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Semidistributive Modules and Rings and Its Applications

Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 449 Semidistributive Modules and Rings

by

Askar A. Tuganbaev Moscow Power Engineering Institute, Technological Univ t!rsiry, Moscow, Russia

SPRINGER SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-6136-0 ISBN 978-94-011-5086-6 (eBook) DOI 10.1007/978-94-011-5086-6

Printed on acid-free paper

Ali Rights Reserved © 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 Softcover reprint of the hardcover 1st edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Contents

Introduction vii

Symbols x

1 Radical:-;, local and semisimple modules 1 1.1 Maximal submodules and the 1 1.2 Local and uniserial modules .... 6 1.3 Semisimple and Artinian modules. 11 1.4 The prime radical...... 21

2 Projective and injective modules 25 2.1 Free and projective modules. 25 2.2 Inje,tive modules 31 2.3 Inje,tive hull . 35

3 Bezout nnd regular modules 47 3.1 Regular modules ...... 47 3.2 Unil-regular rings . 52 3.3 Semilocal rings and distributivity 54 3.4 Strongly regular rings 59 3.5 Bezout rings...... 68

4 Continuous and finite-dimensional modules 73 4.1 Closed submodules . 73 4.2 Cont.inuous modules . 78 4.3 Finil,e-dimensional modules . 86 4.4 Nonsingular rr-injective modules 96

5 Rings of quotients 101 5.1 Ore sets ...... 101 5.2 Denominator sets and localizable rings . 111 5.3 Maximal rings of quotients 121

v VI

6 Flat modules and semiperfect rings 133 6.1 Characterizations of flat modules 133 6.2 Submodules of flat modules . 139 6.3 Semiperfect and perfect rings .. 149

7 Semihereditary and invariant rings 159 7.1 Coherent and reduced rings .... 159 7.2 Invariant rings 168 7.3 Rings with integrally closed factor rings 175

8 Endomorphism rings 187 8.1 Modules over endomorphism rings and quasi injective modules 187 8.2 Nilpotent endomorphisms ..... 195 8.3 Strongly indecomposable modules...... 205

9 Distributive rings with maximum conditions 209 9.1 Arithmetics of ideals . 209 9.2 Noet.herian rings . 213 9.3 Classical rings of quotients of distributive rings 221 9A Rinl?;s algebraic over their centre ... 225

10 Self-injective and skew-injective rings 237 10.1 Quasi-frobenius rings and direct sums ofinjective modules 237 10.2 Cyclic 1r-injective modules ..... 240 10.3 Intel?;rally closed rings. 243 lOA Cyclic skew-injective modules 250 10.5 Countably injective rings .. .. 257

11 Semidistributive and serial rings 261 11.1 Sem idistributive modules 261 11.2 Semidistributive rings . 272 11.3 Serial modules and rings. 285

12 rings and related topics 301 12.1 Seri('s and polynomial rings . 301 12.2 Quaternion algebras . 311 12.3 Subgroups, submonoids, and annihilators 318 12A Regular rings . 322 12.5 Can(~ellative . 327 12.6 and regular monoids 334

Bibliography 337

Index 350 Introduction

A M is called distributive if the Lat(M) of all its submodules is distributive, i.e., Fn(G + H) = FnG + FnH for all submodules F,G, and H of the module M. A module M is called uniserial if all its submodules are comparable with respect to inclusion, i.e., the lattice Lat(M) is a chain. Any direct sum of distributive (resp. uniserial) modules is called a semidistributive (resp. serial) module. The class of distributive (resp. semidistributive) modules properly cont.ains the class of all uniserial (resp. serial) modules. In particular, all simple (resp. semisimple) modules are distributive (resp. semidistributive). All strongly regular rings (for example, all factor rings of direct products of division rings and all commutative regular rings) are distributive; all valuation rings in division rings and all commutative Dedekind rings (e.g., rings of integral algebraic numbers or commutative principal rings) are distributive. A module is called a Bezout module or a locally cyclic module if every finitely generated submodule is cyclic. If all maximal right ideals of a A are ideals (e.g., if A is commutative), then all Bezout A-modules are distributive. The class £. of rings such that for every A E £., each finitely generated A-module is semidistrihutive is wide (e.g., all Noetherian serial and Dedekind rings belong to £.)). If A is a commutative regular ring (e.g., a ) and G is a locally cyclic i group, then t.he group ring A[G] and the polynomial rings A[x] and A[x, x- ] are distributive Bezout rings. We highlight [23], [16], [101], [102], [17] among the first works concerning dis• tributive modules and rings in the noncommutative case. The systematic study of distributive modules over noncommutative rings was initiated in papers [1], [25], [26], [36], and [131]. Distributive modules were considered in [43, §4.1]' [19, Ch. 9], and [116, §2.2]. In [214] distributive modules are applied to complex analysis. In [203], distributive rings are used for a study of rings of continuous functions in topological spaces. In [211], distributive modules are applied to study rings with duality. In [80], [155], [161], [162J distributive rings are used to investigate rings of weak one and hereditary rings. In [159], [161], [168J some applications of distributive rings and modules to rings were obtained. Dis• tributive group and rings were studied in [41]' [73], [89], [158], [160], [161], [163], [165], and [174J. Distributive quaternion algebras were studied in [175], [176], and [177J. In [117], [156], [163], and [180J modules which are distribu• tive over their endomorphism rings were studied. Topological aspects of properties

VB Vlll of distributive modules and rings were considered in [163], [204], and [202]. Dis• tributive modules over incidence algebras were studied in [55]. Semidistributive modules and right semidistributive rings were studied in [22], [35], [58], [59], [86], [88], [87], [191], [193], [138], [210], [212], and [211]. Distributive graded modules were considered in [38]. Rings possessing faithful distributive modules were stud• ied in [16] and [17J. Conditions sufficient for the distributivity of some lattices of linear subspaces were considered in [75] and [84J. Noncommutative rings whose lattices of two-sided ideals are distributive were studied in [23], [20], [92], [103], and [85J. Distributive modules and rings were addressed in surveys [96J, [97J, and [21J. Distributive modules are closely related to multiplication modules (a module is called mull.iplication if M(M : N) = N for any N E Lat(M)). For example, a module M over a A is distributive,¢=> all finitely generated sub• modules of M are multiplication [5J. Multiplication modules over noncommutative rings were considered in [140], [198], [199], [200], [126], and [190J. Multiplication modules over commutative rings were studied in [5], [6], [7], [12], [34], [40J [47], [48], [51], [60], [70], [95], [105], [106], [110], [111]' [112], [113], [123], and [125]. In this book, as a rule, we consider distributive modules over noncommutative rings. Since the distributivity of a commutative ring A is equivalent to the fact that all localizations of the ring A with respect to its maximal ideals are uniserial rings [80J (in particular, all Priifer domains are distributive), the commutative case deserves a special consideration. Here, ~e just highlight papers [5], [80], [81], [122], [173], [204J. Moreover, it is worth noting that uniserial and serial modules and rings (e.g., valuation rings), which are addressed in considerable number of papers, are little touched here. We only mention papers [45J and [207J. Also, in this book we pursue the following two additional aims. The first is to provide the reader with an introduction to the homological and structural methods of the . This book contains the basic facts on projective, injective, flat, semisimple, regular, and finite-dimensional (in the sense of Goldie) modules. The second aim is to develop some tools, which are not covered in monographs. In particular, in Chapter 5 we extend the basic facts on classical localizations of commutative rings to some wider class of rings (cf. [155], [168]). Also, in Chapter 6 we use Hattori -free modules [74], [76J to study flat modules. In Chapter 12 we study power series rings of weak global dimension one. The background required for this book (e.g., definitions of a ring, a module, a homomorphism etc) is standard, and can be found in most graduate level texts on algebra. Before going to the main presentation we cite some remarks, notations and definitions.

0.1 All rings are assumed to be associative and (except for nil-rings and for some stipulated cases) to have nonzero identity elements. Expressions like "a Noetherian ring" mean that the corresponding right and left conditions hold. We denote by End(M) and Lat(M) the endomorphism ring and the lattice of all submodules of the module M, respectively. IX

A simple module MA is any module satisfying the following equivalent condi- tions. (i) M has no nonzero proper submodules. (ii) Each nonzero homomorphism NA -+ M is an epimorphism. (iii) Each nonzero homomorphism M -+ NA is a monomorphism. (iv) M =mA for any nonzero m EM. (v) For any nonzero elements m and n of M, there exists a E A such that ma=n. M 1 and M(I) denotes the direct product and the direct sum of I copies of a module M. Let M be a right module over a ring A. The set of all endomorphisms 'P of M such that Ker('P) is an essential submodule of Af is denoted by sg(M). The set of all m EM such that r(m) is an essential right ideal of A is denoted by Sing(M). A singular module is any module M such that Sing( M) =- M. A nonsingular module is any module M such that Sing(M) = O. 0.2 For a ring A, we denote by C(A) its centre, and for a B ~ A, we denote by rA(B) and £A(B) the right and left annihilators of the subset B, respectively. We can omit the subscripts if the situation is obvious. For a subset B of a ring A, we denote by Id(B) the ideal of A generated by B. A faithful module is any module M A such that r A (M) = O. The ring of all n x n-matrices over a ring A is denoted by An. The subset of An which consists of the matrices whose entries belong to a subset B of A is denoted by Bn . An element a of A is called right regular (resp. left regular, right invertible, left invertible) in A if r(a) = 0 (resp. £(a) = 0, aA = A, and Aa = A). A right and left regular (resp. invertible) element is called a regular (invertible) element. Right (resp. left) invertible elements are also called right (resp. left) units of A. The group of units of a ring A is denoted by U(A).

0.3 A submodule of the factor module of it module M is called a subfactor of M. A module MA over a ring A is called divisible if given any x E M and any regular element a E A, there exists m E M such that .'1: = ma. For any module MA, the set T(M) of all elements m E M such that '/'A(m) contains a regular element of A is called the torsion part of M. The set T(M) is not always a submodule of M. A module MA is called torsion if M =T(M). A module MA is called torsion-free if T(M) = O. 0.4 A is any ring A such that each nonzero element of A is regular. A module MA over a domain A is divisible <¢:::=> M = Ma for any nonzero a E A. A module MA over a domain A is torsion-free <¢:::=> ma 1= 0 for any nonzero m E M and any nonzero a E A. A principal right ideal ring is any ring A such that all right ideals of A are principal. An orthogonally finite ring is any ring which contains no infinite sets of nonzero orthogonal idempotents. A ring is normal if all its idempotents are central. A ring A which has no nonzero nilpotent elements is called a . A ring without nonzero nilpo• tent ideals is called a semiprime ring. A right (left) ideal I is called a right (left) nil-ideal if all elements of I are nilpotent. x

Symbols

End(M) the endomorphism ring of M Vlll Lat(M) the lattice of all submodules of M Vlll C(A) the centre of A IX r A (B) the right annihilator of B ~ A IX eA (B) the left annihilator of B ~ A IX max(M) the set of maximal submodules of M 1 U(A) the group of units of A IX (F:C) therightideal{aEAIFa~C}ofA 8 H(M, T) the sum of all submodules isomorphic to a simple module T 14 Soc(M) the of M 13 cl(N) the set of all maximal submodules of M containing N E Lat(M) 63 max(M) the topological space defined on max(M) if M is spectral 63 op(N) the set of all maximal submodules not. containing a submodule N 63 sg(M) the set of all endomorphisms of M with essential kernels IX Sing( M) the singular submodule of M IX Kdim(M) of M 90 M I the direct product of I copies of M IX M(I) the direct sum of I copies of M IX c(I) the set of all elements a E A such that a + I is regular in A/I 101 c(O) the set of all regular elements of A 101 Qcl(A) a classical right ring of quotients of A 102 clQd(A) a classical two-sided ring of quotients of A 103 gs(M) the set of all f E End(M) such that f(M) is superfluous in M 195 Qmax(A) the maximal right ring of quotients of A 124 maxQ(A) the maximal left ring of quotients of A 124 (a, b/A) the (generalized) quaternion algebra over A 311 (-1, -1/A) the hamiltonian quaternion algebra over A 311 Ad[x, iP]] the left skew (power) series ring 301 Ar[[x, iP]] the right skew (power) series ring 301 deg(g) the degree of a polynomial f 302 B* the annihilator of an ideal B of a semiprime ring A 278 supp(X) the support of a subset X of A[C] 318 (X) the subgroup of C generated by X ~ C 318 IHI the order of a subgroup H of C 318 wH, WA[GIH the right ideal L:hEH(1 - h)A[C] of A[C] 318