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Semidistributive Modules and Rings Mathematics 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 Jacobson radical 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 Noetherian 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 Monoid 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 group rings . 322 12.5 Can(~ellative monoids . 327 12.6 Semilattices and regular monoids 334 Bibliography 337 Index 350 Introduction A module M is called distributive if the lattice 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 ideal 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 ring 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 field) 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 global dimension one and hereditary rings. In [159], [161], [168J some applications of distributive rings and modules to formal power series rings were obtained. Dis­ tributive group and semigroup 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 commutative ring 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 ring theory. 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 torsion-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.
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