k with ak' s;; k, Tak' = T = Tk' ~ k' s;; m since M*T = M*. Thus, any equivalent order containing k is contained in m implying that m is the maximal such order. D 4.22

PROPOSITION

(a) (Faith [64]) Any maximal order in

Q = D n• D a field, is isomorphic to the biendomorphism ring of any right ideal U, say R : : : : End k U where k = End U R· (b) If URis a maximal uniform right ideal, then k = End U R is a maximal order in D. Proof (a) Let U be any right ideal of R. By 1.6, T = trace U R satisfies T R c R where R = Biend U R· Since T contains a regular element of R, R is a right order of Q which is ..!:.. to R. Thus, R = R by maximality of R. (b) Clear. D Examples In this section we show that if A = End 8 V is simple with 8 V finitely generated projective and B right Ore, B can be quite 'bad' when A is quite 'good'.

Orders in simple Artin rings

81

In the following examples, we shall freely use results from Robson [72] and certain generalizations which can be found in Goodearl [73]. These references will be abbreviated to [72] and [73] respectively. ExAMPLE I For any n 2: 2, there exists a right but not left Noetherian domain B with l.gl.dim B = r.gl.dim B = n, a finitely generated projective left B-module 8 V such that A = End 8 V is a simple pri(pli)-domain. Let k ::::> k' be fields with l.dim k'k = oo and r.dim kk' finite (see Cohn [6Ia] for such examples). Set A = !!)k<x> where S is defined in the obvious fashion: S(x) = I, S(a) = 0

Va

E

k.

By Example 7.3 in [72], the subring B = k + SA of A is a hereditary Noetherian domain. By Theorem 4.3 of [73], B' = k' + SA is a two-sided order in A with l.gl.dim B' = r.gl.dim B' = 2. Moreover, B' is right Noetherian and not left Noetherian. Finally, set V = SA. First, Vis finitely generated and projective in B'-mod. For V* 8 , 2 A and A V = A. Thus, IE V*V implying finite projectivity of 8 , V by the dual basis lemma. Moreover, End 8 , V is an equivalent order containing A and hence, A = End B' V. By an obvious modification of this technique, we can choose B' such that B' is a two-sided order in A, B' is right but not left Noetherian and l.gl.dim B' = r.gl.dim B' = n. Moreover, B' can be chosen to be neither right nor left Noetherian but still a two-sided order, having any prescribed global dimension 2: 2. ExAMPLE 2 (Real Pathology) Let R be a pri-domain, D its right quotient field and suppose that DJR is semi simple in mod-R. Then there exists a right hereditary, right Noetherian domain B, which is, however, a two-sided order, a finitely generated projective 8 V such that A = End 8 V is a simple pri(pli)-domain.

82

Simple Noetherian rings

Set S = !!)D<x> where a is the standard formal derivative of D(x), T = D + as and k = R + as. As before, T is a hereditary Noetherian domain. Also k £ S since as c k. Thus, k is a two-sided order in F = right quotient field of S. We claim that k is right hereditary. First, one readily verifies that the exact sequence of k-modules 0-+ TjaS -+SjaS ~ SJT-+ 0

is split exact (the obvious map is the desired right inverse to 1r) and hence, pd(SjT)k ~ pd(SjaS)k· Since Sk is projective by Lemma 2.I in [72], pd(SjaS)k ~ I and hence, pd(SjT)k ~ I, implying that Tk is projective. Since k is an order in T, it is easy to see that T ® kT = T. As an immediate consequence, we have pd Mk = pd MT for all ME mod-T by a proof similar to that of Lemma 2.8 in [72]. Moreover, since DJR is semisimple in mod-Rand isomorphic to Tfk, Tfk is semisimple in mod-R and hence, in mod-k. Finally, let I be any right ideal of k. Then IT is projective in mod-k and ITJI is a direct summand of a coproduct of copies of Tfk. Since Tk is projective, pd Tfk ~ I implying that pd ITJI ~ I. Hence, pd I k = 0. Thus, k is right hereditary and since it is an order in a field, k is right Noetherian. Note that if we set V = as, S = End k V. We also remark that such rings as R exist. For example the rings discussed in Chapter 5 work. The existence of an example principal on the right but not on the left remains an open problem. However, we feel very strongly that such examples do exist. Note that R in Example 2 does not have to be left Ore! Finally, if such an R exists (which is not left Noetherian), then k affords us with an example of a two-sided order which is right hereditary but not left hereditary, thus answering a question raised in Camillo-Cozzens [73].

5. V-rings

In 1956, Kaplansky showed that a commutative ring A is Von Neumann regular iff each simple A-module is injective. Thus, a very natural and important class of commutative rings could be completely characterized by their siiP_ple modules and a fundamental homological property, namely, injectivity. However, this property does not characterize noncommutative regular rings, since, as Faith showed, some regular rings have injective simple modules, while others do not, whereas Cozzens [70] constructed simple pH-domains, not fields having simple modules injective. We present these examples in this chapter. We also show that any right Goldie V-ring is a finite product of simple V -rings. Dual to the notion of projectivity and of interest to us in this and succeeding chapters will be the concept ofinjectivity. For a proof of 5.1 and the statements regarding the injective hull, consult Faith [73a] or Lambek [66]. 5.1 PROPOSITION AND DEFINITION A module MR is called injective if it satisfies any of the following equivalent conditions: (I) The functor hom R(- , M) : mod- R "-" Ab is exact. (2) Every exact sequence 0 -+ M

-+

M'

-+

M"

-+

0

splits. (3) (Baer's criterion) Iff: I -+ M is a map of a right ideal I of R into M, there exists an mE M such that f(x) = mx for all

x

E

I.

D

Given any module M, there exists an injective module 1fiR• unique up to isomorphism, satisfying the following equivalent properties: (l) MR is the maximal essential extension of M. [ 83 ]

Simple Noetherian rings

84

(2) 1ftR is the minimal injective extension of M. 1ftR is called the injective-hull of M and is denoted variously by 1ft or inj hull M R· There is an interesting connection between the ascending chain condition and injectivity, namely, 5.2 PROPOSITION The following are equivalent for any ring R: (I) R is right Noetherian. (2) If {St : i E I} is any family of simple modules, L E8 sl is tel

injective. (3) Each direct sum of injective modules is injective. Proof (After Bass) We shall show that (2) ~ (l) leaving the other implications for the reader. Let I 1 £ I 2 £ . . . be an ascending chain of finitely generated right ideals. Since the I 1 are finitely generated, we can embed the above chain in

I1 £ M2 c I2 £ M3 c I3. · ·, whereitfMtissimple'v'j

~

2.SetE1

= ------I 1fM1 andE = .L j;,

2

®E1 •

By hypothesis, E is injective. If I = ui1, there is a map -+ E defined as follows:

f :I

f(i) =

2: i + M

1

Vi

E

I.

j;,2

i

The sum on the right is of course finite since eventually, I 1, 'v'j ~ n. By 5.1, f extends to all of R ~ imf £ finite

E

summand of E, say

n

.L E8 E 1• This readily implies that In+ 1

=

i= 2

In+ 1, 'v'j

~

2 and hence the chain terminates.

D

There is another useful characterization of Noetherian rings, due to Matlis [58] and Papp [59] (generalized by Faith and Walker [67]) which states that R is right Noetherian if and only if each injective module is a direct sum of indecomposable injectives. We remark that the necessity is an immediate consequence of 5.2, and the obvious fact that over a Noetherian

85

V-rings

ring, every nonzero module has a uniform submodule. The sufficiency is however, more difficult. By modifying 5.1(1) we obtain the more general but very useful concept of quasi-injectivity. 5.3

DEFINITION

ME mod-R is called quasi-injective if

exact implies 0-+ homR(MfN, M)-+ homR(M, M)-+ homR(N, M)-+ 0 exact, i.e., every map f: N-+ M where N s;; M is induced by an endomorphism of M. Note that every semisimple module is quasi-injective, thus, attesting to their importance. There is a very easy way to identify these modules: they are the fully invariant submodules of their injective hull. 5.4 PROPOSITION (Johnson-Wong [61]) MR is quasiinjective if and only if AM = M where A = End 1ftR· Proof That AM is quasi-injective is obvious. Conversely, suppose that MR is quasi-injective and that ,\EA. We must show that .\M s;; M. Set N = {mE M I ,\mE M}. If N =F M, by quasi-injectivity, there exists a .\1 E End M R such that .\ 1m = .\m, Vm E N. By injectivity of 1ft R• there exists a ,\~ E A such that ,\~m = .\ 1m, Vm EN. Since .\ 1 M s;; M, .\M s;; M if (.\~ - .\)M = 0. Now (.\~ - .\)M =F 0 ~ (.\~ - .\)M n M =F 0 since 1ft\} M. However, if 0 =F m' = (.\~ ·- .\)mE (.\~ - .\)M n M, .\m = .\ 1m - m' EM ~ mEN ~ .\ 1m = .\m a contradiction. Thus, N = M and the proof is complete. D Although it will be used only briefly, we shall introduce the concept of a cogenerator which is dual to the notion of a generator. 5.5 PROPOSITION AND DEFINITION A module CR is called a cogenerator if it satisfies any of the following equivalent properties: (I) The functor homR(-, C) : mod-R "-" Ab

86

Simple Noetherian rings

is faithful, i.e., given a nonzero f: M--+ N there exists g: N--+ C such that g of =F 0. (2) For each ME mod-R there exist a set I and an embedding M--+ C1 (product of I copies of C). (3) For each ME mod-R the canonical map M--+ ChomB<M.CJ is a monomorphism. (4) Given any simple moduleS, there is a monomorphism S--+

c.

Proof The proofs of (l) ~ (2) ~ (3) are dual to those given for the dual statements in I. I and (l) ~ (4) is obvious. (4) ~ (l) Let f: M--+ N be any nonzero map and f(x) = yEN be nonzero. yR ::::; Rfyl. and yl. s;; J, a maximal right ideal.

By assumption, there exists a monic h : RfJ--+ ------ C. Thus, we obtain a map g' : yR--+ RfJ ------ defined by the composition yR--+ Rfyl.--+ RfJ--+ RfJ. ------

By injectivity of RfJ, -------- g' extends to a map g": N--+ RfJ. ------ It is easy to check that the composition g = h o g" : N --+ C is the required map satisfying go f =F 0. D 5.6 DEFINITION A ring A is called a right V-ring if each simple right A-module is injective. 5.7 THEOREM Let R be a right V-ring. Then the following conditions are equivalent: (I) Each simple right R-module is injective. (2) Each right ideal is the intersection of maximal right ideals. (3) The radical of M, Rad M = 0 for all ME mod-R. Proof (l) ~ (3): Let 8 1, i E I be a set of representatives of the distinct isomorphism classes of simple R-modules. Then, 8 1 is an injective cogenerator in mod-R. Hence, clearly, C =

n

tel

for ME mod-R, there exists a monomorphism 0--+ M --+ C1 (product ofC, I times). This readily implies that rad M = 0. (3) ~ (2): Obvious. (2) ~ (1): Let f be a map of a right ideal I into RJM where M is an arbitrary maximal right ideal. To show that RJM is

87

V-rings

injective, it suffices to show that f can be extended to R. Kerf = I 1 is such that xR + I 1 = I for some x E I. By (2), there exists a maximal right ideal M 1 of R such that I 1 s;: M 1 and M 1 +I= R. Now M 1 n I = M 1 n (xR

+

I 1) = I 1

+ (xR n

M 1) = I 1 •

Therefore, by defining J to be zero on M 1 and J = f on I, we obtain the desired extension off to R. 0 Remark (I) ~ (2) is credited to Villamayor (unpublished). As an obvious corollary to the above argument, we have 5.8 CoROLLARY If R is a right V-ring, and I is any right ideal of R, then J2 = I. 0 Note that this property is also enjoyed by any simple ring R. For I = I R = I RI R = J2. Finally, we obtain 5.9 THEOREM Let R be commutative. Then R is Von Neumann regular iff R is a V-ring. Proof If R is regular, so is RJI for any ideal I. Since rad(R/I) = 0, condition (2) of 5.7 is trivially satisfied. Conversely, if R is a V-ring and a E R, then aRaR = aR by 5.8. Thus a 2x = a for some x E R and hence R is regular. 0

In order to present Faith's example of a regular ring which is not a left V-ring (it is however a right V-ring!), we first summarize several useful categorial results. 5.I0 PROPOSITION If T:
right A -module. (2) The inclusion functor mod-AJB "-"mod-A preserves injectives.

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Simple Noetherian rings

(3) The inclusion functor mod-AJB "-"mod-A preserves injective hulls of simple right AJB-modules. (4) AJB is flat as a left A-module. D 5.12 Co RO LLAR Y If A is a right V-ring, then, for any ideal B, AJB is a flat left A-module. D 5.13 CoROLLARY If A is a regular ring, then every injective right AJB-module is canonically an injective right A-module, for any ideal B of A. D 5.14 ExAMPLE Let L = End V F be the full right linear ring of an infinite dimensional right vector space V over a field F, let S be the ideal consisting of linear transformations of finite rank, and let R = S + F be the subring generated by S and the subring F consisting of scalar transformations (sending every v r-+ va for some a E F). Clearly, R(L) is a regular ring. By standard functorial arguments (LV is flat, V F is injective and L = End Vp), LL is injective, and since VL is a summand, VL is likewise injective. Moreover, LV is injective iff dim V F < oo (see Faith [72b], p. 163). First we show that R is a right V-ring. Let W be a simple right (or left) ideal of R, and let f: I--+ W be a mapping of a right (or left) ideal of R. In order to determine if W is injective, it suffices to assume that I is an essential one-sided ideal, and thus, that I contains the socle S of R (the sum of the simple right ideals of R). Since RJS is simple, either I = S, or I = R. If I = S, and f is a morphism of mod-R, then f is a morphism of mod-L, since

[f(xa) - f(x)a]s = 0 Vs

E

S, x

E

I, a E L.

Since W is a summand of Lin mod-L, then W is injective along with L, and so f has an extension to a mapping L --+ W which induces a mapping R--+ W extending f. This proves that W is injective in mod-R. If W is a simple right R-module not contained inS, then W::::::: RJS is injective over RJS, whence by 5.13 W is injective over R. This proves that R is a right V-ring. In order to prove that R is not a left V-ring, assume for the moment that V is injective in R-mod, and that f: I--+ V is a

89

V-rings

morphism in L-mod, where I is a left ideal of L. In order to extendf, we may assume that I is an essential left ideal, that is, that I 2 S. Then, f has an extension f' in R-mod, and f' is a morphism in L-mod, since

s[(ax)f' - a(x)f'] = 0 Vs

E

S, x

E

I, a E L.

But this implies that V is injective in L-mod, a contradiction. Thus, R is not a left V-ring. (This fact is also a consequence of Faith [67a], p. 103, Th. 3.1, which implies that the maximal left quotient ring of R is also a right quotient ring. This involves a contradiction.) Next we prove a lemma preparatory to the structure theorem for a right Goldie V-ring. 5.15 LEMMA If A is a right V-ring and I is an ideal of A, then I does not contain a right regular element of A. Proof Let M be a maximal right ideal containing I. Then AJM is injective and hence,

AfMx = AJM V right regular x. For, if x.L = 0 is arbitrary, there is a map xA --+ V defined by xa ~ va, Va EA. By injectivity, 3v' E V such that v = v'x ~ V s;; Vx. Finally, since AfMx = 0, Vx E I, I cannot contain a right regular element of A. D 5.16 THEOREM Let A be a right Goldie V-ring. Then A is a finite product of simple Goldie V-rings. Proof Since A is a right V-ring, rad A = 0, i.e., A is semiprime. Let I be a maximal ideal of A. If I = 0, A is already simple. Otherwise, I (") K = 0 for some K # 0 by the lemma (as seen in the proof of 4.4, essential right ideals contain regular elements), and since A is semiprime, J.L # 0. Consequently, A = I E8 J.L and AJI is clearly a simple Goldie V-ring. The proof is now completed by induction on I, a Goldie V-ring having dimension less than dim A. D

It is instructive to give an alternate proof of this theorem using another useful result.

Simple Noetherian rings

90

5.I7 PROPOSITION Let A be a prime Goldie V-ring. Then A is simple. D A ring R is the subdirect product of rings RfMt, i E I, if

nMt tel

=

o.

Alternate Proof If R is semiprime Goldie, the canonical ring homomorphism R--+

nn RfMt is an embedding, where RfMt is

t=l

a prime Goldie ring (see Jacobson [64a], p. 269). If R is a V-ring, so is each RfMt and hence, each RfMt is simple by the previous result. However, any subdirect product of finitely many simple rings is a direct product of simple rings. D 5.I8 PROPOSITION If A is a V-domain, then A is simple. Proof Let I be a nonzero ideal of A and 0 # a E I. aAaA = aA by 5.8. Thus, IE AaA which implies that I = A. D (For the background of 5.I6-I8, see Ornstein [67] and Faith [67a, 72a, b].) Let A be a hereditary 2-sided Noetherian right V-ring. By 2.33 each finitely generated torsion A-module (either side) is semisimple, and, since A is a right V-ring, each finitely generated torsion right A-module is injective. If in addition, A is a domain, we have, 5.I9 PROPOSITION Let A be a hereditary, two-sided Noetherian right V-domain. Then (I) each cyclic right A-module not isomorphic to A is injective; (2) each indecomposable injective right A -module is either simple or isomorphic to A; (3) each quasi-injective right A-module is injective. Proof (I) Clear. (2) Let E be an indecomposable injective. Then E = () for any cyclic submodule C. If C::::::: A, then E::::::: .A. Otherwise, C is injective and E = C which implies, of course, that E is simple. (3) Let Q be quasi-injective. Then Q = .L E8 Et, where each tel

Et is indecomposable and injective, and Et (") Q # 0, ViE I.

V-rings

91

Thus, we can assume that Q is indecomposable because Q = .L E8 (E; (") Q) and the obvious fact that each summand of a tel

quasi-injective module is quasi-injective. By part 2, either Q is simple which implies that Q = Q or Q = A. Since Q is a left A submodule of Q, then Q = A since A is a field. D If A is not a domain, A ""' K where K is a right Ore domain since A is hereditary. Clearly, K is also a right V-ring satisfying (3) of 5.19. Moreover, since each finitely generated torsion left K-module is semisimple, (2) of 5. 7 is trivially satisfied. Thus, K is also a left V-ring implying that A is likewise. We next consider a fundamental problem in the theory of formal differential (or difference) equations namely the problem of trying to determine when an arbitrary consistent system of homogeneous linear differential equations has a solution in an arbitrary differential ring A. As we shall see, a necessary and sufficient condition for the solvability of any consistent system is that ~AA be an injective module. In particular, when A = k is a field, this is equivalent to !!)k being a V-ring (5.21). Let A be a differential ring (commutative) with derivation () and !!)A the ring of differential polynomials. Each element n

2: a

d =

1 1/) E !!)A

1=0

determines a linear operator on A defined by a~

n

2: a 8 (a) = 1

1

d(a).

1=0

To say that f3

E

im d is equivalent to saying that the n-th order

linear differential equation (l.d.e.) d(x) =

n

.L 1=0

a 181(x) =

f3

has a

solution in A, with ker d = solution space of d(x) = 0. More generally, let d11 E!/)A, i = 1, ... ,m,j = 1, ... ,n and D = (d11 ) E Matmxn(!!}A)· D defines a map (also denoted D)

92

Simple Noetherian rings

as follows:

(Note that we are viewing elements of the as column vectors of length n.)

For

(!j

e A •, there is an

(~A•

..,.,emted

A)-bimodule An

system of linear

differential equations n

2: du(x) =

b1

~ = I, ... , m

(I)

i=l

or more succinctly D(x) =b.

Naturally we would like to know when system (I) has a solution. By a solution to (I) we mean any column vector

G:)

such that

n[(!J] (!J· To that end observe that D induces D*

(ordinary matrix multiplication on the right by D). Here, we have agreed, for functorial reasons, to consider elements of the module ~Am as row vectors having length m. Let C = cok D*. Then

V-rings

93

is exact. Applying the functor hom~ A(-, A) we get (after some obvious identifications) the exact sequence 0 ~ hom~A(C, A)~ An~ Am.

Suppose g E Am = hom~J~Am, A) and D(f) = g for some fE An, that is, f is a solution of the system D(x) = g. Then, clearly, g induces the zero element of hom~A(K, A) where K = ker D*. For 0 ~ K ~~Am~ ~An

lg A

l'

~A

commutes and has exact rows. Finally, let J = {g E Am I g induces the zero element of hom~A(K, A)}. Then, clearly, Jfl::::::: Ext~A(C, A)

where

I = im D.

Thus, we can think of the extensions of A by C, Ext~A(C, A) as obstruction classes to the solvability of D(x) = g, their triviality equivalent to solvability. Similarly, hom~A(C, A) measures the number of different solutions. If Ext~A(C, A) = 0 for all finitely generated left ~A-modules C, then each' consistent system' D(x) = g (g E J) has a solution in A and conversely. This condition is of course equivalent to the injectivity of A. When A = k is a field, k as a left ~k-module is clearly simple and hence, there is an intimate connection between the 'real world' and the injectivity of certain simple modules. The next proposition is a trivial consequence of 3. 7 and will be used in the proof of the main results of this section. 5.20 PROPOSITION Let M, N E ~-mod, MR flat and ~N injective. Then homR(M, N) is an injective differential module. D 5.21 THEOREM Thefollowing are equivalent for~= ~k· (I) ~k is injective; (2) each M E ~-mod which is finite dimensional over k is injective in ~-mod; (3) each linear differential equation has a solution in k.

Simple Noetherian rings

94

Moreover, ~k is the unique simple left £!)-module up to isomorphism iff for each homogeneous l.d.e. has a nontrivial solution ink. Proof (I)~ (3) and (2) ~(I) are clear. (I)~ (2): Let M be finite dimensional and M* = homk(M, k). Clearly, M*k is flat and since ~k is injective, M** is injective in f!}-mod by 5.20. However, since dim kM < oo, M : : : : M** in f!}-mod by 3.8, completing the proof of the equivalence of (1) and (2). Since £!) is a pli-domain, to show that ~k is unique, it suffices to show that hom~(£!) jf!)d,

for all dE£!), d =

n

.L 1=0

k)

=1=

0

a 181• Consider the homogeneous linear

differential equation n

L a 8 (x) = 0 1

1

1=0

and let a be a nontrivial solution. Clearly, the mapf: f!}jf!)d--+ k defined by r + f!)d r-+ r ·a is £!)-linear and nonzero. Conversely, if ~k is unique and

n

.L 1=0

a 181(x) = 0 is an arbitrary

homogeneous linear differential equation, set d

n

=

.L 1=0

a 181 and

let f : £!)f f!)d --+ k be nonzero (f exists since ~k is unique). If (I + f!}d)f = a =1= 0, d(a) = 0 and hence, we have found the desired nontrivial solution of d(x) = 0. D Constructing fields k satisfying condition (3) is quite easy (at least in theory). For example, if k is universal (cf. Cozzens [70]), then condition (3) is trivially satisfied. Thus, we have our first nontrivial examples of simple Noetherian V-domains which are not fields. Note that each cyclic £!)-module, not isomorphic to£!), is injective. This is because such a module is necessarily finite dimensional over k (this also follows from 5.I9(I)). Analogous results hold for partial differential rings (see Cozzens-Johnson [72]). Of course, these rings can have any

95

V-rings

prescribed global dimension. Hence, nontrivial examples of Vdomains exist in all dimensions. A class of simple Noetherian V-domains with similar properties can be constructed via the twisted, finite Laurent polynomials L = LP(k) defined on p. 51. By considering difference n

.L

equations of the form

atpt(x)

b, p the automorphism of k,

=

t=O

we obtain results analogous to Theorem 5.21. We shall refer to this Theorem by 5.21 *.However, in this case, it is quite easy to give rather explicit examples of fields satisfying (3) of 5.21 *. 5.22 ExAMPLE Let k be a field of characteristic p and p : k--+ k be defined by p(a) = aP for all a E k (the Frobenius map). Clearly, pis an automorphism of k, having infinite period whenever k, is, say, separably closed. In this context, linear difference equations are polynomials of the form n

L atXP'

at

E

k.

(2)

t=O

By 5.21 *, if each polynomial of the form n

L atXP' =

b

at, b

E

k,

t=O

has a root ink, then Lis a V-ring. Moreover, in order to have a unique simple £-module, each polynomial of type (2) containing at least two terms must have a nontrivial root in k. By letting k = ZP, these properties are obviously satisfied by algebraic closure of ZP. Recently, Osofsky [71] showed that by suitably modifying the field k in Example l, examples of simple Noetherian V -domains could be obtained with infinitely many nonisomorphic simple modules in contrast to the above example which has only one. In order to make all of this seem plausible, we shall exhibit another interesting property of the ring L which at the same time enables us to precisely determine the number of !-dimensional simple £-modules. Let .9'(1) denote the set of all isomorphism classes of one-

Simple Noetherian rings

96

dimensional (k-dimension, that is) £-modules. If M is onedimensional, (M) will denote the class of M in .9'(1). For M, M' and M" all one dimensional, M ®k M' is an £-module and clearly, dimk(M ®k M') = l. Moreover, it is quite easy to show that M ®k M' : : : : M' ®k M and (M ®k M') ®k M" : : : : M ®k (M' ®k M") as £-modules (of course the £-action is that defined in Chapter 3). Since (k) E .9'(1) and k ®k M : : : : Min £-mod, the operation (M) o (M') = (M ®k M') on .9'(1) is commutative, associative, and has identity (k). Thus, (.9'(1), o) is a commutative monoid. More can be said:

ForME L-mod, M* = homk(M, k)

5.23 PROPOSITION

M* ®k M ~ homk(M, M) as £-modules, where cp is defined by cp(f ® m)[m'] = f(m')m for allfEM*, m, m' EM.

D

It follows immediately from Proposition 5.23 that (.9'(1), o) is in fact, a group if (M) - l is defined to be (M*).

5.24 THEOREM Let k* be the multiplicative group of units of the field k and define cp: k*--+ k* by cp(a) = a- 1 p(a). Then .9'(1) : : : : k*fim cp.

Proof Let (M) E .9'(1) and mE M. Then p(m) = a(m)m for a unique a(m) E k. Since M = km for some mE M, a(f3m)f3m = p(f3m) = p(f3)p(m) = p(f3)a(m)m. Thus, a(f3m) = a(m)mod im cp. Hence, the correspondence (M) r-+ a(m) im cp is a map ifJ: .9'(1)--+ k*fim cp. The verifications that ifJ is a surjective group homomorphism are routine and will be left to the reader. Injectivity of ifl: Suppose (M) E .9'(1), M = km and a(m) im cp = 1/J((M)) = g- 1 p(g) for some g E k. By the above, for each m' EM, a(m') = {3- 1 p(f3) for some f3 E k. Since p(m) can be written in the form f3p(f3) - 1 m, we see that the map M --+ k

97

V-rings

defined by {3m ~ 1 is an L-linElar isomorphism, implying that (M) = (k). D To exhibit a field k where (1) of 5.21 * is satisfied but where Lk is not unique, we proceed as follows. Let q be a prime integer and let k be any field with algebraic closure Tc. Let .fF = { L

I L is a field, k s;; ~

5.25 DEFINITION

L s;; Tc and l E L degree l over k is relatively prime to q} A q-field over k is a maximal element of

.fF.

The utility of q-fields stems from 5.26 PROPOSITION (Osofsky [71]) Let L be a q-jield over k, and let f E L[ X] have degree relatively prime to q. Then f has a root in L. If g E k[X] is irreducible and has degree divisible by q, then g has no root in L. D To obtain the above mentioned example, let k 2 F = Zp(X) be a 2-field, 2 =F p, p : k --+ k defined by a ~ aP and {1r1 I i E w} an infinite set of primes of Zp[X]. Consider the cosets 17-1 = 1r1 im cp and 17- in k*fim cp. We claim that they are distinct. For 1 if 17-1 = 17-1, then 1r11r1 - 1 = aP- 1 for some a E k*. Thus, the polynomial f(Y) = 1r1 YP- 1 - 1r1 has a root in k*. However, by Eisenstein's criterion, f is irreducible over F[ Y] and hence over k[Y] by Proposition 5.26, since k is a 2-field and 2 I deg f. Thus, 17-1 =F 17-1 which implies that the cardinality of (.9'(1)) ~ Xo·

6. PCI-rings

A 'good' theory of ring theory is one which generalizes the Wedderburn-Artin theorems for a ring R with radical N, and a 'good' theory of modules is one which generalizes the basis theorem for abelian groups. In this chapter we are concerned with the latter, in the setting of Dedekind domains, particularly the aspect which states that every finitely generated module M is a direct sum of finitely many ideals and cyclic modules. A 'conceptual' way to effect such a decomposition of M is to notice that M modulo its torsion submodule t(M) is a torsion free module F embeddable in a free module P. Then, the canonical projections of P--+ R induce projections ofF into R which furnish the requisite ideal summands. Moreover, Mft(M) being projective implies that the torsion module t(M) is a summand of M, and the cyclic decomposition for t(M) is a consequence of the fact that R modulo any ideal A # 0 is an Artinian principal ideal ring, hence a uniserial ring over which every module is a direct sum of cyclics (as Kothe [35] showed). Another way to obtain the cyclic decomposition is to observe that every proper cyclic module RJA is an injective RIAmodule. Thus, every cyclic submodule 'of highest order' splits off! We adopt this point of view in this chapter. However, instead of requiring that every proper cyclic right module RfA be injective modulo its annihilator ideal, we make the stronger hypothesis. (right PCI) Every proper cyclic right R-module Cis injective.

Proper cyclic means that C is cyclic but C 1t R. We show that any right PCI-ring is either a semisimple ring, or a simple right semihereditary, right Ore domain. This [ 98 ]

99

POI-rings

reduces the entire theory to the simple right PCI-domains. As we have already observed, such rings occur freely in nature, constituting a broad class of the known simple domains. In order to obtain these results however, we repeatedly use the following characterization of semisimplicity in terms of the class of cyclic modules, namely, 6.1 THEOREM (Osofsky) A ring R is semisimple (Artinian) if and only if each cyclic right R-module is injective. D

Thus, in terms of our development, the limiting case of a PCI-ring (hence V-ring) is a semisimple ring. For a short proof, see Osofsky [68]. Recall that if I is a right ideal of a ring R, then a right ideal K which is maximal in the set of all right ideals Q such that I n Q = 0 is called a complement right ideal, and K is said to be a relative complement of I. In this case, I + K is an essential right ideal of R. Moreover, then, I is a relative complement of K if and only if I is a complement right ideal. In this case, I and K are said to be relative complement right ideals. A ring A is right neat provided that the right singular ideal of A is zero. In this case, the injective hull A of A in mod-A is a right self-injective regular ring containing A as a subring. Moreover, for any right ideal I of A, a unique injective hull1 of I in mod-A is contained in A. Furthermore, 1 is a right ideal of A, generated by an idempotent. Every annihilator right ideal of a right neat ring A is a complement, and every right ideal I of A is contained as an essential submodule in a unique complement 1 which is in fact 1 n A (see R. E. Johnson [51] or Faith [67a]). 6.2 LEMMA If R is a right neat ring, and if I is a right ideal of R, then I is a complement right ideal of R if and only if RJI embeds in R. Proof The complement right ideals of fl are those of the form efl, for some idempotent e E fl, and those of R are of the form eil n R. Thus, if U = eil n R, then the isomorphism

Rjeil n R = RJU::::::: eR

100

Simple Noetherian rings

embeds RJU into eR s;; fl. Conversely, iff: RJU--+ fl, is an embedding, and if f(1 + U) = y, then U = yl. n R. Write fly = fl(1 - e), for some idempotent e. Then yl. = ell, and hence, U = ell n R. D 6.3 THEOREM Let A be a right POI-ring. Then, either A is a regular ring, or else A is a simple ring. Moreover, A is a right V-ring, and every indecomposable in}ective right A-module is either simple or isomorphic to A. Proof By Brown-McCoy [50], A has a maximal regular ideal M, and AJM has no regular ideals ,eO when A # M. If I is any ideal ;60, then every cyclic module C of Afi is a proper cyclic module over A, hence C is an injective module over A, and therefore injective over AJI. Then, 6.1 implies that AJI is semisimple Artinian. Hence, if A is not regular, then M = 0. Since every simple right module is injective, A is a right V-ring, and so by 5.15 a nonzero ideal B contains no right ideal isomorphic to A. Then, for any y # 0 in B, yA is injective, hence a summand of B generated by an idempotent. This implies that B is a regular ideal, contrary to the assumption that M = 0. Thus, A must be simple when A is not regular. If E is an indecomposable injective right A -module, then E = () for any nonzero cyclic submodule C. Thus, if C::::::: A, then E::::::: .A, and if C 1t A, then C is injective, whence E = C is simple. D 6.4 PROPOSITION A right POI-ring A is right neat. Moreover, A has a nonsimple indecomposable in}ective right module E if and only if A is a right Ore domain, not a field. In this case, A is a simple ring. Proof Simple rings and regular rings are right neat, so the first sentence follows from 6.3. Since, by the remarks preceding 6.3, A is a regular ring, then A is indecomposable as a right A-module if and only if A is a field. In this case, A is an integral domain, and A is the right quotient field of A. Since AA is not simple, then A # A. This proves one part of the second sentence, and the converse part is trivial. Then A is either regular or simple by 6.3 and the

POI-rings

101

former can hold only if A is a field. Thus, A is simple in either case. D 6.5 PROPOSITION A right-POI domain R is right semihereditary, that is, each finitely generated right ideal of R is projective. Proof Let P be any nonzero projective right ideal, and let a E R. If C = aR + P is such that C f P : : : : R, then there exists x E R such that x.L = {r E R I ar E P}. Since R is a domain, this would imply that aR n P = 0, and then C = aR E8 P is projective. IfCJP 1t R, then CJP is injective, hence a summand of RJP. Let D be a right ideal of R containing P such that DJP is a summand of RfP complementary to CJP. Then there is an exact sequence 0

--+

P--+ C

E8 D --+ R --+ 0,

since R = C + D, and P = C n D. Thus, C E8 D : : : : R E8 P is projective, and hence C (also D) is projective. An obvious induction on the number of elements required to generate a right ideal shows that any finitely generated right ideal of R is projective, so R is right semihereditary. D The next several results are preparation for the main results stated in Propositions 6.11 and 6.12.

A right ideal A of R is either semisimple, or else A contains an essential right ideal A' of R with A' =F A. Moreover, A' is not a complement right ideal of R. Proof A module is semisimple iff it has no proper essential submodules. Thus, if A is not semisimple, there is an essential submodule A' =F A. Let B be a relative complement of A' in R. If A' were a complement right ideal, then A' would be a relative complement of B, in which case A n B =F 0, and (A n B) n A' = 0 contradicting the essentiality of A' in A. D 6.6

LEMMA

6.7 LEMMA Let I and K be two right ideals of a ring R such that I n K = 0, and I + K is essential. Then, the injective hull of RJI E8 RJK contains fl as a summand. Proof Clearly, RJI E8 RJK contains the direct sum I E8 K,

Simple Noetherian rings

102

which is essential in R, hence the injective hull contains the summand R. D 6.8 PROPOSITION Let R be a right POI-ring. Then, either R is a right Ore domain, or else there is an exact sequence R 2 --+ fl --+ 0, or else R has nonzero right socle. Proof By 6.4, R is a right neat ring, with injective hull fl which is a regular ring containing R as a subring. If every nonzero right ideal of R is essential, then fl is a field, in which case R is a right Ore domain. Otherwise, there are right ideals I and K, neither of which are 0 orR, such that I + K is essential and I n K = 0. If I' and K' are essential submodules of I and K, respectively, then I' + K' is an essential right ideal of R, and by 6.7, the injective hull of Rfl' E8 RJK' contains fl as a summand. Now Rfl' : : : : R would imply that I' is a complement right ideal of R by 6.2. However, by 6.6 I' is a complement right ideal essential in I only if I' = I. Thus, by 6.6 either I or K is semisimple, or else RJI' E8 RJK' is an injective module containing fl as a summand. In the latter case, there is an exact sequence R 2 --+ il-+ 0, as asserted. D A right R-module E is completely injective provided that every factor module EJK is injective.

If R is a right POI regular ring, and if e is an idempotent such that eR is completely injective (e.g., if e is contained in a nonzero idealS # R), then eRe is a semisimple ring, and eR contains a minimal right ideal of R. Proof First, the canonical map eR @ Re --+ eRe is an iso6.9

LEMMA

R

morphism. To see this, let {a1}f= 1 and {b1}r=l be finitely many elements of R such that

n

.L 1=1

ea1b1e = 0. Then:

Henceforth, let Q = eRe, and let M = QJK be a cyclic right Q-module. Since Q is a regular ring along with R, then every

POI-rings

103

Q-module is flat, and therefore the canonical exact sequence 0 --+ K --+ Q --+ M --+ 0 implies exactitude of 0--+ K

0Q eR--+ Q 0Q eR--+ M 0Q eR--+ 0.

Since Q 0 eR ::::; eR as a (Q, R)-bimodule, this implies that Q

C = M

0

eR is an epimorphic image of eR in mod-R, and

Q

hence is injective. (If e is contained in a nonzero idealS # R, then eRJL ::::; R, for some right ideal L, would imply via projectivity of R that eR contains a submodule ::::; R, contrary to 5.15 which states that S contains no right ideals ::::; R. Thus, eR is completely injective in this case.) In order to show that M is injective in mod-Q, it suffices to show that M is isomorphic to a summand of any over-module. Since C = M 0 eR is injective in mod-R, if Y is any right Q

Q-module containing M, then the induced inclusion C--+ Y splits. Write Y

0

eR = C

E8 X,

0

eR

Q

for some right R-module X.

Q

Now, (Y

0Q

eR)

0R

Re::::; (C

0R

Re)

E8 (X 0R

Re).

Since eR 0 Re ::::; Q canonically, it follows that R

C

0R

Re ::::; (M

0Q

eR)

0R

Re ::::; M

0Q

(eR

0R

Re)

::::;M0Q::::;M. Q

Thus, M is isomorphic in mod-Q to a summand of Y::::; ( Y 0 eR) 0 Re and so M, hence every cyclic right Q-module, is Q

R

injective. Then, Q is semisimple. Let I be a minimal right ideal of Q. Since Q is semisimple, I = fQ for an idempotent f E Q, and thenfQf = fRf is a field ::::; EndfQQ. Since R is semiprime, this implies that fR is a minimal right ideal of R contained in eR (Jacobson [64a], p. 65). D 6.10

LEMMA

If R is a right POI-ring, and if Sis any ideal

# 0, R, then S contains any right ideal I for which there is an

Simple Noetherian rings

104

isomorphism f: RJI--+ R. Moreover, I = x.L for x = f(1 + I), and xy = 1, for some y E R. Proof Let f: RJI--+ R be an isomorphism. Then, I = x.L, where x = f(1 + I), and xR = R. Thus, xy = 1 for some y E R. Since every cyclic right RJS-module is injective, then RJS is semisimple Artinian by 6.1. Therefore, every one-sided inverse in RfS is two-sided, and consequently, yx = 1 - s, for somes ES. Thus, if xa = 0, then a= sa ES. This proves that I = x.L is contained inS. D 6.11 PROPOSITION If R is a right POI-ring, and if R is regular, then R is semisimple. Proof Let S denote the intersection of all nonzero ideals of R. The proof falls into three cases: (1) S = 0 but R is not simple. By Lemma 6.10 if Rfi is a cyclic right R-module isomorphic toR, then I is contained in every nonzero ideal of R. Thus, (1) implies that RJI is injective for every right ideal I =1= 0. If R has idempotents e =1= 0, e =1= 1, then RfeR : : : : (1 - e)R is injective, and, similarly, eR is injective, in which case, R = eR E8 (1 - e)R is injective. Thus, every cyclic right R-module is injective, so R is semisimple. (2)

s

=I= 0.

First, RfS is semisimple, since every cyclic RJS-module is injective. Second, by Lemma 6.9 R has nonzero socle. The fact that S = 8 2 is the least ideal of R implies that R is prime, and hence that S is the right socle of R. If V is any minimal right ideal of R, then V is injective, and hence V fl = V is a right ideal of fl, and in fact, V is a simple right ideal of fl. Thus, right socle fl 2 S. Furthermore, if W is a minimal right ideal of fl, then W n R = U is a minimal complement right ideal of R such that W = U fl. But every nonzero right ideal of R contains a minimal right ideal, and the latter is a minimal complement. Thus, U is a minimal right ideal of R, and, as indicated above, U = U fl. Hence, W = U is a minimal right ideal of R. This proves that S = socle fl. Therefore, S is an ideal of fl. If the socle S is finite, then S = R is semisimple, and the proof is complete. Otherwise, S : : : : v for some infinite

POI-rings

105

cardinal a, and hence, there is a right ideal K contained in S such that K::::::: v ::::::: S::::::: K 2 • Then, RJK::::::: R would imply that K = eR for an idempotent e E R, in which case K, whence S, is a finite sum of injective simple modules. But this case has just been discarded. Thus, RJK is an injective module containing a submodule K 1 ::::::: S, and hence containing the injective hull R of S. Since R is then a summand of a cyclic module, R is cyclic, and hence there exists x E R such that R = xR. Let a E R be such that xa = l. Since S is an ideal of R, if c E R, and if ac E S, then c = xac E S. This proves that [a + S] is not a left zero divisor in RJS. Since RJS is semisimple, this means that [a + S] is a unit of RJS, with inverse, say [a' + S]. Then, aa' = 1 - s, for somes ES, and

xaa' = x(1 - s) = a'. Thus, x = xs + a'. Since S is an ideal of R, then x E R, and therefore, R = xR s;; R. Since every cyclic module is therefore injective, R is semisimple in this case too. (3) R is simple. This case follows from the next proposition. 6.12 PROPOSITION A simple right POI-ring R is either semisimple, or else a right semihereditary simple domain. Proof Case (1). R contains a nonzero injective right ideal A. Since R is simple, then A is a generator ofmod-R, which implies that R = R and hence, R is semisimple by 6.1. Case (2). If (1) does not hold, then, since every simple module is injective, R has zero right socle. Then, by 6.8, either R is a right Ore domain, or else R 2 --+ R --+ 0 is exact. In the former case, R is right semihereditary by 6.5. Hence, assume: Case (3). R 2 --+ R --+ 0 is exact. Since R has no injective right ideals =F 0, every principal right ideal aR =F 0 is isomorphic to R. Let f: aR--+ R be an isomorphism. Then, f(a) = xis an element of R such that xR = R, and x.L = a.L. Let y E R be such that xy = l. Then, e = yx is idempotent. If e. =F 1, then eR::::::: Rand (1 - e)R::::::: R, hence R::::::: R 2 • This implies that R --+ R--+ 0 is exact and R --+ R E8 R --+ 0 is exact. Since C = R E8 R is thereby cyclic, then C is either injective,

Simple Noetherian rings

106

whence R is injective, or else 0 : : : : R, whence R contains an injective right ideal A : : : : fl. But, in either case, R is semisimple. This proves that e = yx = 1, whence that x.L = a.L = 0 for every nonzero a E R. Then R is an integral domain, which must be simple and right semihereditary by 6.5. D 6.13 THEOREM A right POI-ring R is either semisimple, or a right semihereditary simple domain. Proof By 6.3, R is either regular, whence semisimple by 6.11 or else R is simple, in which case the last proposition applies. D 6.14 PROPOSITION Let R be a right POI-domain which is not right Ore, then: (1) fl is a finitely presented cyclic module. (2) If I is any right ideal, then RJI : : : : fl iff I is a complement right ideal # 0, R. Furthermore, any complement right ideal I is finitely generated (by 3 elements). (3) Any finitely generated torsion right module M is injective, and a-cyclic (that is, a finite direct sum of cyclic modules). (4) If I is a right ideal, then there is a unique least complement i containing I, and i is the unique complement containing I as an essential submodule. Moreover (any finitely generated submodule of) iji is cyclic and injective. (5) If x E fl, and xi R, then xR = fl iff xa = 1 for some a E R. Moreover, in this case, J a = x.L + aR is a finitely generated essential right ideal, and Rf J a ::::::: flj R. Proof of (1) If I is any complement right ideal #0, and if K is the relative complement of I, then K embeds canonically as an essential submodule of RJI. Since RJI is injective, then RJI : : : : 1?., so we must show that K : : : : fl. Now K contains a right ideal aR::::::: R, and hence we have

fl

2

K.

2

(;R : : : : fl,

so by a theorem of Bumby and Osofsky (see Bumby [65]) we have K : : : : fl as desired. This will prove (1), once we complete the proof of (2) showing that I is finitely generated. Proof of (2) Let aR be a nonzero cyclic right ideal which is not essential. Then RfaR is an injective module containing a submodule X::::::: fl. Write RfaR = YfaR E8 X, for some

POI-rings

107

module Y 2 aR. Then, RJY::::::: X::::::: fl, so that Y is a complement right ideal by 6.2. Furthermore, Y faR is cyclic, so that Y = aR + bR, for some b E R, is finitely generated, hence projective. Therefore, fl is finitely presented. If I is any complement right ideal =F 0, R, then RJI : : : : fl, by l. Then Schanuel's lemma implies that R E8 Y::::::: R E8 I, so I is generated by 3 elements. Conversely, if I is a right ideal such that RJI : : : : fl, then I is a complement right ideal by Lemma 6.2. This proves (2).

Proof of (3) If M = a 1 R + ... + anR is finitely generated and torsion, then a 1 R is injective, so M = a 1 R E8 M 1 for some submodule M 1 . Now M 1 is generated by a;, ... , a~, where b' is the image of any b E M under the canonical projection M --+ M 1 . By induction, M 1 , hence M, is a direct sum of finitely many injective cyclic modules. Therefore, M is injective. Proof of (4) By the remarks preceding 6.2, in any right neat ring R, any right ideal has just one injective hull! (=maximal essential extension) contained in fl. Then i = 1 n R is the unique complement containing I as an essential submodule. Moreover, since any least complement containing I is essential over I, then i is the unique least complement containing I. Any finitely generated submodule of ifI is injective by (3), hence a summand of RJI, and therefore cyclic. Proof of (5) If xR = fl, then 3a E R with xa = l. Conversely, if xa = l with x E fl, a E R, x ¢; R, then x.L =F 0 since x(l - ax) = 0. Thus, xR : : : : Rfx.L is a proper cyclic, therefore injective. But xR 2 R and therefore xR = fl. Let J = {r E R I xr E R}. Clearly, xJ = R, and there is a (canonical) isomorphism xRfxJ : : : : RjJ (sending [xr + xJ] r-+ r + J, 'r/r E R). Since xR = fl, we have the desired isomorphism iljR::::::: RfJa provided we show that J coincides with Ja = x.L + aR. Clearly J 2 x.L + aR. Moreover, if y E J, y = (y - axy) + axy and y - axy E x 1 , axy E aR which shows that J = x.L + aR = Ja. D Following from 6.14(3), we have a theorem reminiscent of the situation for hereditary Noetherian prime rings and Dedekind rings (cf. Kaplansky [52], Levy [63]).

Simple Noetherian rings

108

6.15 CoROLLARY If R is a right POI, right Ore domain, then every finitely generated right ideal is projective and generated by two elements. Furthermore assuming that R is left Ore, then any finitely generated module M is a direct sum of a finite number of right ideals of R, and cyclic injective modules. Proof Over a two-sided Ore domain, any finitely generated torsion free module F is embeddable in a free module of finite rank (see 2.36). Thus, since R is semihereditary by 6.5, F = Mft(M) is projective, so the canonical map M--+ F splits. Thus, M : : : : F E8 t(M). Moreover, F is isomorphic to a direct sum of right ideals. Finally, the proof of 6.14(3) suffices to establish that t(M) is a-cyclic, and injective. In particular, if I is any finitely generated right ideal, and if a is any nonzero element of I, then I faR, being finitely generated torsion, is injective, and, being a summand of RfaR, is therefore cyclic. Thus, I = bR + aR for some b E R. D The proof of 6.14(2) has the corollary: 6.16 CoROLLARY If R is a right neat ring, and if I is a right ideal such that fl embeds in Rfl, then there exists an element bE R such that Y = bR + I is a complement right ideal of R. D 6.17 THEOREM A right POI-domain is right Ore. Proof If R is not right Ore, then we may assume that fl : : : : Rfl is a finitely presented cyclic module by 6.14(1), that is, that I is finitely generated. Write 0--+ I--+ R--+ il--+ 0

(1)

exact. Now, any finitely generated left R-submodule M of fl can be embedded in R (via a right multiplication by a nonzero element of R), and so fl is a direct limit of projective modules, whence fl is a flat left R-module (all of which is well-known; e.g., Sandomierski [68]). Thus, we have exactness of 0 --+ I

0

il--+ R

0

il--+ il0 il--+ 0.

(2)

Since fl is left flat, and I is finitely generated, then I

0 fl : : : : I fl

R

R

R

R

is a finitely generated right ideal of fl, which by the remark in the proof of 6.14(1) is isomorphic to fl, whence a summand of fl.

109

POI-rings

This proves that flQ9 fl is isomorphic to a summand of fl, R

whence has zero singular submodule on the right. Now the kernel of the canonical map flQ9 il--+ fl is contained in the R

singular submodule, hence must equal 0. This proves that the embedding of R into fl is a ring epic (Silver [67]). This, together with left flatness of fl, suffices to show that a right fl-module M is injective as a right fl-module iff it is injective as a right R-module. In particular, if M is a proper cyclic right fl-module, we conclude that M is injective, since M is a proper cyclic right R-module. Then, by Osofsky's theorem, fl is semisimple, which can happen iff R is a field. This proves that R is right Ore. D It is an open question whether or not a right PCI-domain R is right Noetherian. If it is, then every injective right module M is ~-injective in the sense that any direct sum of copies of M is injective. It is known that to test for ~-injectivity of M it suffices to test injectivity of M (Faith [66]). Moreover, over a PCI-domain, every indecomposable injective module E is either simple, or E : : : : fl is the right quotient field of R. The latter is always ~-injective by a theorem of Faith [66]. So, in order to test ~-injectivity of indecomposable injectives, it suffices to consider just simple modules. This we do in 6.18- 6.19 below. 6.18 PROPOSITION Let R be any ring, and let V be a simple injective right R-module which is not ~-injective. Then, some cyclic right R-module RfJ has essential socle : : : : v<w>, a direct sum of countably infinitely many V's. Proof As stated, an injective module M is ~-injective if and only if M<w> is injective. Assuming that V is not ~-injective, then v<w> has injective hull E # v<w>. Since v<w> is essential in E, this implies that E is not semisimple, and hence there is a cyclic submodule xR of E which is not semisimple. Since E has essential socle, then xR has essential socle H. Since every finite direct sum of injective modules is injective, essentiality of H in xR implies that H does not have finite length. Since H is contained in v<w>, this implies that H : : : : v<w>. D

The proof has the corollary.

llO

Simple Noetherian rings

6.19 CoROLLARY If R is a right POI -ring, and if R is not right Noetherian, then there is a cyclic injective right R-module E with infinite essential socle. Proof By 6.13, R must be a domain, not a field. By 5.2, not every semisimple module is injective. Hence, there is a semisimple module M with M not semisimple, hence containing a cyclic submodule xR which is not semisimple. Then, xR has infinite socle, since any finite direct sum of simple modules is injective. Moreover, since R is not a field, soc R = 0, so xR ~ R, that is, xR is injective. D 6.20 PROPOSITION Let R be a right POI-ring, let I be a finitely generated essential right ideal, and let E = RJI. Then, B = EndER is a right self-injective regular ring. Proof It is clear that E is completely injective. If b E B, thenO = bEisinjective,henceasummansfofE. WriteO = XJI for a right ideal X, and let YJI be the complementary summand of E. Then YJI is cyclic, so Y is finitely generated. This implies that 0::::::: RJY is finitely presented. If kerb = KJI, then there is an exact sequence 0--+ K --+ R --+ 0--+ 0, hence by Schanuel's lemma, K is finitely generated. But, by induction, every finitely generated submodule of E is injective. Thus, ker b is essential only if b = 0. By Utumi's theorem (Appendix 1), this proves the proposition. D 6.21 CoROLLARY Let R be a right POI-domain, with a unique simple right R-module V. Assume that R is not right Noetherian, and let I be the right ideal given by 6.19 such that E = RJI is injective with infinite essential socle. Then I is not finitely generated. Proof E = RJI is not semisimple, hence there exists a maximal right ideal M 2 I such that MJI contains the socle of RJI. The socle S of E : : : : direct sum of copies of V, so there exists a nonzero map RJM--+ E. Thus, MJI is an essential submodule of RJI, and is the kernel of an endomorphism f of E. Then,fis contained in the radical of EndER, so the preceding proposition shows that I cannot be finitely generated. D

POI-rings

Ill

Since B = EndER has nonzero radical J, and BjJ::::::: End SR, BjJ is a full right linear ring. 6.22 DEFINITION A ring R is a right fir provided that every right ideal of R is a free module, and every free right module has invariant basis number (IBN). Thus, an isomorphism ff- 1> ::::::: U-J), for sets I and J, implies card I = card J. 6.23 COROLLARY A right POI-domain R is a principal right ideal domain iff R is a right fir. Proof Any right fir which is also right Ore is necessarily a pri-ring. For if I is any right ideal generated by a 1 and a 2 , a 1 ¢; a 2 R, then since a 1 R n a 2 R # 0, a 1 x = a 2 y # Oforx, y E R. Hence, we obtain a nontrivial linear relation in I, namely a 1 x + a 2 ( - y) = 0 contradicting freeness of I. So, I = a 2 R. The proof of the corollary is now clear. D 6.24 PROPOSI'l'ION If R is a right Noetherian right POIdomain with right quotient field Q # R, then R is right hereditary and (1) QJR is semisimple; (2) R embeds canonically in End QJR under a map: r r-+ r', where r' sends [x + R] r-+ [rx + R]; (3) for any right ideal I, the factor module RJI is a semisimple module of finite length; (4) any finitely generated torsion module is semisimple of finite length. Proof A right PCI-domain R is right semihereditary by 6.5, and hence Noetherian implies right hereditary. (1) The module QJR is a torsion module, and by 6.14(3) every finitely generated torsion module is injective. Since R is right Noetherian, this implies that for any finitely generated submodule M ofQJR, every submodule is a summand, and hence that M is semisimple. It readily follows that QJR is semisimple. (2) Obvious. (Note that since R is simple, and the map is nonzero, the map is an embedding.) Proofs of (3) and (4) are similar to that of (1) (cf. 6.14(3).

D

Simple Noetherian rings

ll2

6.25 THEOREM A left Ore right Noetherian right POIdomain is left Noetherian. Proof Let (3)

be an ascending chain of finitely generated (hence projective) left ideals of R, c E I 1 and D, the 2-sided quotient field of R. The set It - 1 = {dE D I ltd s;; R}

is a right R-submodule of D, canonically isomorphic to It* = homR(It, R) since Rlt is essential. By applying ( ) - 1 to (3) we obtain (Rc)- 1 2 1 1 -

1

212 -

1

2 ... 2

R.

Since (Rc) - 1 = c - 1 R, multiplication by c yields R 2 cl 1 -

1

2 cl 2 -

1

2 ... 2 cR.

By 6.24(3), RfcR is Artinian. Thus for some n > 0, cl n- 1 = cln+i- 1 for all j ~ 1 and hence In - 1 = In+i- 1 • However, In - 1 - 1 = In for all n by projectivity of In and hence, In = In+i for all j ~ 1. Thus, chain (3) terminates in finitely many steps.

D 6.26 THEOREM (Boyle [74]) A right and left Noetherian ring R is a right POI-ring iff R is a right hereditary right V-domain. Furthermore, in this case, R is a left POI -domain. Proof Since a two-sided Noetherian, right hereditary ring is left hereditary, (Jans [64], p. 58), the first statement follows trivially from 5.19(1). By 6.24(4), each finitely generated torsion right R-module is semisimple and hence, each finitely generated torsion left R-module is semisimple by 2.32. Consequently, each left ideal is a (finite) intersection of maximal left ideals. Thus, R is a left V -domain and hence, a left POIdomain by 5.19.1. D

7. Open problems

Let A = End 8 U be a simple right Goldie ring, with U A a uniform right ideal finitely generated projective and faithful over the right Ore domain B, with T = trace 8 U a least ideal of B. (Some of these requirements are redundant.) (1) Is A similar to a domain R?

This can happen iff A contains a (finitely generated) projective uniform right ideal W. (2) Is A ::::; Rna full n x n matrix ring over a domain R?

Clearly yes to (2) implies yes to (1). (3) Does A have nontrivial idempotents when A is not a domain?

If e = e2 E A is nontrivial, then eA is a (B, A)-progenerator, where B = eAe ::::; End eAA- Suppose that (3) has an affirmative answer. Since B has Goldie dimension less than that of A, an induction on the Goldie dimension shows that there exists e E A such that B = eAe is a domain. Then A is similar to a domain, since eA is a (B, A)-progenerator. Thus, yes to (3) implies yes to (1). (Note added in proof: A. Zalesski has announced an example giving a negative answer.) (4) Let R be a right Ore domain with a finitely generated projective left module W such that T = trace R W is a least ideal of R. Can W be chosen such that W is indecomposable of rank > 1 ? (The rank of W is defined to be to the dimension over D of the left vector space D@ W, where D is the right quotient field R

of R.) (5) If R is a simple right Ore domain, is every finitely generated projective left R-module W a direct sum of modules of rank 1 ? [ 113 ]

Simple Noetherian rings

114

By induction on the ranks, one can see that the answer to (5) is yes iff that to (4) is no for simple R. (In this case traceR W = R for every W.) Moreover, yes for (5) holds iff yes to (3) holds. (6) Assume that the answer to (2) is yes, and let B be a simple right Ore domain. If W is any finitely generated projective module over B, then there exists a projective module U of rank lover B, and an isomorphism W::::; Un, for some integer n ~ l. For then Rn = A = End 8 W is a simple right Goldie ring, and U = We 11 has the desired property, where e11 is the matrix unit of Rn (in the usual notation). Clearly U has rank l because End 8 U::::; e11 Ae 11 ::::; R is right Ore domain. Moreover

w = L Weu::::;

Weu

®· .. E8

Weu ::::;

un.

I= 1

(7) Does there exist a right V-domain with a prescribed (finite) number of simple right modules? In Chapter 5, we presented Cozzens' examples of simple V-domains, each having a unique simple module (on either side). We also constructed Osofsky's example of a V-domain with countably many simple modules. But so far we have neither an example of a right V-domain which is not a left Ore domain, nor one having a finite number n of simple modules for any n>l.

(8) Determine new examples of simple Noetherian (domains) rings unlike those described in Chapter 3. (9) Is the ultraproduct (defined presently) of countably many copies of a simple Noetherian V-domain such as that described in Chapter 5 necessarily a V-domain? a POI-domain? Recall that a filter .fF of subsets of a set I is a family of subsets of I satisfying (a) 0, the empty set, is not in .fF, (b) 81,82 E.fF ~ 81 (") 82 E.fF, (c) SE.fF,S s;;; T s;;; I~ TE.fF.

An ultrafilter is defined to be a maximal filter which obviously exists by Zorn's lemma.

Open problems Suppose {Ra : a E

ll5

I} is a family of rings and R

n

Ra = ael Ra'rla}. If .fF is any ffiter on I, we =

{!If: I--+ U Ra and f(a) E ael define f = g(mod .fF) iff {a I f(a) = g(a)} E .fF. Clearly, 1 = {! E R If =O(mod .fF)} is a two-sided ideal of R. The factor

ring Rji is usually denoted

n Raf.tF and is called an ultraael

product whenever .fF is an ultrafilter.

It is known that ultraproducts preserve certain properties of the rings Ra, namely, those which can be described by first order statements (see, e.g., Robinson [63]). For example, the properties of being a domain, a field, an Ore domain, semihereditary are all first order properties; being Noetherian and hereditary however, are not first order properties. If Ra is the domain described above for all a E w, set U = nRaf~ U is necessarily a non-Noetherian, semihereditary Ore domain. Moreover, U is simple since the following first order property, equivalent to simplicity, holds in each Ra: 'r/a E Ra, 3x, y E Ra such that

xa+ay=l. Find an example of a right Noetherian right V-(PCI-) domain which is not a left V-(PCI-) domain. Here is a possible method for generating the desired nonsymmetric examples. Let () be a p-derivation of a noncommutative field k with p-nonsurjective. Extend 8(p) to a field Tc ::::> k in such a manner that (a) every linear differential equation in () has a solution in Tc, (b) p is still nonsurjective. Clearly, if this is possible, the ring of p-differential operators with coefficients in Tc will be the desired one-sided example (see Theorem 5.21). (10)

(ll)

Find an example of a non-Noetherian POI-domain.

(12) Find a left Noetherian right Ore domain which is not right Noetherian (see example 2, Chapter 4).

Simple Noetherian rings

116

(13) Is the tensor product of two simple V-rings A ®c B over the center C of A necessarily a V -ring, e.g., is it true for fields? e.g., if D is a field with center C, is D ®c C(X) a V-ring? (If DJC is an algebraic algebra, then D ®c C(X) is a field whenever D[X] = D ®c G[X] is not a primitive ring.)

A ring R is a right QI-ring if every quasi-injective right Rmodule is injective. The examples of Noetherian V-rings of Cozzens [70] are all QI-rings, and principal (right and left) ideal domains. (See Chapter 5.) These domains are thus hereditary. Boyle [74] proved that Noetherian hereditary V-rings are QI, and conjectured that every right QI-ring is right hereditary. Now a theorem of Webber [70] as generalized by Chatters [71] states that any (2-sided) Noetherian hereditary ring satisfies the restricted right minimum condition, RRM: R satisfies the d.c.c. on right ideals containing any essential right ideal. Thus, for (right and left) QI-rings, RRM is a necessary condition for the truth of Boyle's conjecture. A ring R has right Krull dimension K -dim ~ 1 if each properly descending chain Al

::::>

A2

::::> ••• ::::>

An

::::>

of right ideals of R, such that for each i = 1, 2, . . . the right R-module AtfA1 + 1 is not Artinian, has only finitely many terms. A structure theorem of Michler-Villamayor [73] classifies right V-rings of right Krull dimension (K-dim) ~ 1 as finite products of matrix rings over right Noetherian, right hereditary simple domains, each of which are restricted semisimple. These rings are thus right hereditary, verifying Boyle's conjecture for these rings and in particular for RRM-rings. This introduces the problem: (14)

Prove Ann Boyle's conjecture.

For an extension of the Michler-Villamayor result, using different methods, consult Faith [74b], who proves that any right QI simple but nonsemisimple ring R possesses an injective non-

Open problems

117

simple indecomposable right module E such that every factor module of any direct sum of copies of E is injective, and any proper factor module of E is semisimple. Moreover, if every nonsimple indecomposable injective right module E has this property, then R is right hereditary (Faith [74b]). A right QIring has the property that the endomorphism ring of an indecomposable injective module is a field. This property for a right Noetherian ring has been characterized by Faith [74a].

Afterword

The most important class of simple rings is the category of fields, denoted FIELDS. These are the rings in which every nonzero element is a unit, and, like the quaternions, may be noncommutative. They have also been called division rings, but we retain this terminology only if R is a finite dimensional algebra over its center, in which case we say division algebra. The literature on division algebras is vast, and we refer the reader either to Albert [39], or to Deuring [48] for an account up to about 1936. Recently, Amitsur [72] solved the question of the existence of division algebras which are not crossed products in the negative, so the reader may refer to this paper for new perspectives.

Steinitz extensions Steinitz [48] determined that any commutative field F can be obtained as an extension of the prime subfield P by two intermediate extensions: first, a purely transcendental extension TfP (meaning that Tis generated by a set {x1} 1e 1 of elements any finite subset of which generates a rational function field over Pin those variables); second, F is an algebraic extension of T in the sense that every element y E F satisfies a nonzero polynomial overT. Cartan-Brauer-Hua Unfortunately, a Steinitzian simplicity cannot hold for a noncommutative field F, since a very elementary theorem, the Cartan-Brauer-Hua theorem (see Jacobson [64a]), implies that F is generated as a field by the conjugates of any noncentral subset. Therefore, if some element y of F is transcendental, that is, not algebraic, over its center, then F is generated by [ 118 ]

Afterword

ll9

{x- 1yx}xeF*• where F* denotes the units group ofF, so F is

then generated by transcendental elements. Similarly, ifF is not purely transcendental, that is, if some noncentral element y of F is algebraic over its center, then F is generated by the conjugates of y, hence by algebraic elements! If, as we have indicated, the category FIELDS admits no quick classification in either of the two classical modes, i.e., the classification of all rational division algebras as 'cyclic division algebras' as introduced by L. E. Dickson, or in Steinitzian style, then how can we expect a rapid purview of the much larger class of simple rings? VVedderburn-~in

Of course, great simplifications are possible if we choose the Wedderburn-Artin gambit: any simple ring R with descending chain condition on right ideals, that is, any simple right Artinian ring, is a full n x n matrix ring Dn over a uniquely determined field D. This 'reduces' the structure of R to that of an 'underlying' field (i.e., forget the field!). The power of the Wedderburn-Artin theorem lies in the fact that the structure of any module over a semisimple ring is transparent, namely, a direct sum of simple modules. This fact makes it the fundamental tool in the structure of right Artinian rings, since if J denotes the radical of such a ring R, then any RfJ-module, e.g., JfJ 2 , is semisimple. In contrast, the existence of so many types of simple Lie algebras causes the proofs of many theorems on Lie algebras to devolve into seemingly countless special cases. Thus, while the variety of simple Lie algebras lends interest to the subject, the application of their structures makes the ensuing mathematics more boring! The same observation can be made about the structure theory of simple rings inasmuch as there are many seemingly different, i.e., nonisomorphic types, and, as we have shown, module theory over them is quite complex even for the most explicit types of simple rings. The magnitude of the task of classifying simple rings is the raison d'etre of this volume. While primitive rings and prime

120

Simple Noetherian rings

rings have evolved as fntitful generalizations of simple rings, and (in the latter case) of integral domains, there are just too many to classify. At present, the determination of the structure of simple rings with ascending chain conditions offers challenges enough to occupy the talents of mathematicians in the foreseeable future. (See Open Problems, Chapter 7.)

Appendix

I. Homological dimension If ME mod-R, a projective resolution forM, P(M), is an exact sequence

· · .--+ Pn--+ Pn-1 --+. · .--+ P1--+ Po--+ M--+ 0, where each P 1 is projective. Clearly, each module has a projective resolution. If P 1 = 0, Vi ~ n + l, we say that P(M) has length n. The greatest lower bound of the lengths over all projective resolutions for M R is called the projective or homological dimension of M R and is denoted pd M R, pd M or hd M. By a trivial extension of Schanuel's lemma, one can show that ifpd M =nand 0--+ K--+ Pn_ 1 --+ ... --+ P 0 --+ M--+ 0 is exact with P 1 projective for all i then K is projective. sup{pd M I ME mod-R}is called the right global dimension of R and will be denoted r.gl.dim R. The Auslander global dimension theorem reduces the computation of r.gl.dim R to that of the projective dimensions of the right ideals. THEOREM (Auslander)

If R is not a semisimple ring, then

r.gl.dim R = sup{pd I

II

a right ideal}+ l.

If M and Care any two R-modules and 0--+ K 0 is exact with P projective, then

-4. P 4 M--+

0--+ homR(M, C)~ homR(P, C)~ homR(K, C) --+ cok f* --+ 0 is exact. cokf* is usually denoted ExtR 1 (M, C). ExtR 1 (M, C) is a functor: mod-R "-" Ab, contravariant in the first variable and covariant in the second. Moreover ExtR 1 (M, C) = 0 VC(VM) iff M is projective (Cis injective). (See Lambek [66] or CartanEilenberg [56].) [ 121 ]

122

Simple Noetherian rings

Flat Modules and Regular Rings PROPOSITION AND DEFINITION A module MR is called flat if it satisfies any of the following equivalent properties: (l) M ®R (- ): R-mod "-" Ab is an exact functor; (2) for all (finitely generated) left ideals of R,

M ® Jc~n MI where the canonical map is the one induced by M ® I -. M ® R =M. PROPOSITION AND DEFINITION R is called (Von Neumann) regular if R satisfies any of the following equivalent properties: (l) each ME mod-R is flat; (2) each finitely generated right ideal of R is generated by an idempotent; (3) 'r/a E R, 3x E R such that axa = a. UTUMI'S THEOREM (Utumi [56]) Let MR be injective, A = End M R and N, the ideal of A consisting of all ,\ E A satisfying M V ker ,\. Then (l) AfN is regular; (2) N = rad A.

2. Categories, functors and natural equivalences A category G consists of a class of objects obj G; for each ordered pair of objects, (A, B), a set Mora(A, B), called the morphisms or maps from A to B; and for each ordered triple of objects, (A, B, C), a map (called composition) Mora(B, C) x Mora(A, B)-. Mora(A, C)

satisfying (l) Mora(A, B) fl Mora(A', B') =F 0 iff A = A' and B = B'; (2) eachf E Mora(A, B) has a left identity in Mora(B, B) and a right identity in Mora(A, A);

(3) composition is associative.

Appendix

123

mod-R where objects are right R-modules and maps, R-linear homomorphisms. ExAMPLE

A map f from A f: A---+ B.

to B will be denoted by the standard symbol

A map f: A---+ B is called an equivalence if there exists g: B---+ A such that fg = 1 8 and gf = lA. (Of course, fg is the composition off and g and 18 is the (necessarily unique) left identity for f). Let G and G' be categories. A (contravariant) covariant functor T: G "-" G' is a rule associating with each A E G, an object T(A) E G', and for each mapf: A---+ B, a map T(f): T(A) ---+ T(B). (T(f): T(B)---+ T(A)) satisfying (l) T(lA) = lT(A)• VA E G; (2) T(fg) = T(f)T(g), Vf, g,

(T(fg) = T(g)T(f), Vf, g). VA E G, we have two functors (l) Mora(A, -): G "-"SETS, (2) Mora(-, A): G "-"SETS, defined as homR(A, -) and homR(-, A) respectively. (l) is a ExAMPLE

covariant functor and (2) is a contravariant functor. The unadorned term functor means covariant functor. Two functors S, T : G "-" G' are said to be naturally equivalent, S ::::= T, iffor each A E G there exists an equivalence "Y/A: S(A)---+ T(A) such that for allf: A---+ B the diagram S(A) ~ T(A)

!S(f)

!T(/)

S(B) ~ T(B)

commutes. We call a covariant functor T : G "-" G' an equivalence if there exists S: G' "-" G such that To S ::::= la· and So T ::::= la. In this case, we say that G and G' are equivalent and indicate this by G::::; G'. Finally, a functor T: G "-" G' is called a left adjoint for

124

Simple Noetherian rings

S: G' "-" G if VA E G, Mora-(T(A), ) :::::: Mora(A, S( )) and similarly for the other variable. ExAMPLE If G = mod-A, G' = mod-B, and N, an (A, B)bimodule, then T = ®ANistheleftadjointforS = hom 8 (N, ).

Notes

Chapters 1 and 2

The Introduction indicates a brief history of the subject of non-Artinian simple Noetherian rings, introduced by the Goldie and Lesieur-Croisot Theorems in 1958-60. The Noetherian chain conditions themselves were introduced by Noether in 1921 in carrying over results and techniques of algebraic number theory and algebraic geometry to commutative rings satisfying the ascending chain condition (thus, Noetherian rings). In 1927, Artin generalized some of the Wedderburn Theorems for finite dimensional algebras to noncommutative rings satisfying the (ascending and) descending chain conditions for right ideals. (Later, Hopkins and Levitzki, independently, in 1939, showed that the descending chain condition implies the ascending chain condition, that is, they proved that right Artinian rings are right Noetherian.) In view of the historical aspect of the Introduction, and ad hoc references to key theorems in Chapters 1 and 2, further comments may seem uncalled for. Nevertheless, we have not mentioned the Popesco-Gabriel [64] Theorem which states that for any abelian category C with generator U and exact direct limits, there is a full embedding Homc(U, ) : C "-" mod-B where B = End cU, with exact left adjoint S. This very general theorem provides additional insight for the correspondence theorem for projective modules of Faith [72a] (see 1.13), one of the main tools used in obtaining the structure of simple Noetherian rings given in Chapter 2. To wit, the adjoint functor S induces a category equivalence C : : : : mod-BJK where K = ker S = {X I S(X) = 0}, and mod-BJK is the Gabriel-Grothendieck quotient category (defined in Gabriel [62]). Thus, C is a 'localization' of mod-B in the sense of Gabriel [62], or a 'torsion theory' of mod-Bin the sense of Lambek [71]. [ 125 ]

126

Simple Noetherian rings

It is clear that this can be applied to a simple ring A with uniform right ideal U, since U is a generator ofC = mod-A. In this case, as we have indicated in the Introduction and elsewhere, B = End cU is a right Ore domain, and thus the category mod-A is a quotient category of mod-B. Moreover, the functor Homc(U, ) : C "-" mod-B is a full embedding.

Chapter 3

The best known example of a non-Artinian simple Noetherian domain is what is called the 'algebra of quantum mechanics' or the Jordan-Weyl-Littlewood algebra A 1 • The first systematic study of this algebra which we have found appears in Littlewood [33]. Here, a laborious construction of a quotient field for A 1 is given, anticipating Ore's more general results. Independently, Ore [31, 33] introduced the general ring of skew polynomials k[t; p, 8], established its Euclidean-like properties, and gave the general construction of a classical quotient field for an arbitrary Ore domain. Subsequently, Hirsch [37] generalized some of the Littlewood algebras to 2n generators, providing necessary and sufficient conditions for their simplicity. Twenty-five years later, Rinehart [62] showed that r.gl.dim A 1 = l and in general, r.gl.dim An ~ 2n - l whenever the characteristic of k equals 0; r.gl.dim An = 2n otherwise. Here, An is the ring of linear differential operators in of8X 1 , ••• , ofoXn, over the partial differential ring k[X 1 , ••• , Xn]· Roos [72] has shown that r.gl.dim An = n if the characteristic of k equals 0, settling a special case of a problem posed in Rinehart's paper. Rinehart also showed that these rings can arise in a different algebraic setting, namely, as certain quotients of the universal enveloping algebra of a finite dimensional abelian Lie algebra. In the same direction and attesting to the ubiquity of simple Noetherian rings, Dixmier [63] has shown that if g is a finite dimensional nilpotent Lie algebra over an uncountable, algebraically closed field of characteristic 0, and I is an ideal of the

Notu

127

universal enveloping algebra A of g, then Afl : : : : Am for some m > 0, if and only if I is maximal (see also Cohn [61b]). The necessity of 3.2 (for the case of simple R) appears in Amitsur [57]; the necessity of 3.2(a) is implicit in Hart [71]. Also 3.3- 3.5 are the Krull dimension 1 case of more general results due to Hart (same paper). 3.6(d)-(/) are due to Webber [70]. Finally, the generalities on differential modules are due to J. L. Johnson [71]. Chapter 4 The Goldie and Faith-Utumi Theorems place any simple Noetherian ring R between Fn and Dn where F is a right Ore domain with right quotient field D, and nan integer ;;::: l. This matrix representation has the following drawback: R may properly contain Fn, since in general, F is only a ring-l. The precise relationship between R and F is unknown. For example, does there exist a maximal order F of D such that F n 4 R 1 And if so, are any two of them equivalent right orders 1 Let M denote the n x n matrix units of Dn, let A = {r E R I r M s;; R}, and B = {r E R I Mr s;; R}. Then, as we have shown in the Faith-Utumi theorem, U = B () D = A fl Dis an ideal of P = R () D, and Pn, Un, Un 2 are right orders equivalent to R. Moreover, the proof shows that P, U, and U2 are equivalent right orders of D. Thus far we have not needed the assumption that R is simple, that is, the above holds for any right order R in Dn· However, when R is simple, then a theorem of Faith [65] states that B, A, BA, and U 2 are all simple rings, that is, Fin the FaithUtumi theorem can be chosen to be a simple ring- I. In this case it is easy to see that R is then right Noetherian iff F satisfies the a.c.c. on idempotent right ideals. It would be interesting to know if F can be chosen to be unique up to equivalence of orders. In representing a simple Noetherian ring A = End 8 U, the cut down lemma enables one to choose an Ore domain B with at most 1 nontrivial ideal but then, unlike A, B may not be a maximal order in its quotient ring. On the other hand,

128

Simple Noetherian rings

by the maximal order theorem, we can choose U and B so that B is a maximal order but then B, unlike A, may have an infinite ideal lattice. What properties other than maximality which are inherited when A is simple and Noetherian, is an interesting question which deserves further exploration. We should also point out that the results in the section on maximal orders bear striking similarity to those for maximal orders in the classical sense (see Auslander-Goldman [60] and Ramras [69], [71 ]).

Chapters 5 and 6 Michler-Villamayor [73] have characterized right V-rings having right Krull dimension ~ 1 (that is, right V -rings R such that Rfl is Artinian for all essential right ideals /) as rings similat" to finite products of rings, each of which is a matrix algebra over a right Noetherian PCI-domain. (The theorem is also a consequence of techniques used repeatedly in Chapters 5 and 6.) Goodearl [72] showed that if every proper cyclic right Rmodule is semisimple, then R is right hereditary and a finite product of simple rings (etc.), reminiscent of the original constructions of Cozzens [70]. The observation that the solvability of systems of linear differential equations and the extensions of the base field are intimately connected, is due to Malgrange [66]. The present discussion, as well as (1) ~ (2) of 5.21 and its elegant proof, are due to J. Johnson. (The original proof of the remaining implications given in Cozzens [70] is more complicated.) q-fields were invented by Osofsky to produce examples of Vdomains (PCI-domains) with infinitely many nonisomorphic simples (see Osofsky [71 ]). The example following 5.26 is taken directly from that paper. Most of Chapter 6 is taken from Faith [73b ], and generalizes the results of Boyle [74] (see 6.26) to non-Noetherian PCI-rings.

Reforences

Albert, A. A. (1939) Structure of Algebras, Amer. Math. Soc. Colloq. Publ., Vol. 24, Providence, R.I. Amitsur, S. A. (1957) Derivations in Simple rings, Proc. Lond. Math. Soc. (3), 7, 87-112. Amitsur, S. A. (1972) On central division algebras, Israel J. Math. 12, 408-20. Artin, E. (1927) Ziir Theorie der hyperkomplexen Za.hlen, Abh. Math. Sem. Univ. Hamburg 5, 251-60. Auslander, M. & Goldman, 0. (1960) Maximal orders, TranB. Amer. Math. Soc. 97 (1), 1-24. Bass, H. (1960) Finitistic dimension and a. homological generalization of semiprimary rings, TranB. Amer. Math. Soc. 95, 466-88. Bass, H. (1962) The Morita Theorems, Mathematics Department, University of Oregon, Eugene. Boyle, A. K. (1974) Hereditary QI-rings, Trans. Amer. Math. Soc. 192, 115-20. Brown, B. & McCoy, N.H. (1950) The maximal regular ideal of a. ring, Proc. Amer. Math. Soc. 1, 165-71. Bumby, R. (1965) Modules which are isomorphic to submodules of each other, Arch. Math. 16, 184-5. Camillo, V. & Cozzens, J. H. (1973) A.theorem on Noetherian hereditary rings, Pacific J. Math. 45, 35-41. Ca.rta.n, H. & Eilenberg, S. (1956) Homological Algebra, Princeton Univ. Press, Princeton. Chatters, A. W. (1971) The restricted minimum condition in Noetherian hereditary rings, J. Lond. Math. Soc. (2) 4, 83-7. Cheva.lley, C. (1936) L'a.rithmetique dans les algebres de matrices, Actualitea Sci. Induat., No. 323, Paris. Cohn, P. M. ( 1961a) Quadratic extensions of skew fields, Proc. Lond. Math. Soc. (3), 11, 531-56. Cohn, P.M. (1961b) On the imbedding of rings in skew fields, Proc. Lond. Math. Soc. (3), 11, 511-30. Cozzens, J. H. (1970) Homological properties of the ring of differential polynomials, B1
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References

Faith, C. (1964) Noetherian simple rings, Bull. Amer. Math. Soc. 70, 730-1. Faith, C. (1965) Orders in simple Artinia.n rings, Trana. Amer. Math. Soc. 114, 61-5. Faith, C. (1966) Rings with ascending chain condition on annihilators, Nagoya Math. J. 27 (1), 179-91. Faith, C. (1967a) Lectures on Injective Modules and Quotient Rings, Lecture Notes in Mathematics, Springer-Verlag, New York-Berlin. Faith, C. (1967b) A general Wedderburn theorem, Bull. Amer. Math. Soc. 73, 65-7. Faith, C. (1971) Orders in semiloca.l rings, Bull. Amer. Math. Soc. 77, 960-2. Faith, C. (1972a) A correspondence theorem for projective modules, and the structure of simple Noetherian rings (in Proceedings of the Conference on Associative Algebras, Nov. 1970), Inatitute Nazionale di Alta Matematica, Sympoaium Matematica, 8, 309--45. Faith, C. (1972b) Modules finite over endomorphism ring, Proceedings of the Tulane University Symposium in Ring Theory, New Orleana, 1971, Lecture Notes in Mathematics No. 246, pp. 145-90, Springer-Verlag, New YorkBerlin. Faith, C. (1973a) Algebra: Rings, Modules and Categories, Springer-Verlag, New York-Berlin. Faith, C. (1973b) When are cyclics injective?, Pacific J. Math. 45, 97-112. Faith, C. (1974a) On the structure of indecomposable injective modules, Comm. in Algebra, 2, 559-71. Faith, C. (1974b) On hereditary rings and Ann Boyle's conjecture. Faith, C. & Utumi, Y. (1965) On Noetherian prime rings, Trana. Amer. Math. Soc. 114, 1084--9. Faith, C. & Walker, E. (1967) Direct-sum representations of injective modules, J. Algebra, 5 (2), 203--21. Findlay, G. & La.mbek, J. (1958) A generalized ring of quotients I, II, Can. Math. Bull. 1, 77-85, 155--67. Gabriel, P. (1962) Des categories a.beliennes, Bull. Soc. Math. France, 90, 323--448. Goldie, A. W. (1958) The structure of prime rings under the ascending chain condition, Proc. Lond. Math. Soc. (3), 8, 589--608. Goldie, A. W. (1960) Semiprime rings with maximum condition, Proc. Lond. Math. Soc. (3), 10, 201-20. Goldie, A. W. (1962) Noncommuta.tive principal ideal rings, Archiv. Math. 13, 213-21. Goldie, A. W. (1969) Some aspects of ring theory, Bull. Lond. Math. Soc. 1, 129-54. Goodea.rl, K. R. (1972) Singular torsion and the splitting properties, Amer. Math. Soc. Memoirs, No. 124. Goodea.rl, K. R. (1975) Subrings of idealizer rings, J. Algebra, 33 (3), 405-29. Hart, R. (1967) Simple rings with uniform right ideals, J. Lond. Math. Soc. 42, 614--7. Hart, R. (1971) Krull dimension and global dimension of simple Ore-extensions, Math. Zeit. 121, 341-5. Hart, R. & Robson, J. C. (1970) Simple rings and rings Morita. equivalent to Ore domains, Proc. Lond. Math. Soc. (3) 21, 232-42. Hirsch, K. A. (1937) A note on noncommuta.tive polynomials, J. Lond. Math. Soc. 12, 264--6. Jacobson, N. L. (1943) The Theory of Rings, Amer. Math. Soc. Mathematical Surveys, Vol. 2, Providence, R.I.

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Index

AI, 46, 125 a..c.c., 2 a..c.c.~, 27 a..c.c. EB, 22 adjoint (left), 123 algebra. of quantum mechanics, 125 Amitsur-Littlewood domains, 6, 43, 50 annihilator, 24 a.nnulet, 24 Artinia.n module, xiv Artinia.n ring, xiv ascending chain condition, 2 Auslander global dimension theorem, 121 Ba.er's criterion, 83 balanced module, xiv basis of a. module, xiv Bass' global dimension two theorem, 78 biendomorphism ring, xiv Boyle's conjecture, 116 Boyle's theorem, 112 Ca.rta.n-Bra.uer-Hua. theorem, 118 category, 122 cogenera.tor, 85 complement right ideal, 38 completely injective module, 102 constants of a. derivation, 44 contravariant functor, 123 correspondence theorem for projective modules, 17 covariant functor, 123 cut-down proposition, 10 d.c.c., 32 degree of a. differential polynomial, 43 ll·inva.ria.nt ideal, 44 derivation, 42 descending chain condition, 32 differential module, 49 differential polynomial ring, 43, 50

direct sum, xiv direct summand, xiv division algebra., 118 domain, xvii dual basis lemma., 9 dual (categories), 37 duality, 37 endomorphism ring, xiii endomorphism ring and projective module theorem, 28 endomorphism ring theorem, 1, 28 epi, xiv epimorphism, xiii equivalence (functor), 123 equivalence (map), 123 equivalent categories, 123 equivalent orders, 26 essential extension, 20 exact sequence, xv Ext'(-, - ), 121 Fa.ith-Utumi theorem, 5, 67 field, xvii filter, 114 finite dimensional module, 22 fir, 111 flat module, 122 free basis of a. module, xiv Frobenius map, 95 full (right) linear ring, 88 functor, 123 G·inva.ria.nt subset, 51 generator, 8 global dimension, 121 global dimension two theorem, 4, 41 Goldie dimension, 22 Goldie ring, 27 Goldie-Lesieur-Croisot theorem, 65 Hart's theorem, 28 hereditary ring, 33 hereditary ring theorem, 3, 33 Hilbert basis theorem, xvi

[ 133]

Index

134 homological dimension of a. module, 121 Hopkins-Levitzki theorem, 125 idealizer, '24 idempotent right ideal theorem, 2, 30 inner derivation, 42 injective hull, 84 injection, xiv injective mapping, xiv injective module, 83 Jacobson domains, 6, 61 Johnson-Utumi maximal quotient ring, 21 J orda.n-Weyl-Littlewood algebra., 125 Krull dimensional

~

1, 116

lateral right ideal, 23 l.d.o., 43 least ideal theorem, 2, 25 left equivalent ( ~) orders, 26 map, 122 matrix units, 67 maximal equivalent ( :t) order, 27 maximal left equivalent (~)order, 27 maximal order theorem, 3, 76 maximal rational extension, 21 maximal right equivalent ( ~) order, 27 maximal submodule, xvi maximum condition, xv minimum condition, xvi mono, xiv monomorphism, xiv Morita. equivalent, 15 Morita. invariant (property), 15 morphism, 122 natural isomorphism, xv naturally equivalent functors, 123 neat ring, 99 nil right ideal, 64 Noetherian module, xv N oetheria.n ring, xvi order, 72 Ore condition, 65 Ore domain, 27 Osofsky's theorem, 99 partial differential ring, 50 PCI.ring, 98

pli·(pri·) domain, 6 Popesco-Ga.briel theorem, 124 prime ring, xvii primitive ring, 62 principal right ideal theorem, 5, 70 progenera.tor, 14 projective dimension of a. module, 121 projective module, 9 projective module theorem, 2, 28 projective resolution of a. module, 121 proper cyclic, 98 q.field, 97 QI-ring, 116 quasi-injective module, 85 quoring, 26 quotient ring, 26 ra.d (radical), xiv rational extension, 20 reflexive ideal theorem, 4, 79 reflexive module, 50 regular ring, 83, 122 relative complement, 38 restricted (right) minimum condition, 116 p-deriva.tion, 52 right equivalent ( ~) orders, 26 right order, 26 ring of linear differential operators, 43, 50 ring·l, xiii Ritt algebra., 43 RRM,ll6 same centers theorem, 3, 26 Scha.nuel's lemma., 11 semiheredita.ry ring, 101 semiprime ring, xvii semisimple module, xvi sequence, xv short exact sequence, xv a-cyclic module, 106 L-cyclic module, 109 similar rings, 15 simple endomorphism rings theorem, 4, 29 simple localization, 6, 52 simple module, xvi simple ring, xvii single ideal theorem, 2, 26 singular ideal, 65 singular submodule, 20 skew polynomial ring, 52

Index soc (socle), xvi Steinitz extension, 118 subdirect product, 90 surjection, xiv surjective mapping, xiv summand, xiv torsion element of a. module, 36 torsion free module, 36 torsion module, 36 torsion submodule, 36 torsionless module, 50 trace, 8 twisted, finite Laurent polynomials,

95

135 twisted group ring, 50 ultrafilter, 114 ultra.product, 115 uniform module, 28 uniqueness of orders theorem, 2, 31 universal differential field, 7, 94 Utumi's theorem, 122 V-ring, 86 Von N euma.nn regular ring, 83, 122 Webber's theorem, 35 Wedderburn-Artin theorem, 32

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