Table Of ContentSemidistributive Modules and Rings
Mathematics and Its Applications
ManagingEditor:
M.HAZEWINKEL
CentreforMathematicsandComputerScience,Amsterdam,TheNetherlands
Volume449
Semidistributive
Modules and Rings
by
Askar A. Tuganbaev
Moscow Power Engineering Institute,
Technological Univt!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 1s t 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 Maximalsubmodules 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 ofquotients 101
5.1 Ore sets . . . . . . . . . . . . . . . . . 101
5.2 Denominatorsetsand localizablerings . 111
5.3 Maximal rings ofquotients 121
v
VI
6 Flat modules and semiperfect rings 133
6.1 Characterizations offlat modules 133
6.2 Submodules offlat 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 Modulesoverendomorphismrings andquasiinjectivemodules 187
8.2 Nilpotent endomorphisms . . . . . 195
8.3 Strongly indecomposable modules. . . . . . 205
9 Distributiveringswithmaximumconditions 209
9.1 Arithmetics ofideals . 209
9.2 Noet.herian rings . 213
9.3 Classicalrings ofquotientsofdistributiverings 221
9A Rinl?;s algebraic over their centre ... 225
10 Self-injective and skew-injective rings 237
10.1 Quasi-frobeniusrings anddirectsumsofinjectivemodules 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 Semidistributive 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 ofdistributive (resp. semidistributive) modules
properly cont.ains the class ofall 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 ofintegral
algebraic numbers or commutative principal ideal rings) are distributive.
A module is called a Bezout module or a locally cyclic module ifevery finitely
generated submodule is cyclic. If all maximal right ideals of a ring A are ideals
(e.g., ifA is commutative), then all Bezout A-modules are distributive.
Theclass£.ofringssuch thatfor every A E£., eachfinitely generated A-module
is semidistrihutive is wide (e.g., all Noetherian serial and Dedekind rings belong
to £.)). IfA is a commutative regular ring (e.g., a field) and G is a locally cyclic
group, then t.he group ring A[G] and the polynomial rings A[x] and A[x,x-i] are
distributive Bezout rings.
We highlight [23], [16], [101], [102], [17] among the first works concerning dis
tributive modulesand rings in the noncommutativecase. 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 investigaterings ofweak
global dimension one and hereditary rings. In [159], [161], [168J some applications
ofdistributive 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
tiveover theirendomorphismrings were studied. Topologicalaspects ofproperties
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 ofsome 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 acommutativering A isdistributive,¢=> allfinitely generated sub
modulesofM are multiplication [5J. Multiplicationmodulesover 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 ofthe ring A with respect to its maximal ideals are uniserial
rings [80J (in particular, all Priifer domains are distributive), the commutative
case deserves aspecial consideration. Here, ~ejust 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.
Thesecond aimis todevelopsometools, 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 ofrings (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 ofweak global dimension one.
The background required for this book (e.g., definitions ofa ring, a module, a
homomorphismetc) 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 ofthe module M, respectively.
IX
A simple module M is any module satisfying the following equivalent condi-
A
tions.
(i) M has no nonzero proper submodules.
(ii) Each nonzero homomorphism NA -+ M is an epimorphism.
(iii) Each nonzero homomorphism M -+ N is a monomorphism.
A
=
(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.
M1 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 ofall endomorphisms'P ofM
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 singularmodule 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 subset B ~ A,
we denote by rA(B) and £A(B) the right and left annihilators of the subset B,
respectively. We can omitthe subscripts ifthe situation isobvious. For a subset B
ofa ring A, we denote by Id(B) the ideal of A generated by B. A faithful module
=
is any module MA such that rA(M) O.
The ringofall nxn-matricesover a ring A isdenoted by An. Thesubset ofAn
which consists ofthe 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 ofunits ofa 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 M over a ring A is called divisible if given any x E M and
A
=
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 M is called torsion if M T(M). A module M
A A
=
is called torsion-free ifT(M) O.
0.4 A domain is any ring A such that each nonzero element ofA 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 principalright ideal ringis any ring Asuch that all right
ideals ofA 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 reducedring. A ring without nonzero nilpo
tent ideals is called a semiprime ring. A right (left) ideal I is called a right (left)
nil-idealifall 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
rA(B) the right annihilator of B ~ A IX
e
A(B) the left annihilator ofB ~ A IX
max(M) the set ofmaximal submodules of M 1
U(A) the group ofunits ofA IX
(F:C) therightideal{aEAIFa~C}ofA 8
H(M,T) the sum ofall submodules isomorphic to a simple module T 14
Soc(M) the socle of M 13
cl(N) the set ofall 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 ofall maximal submodules not. containing a submodule N 63
sg(M) the set ofall endomorphisms of M with essential kernels IX
Sing(M) the singular submodule of M IX
Kdim(M) Krull dimension of M 90
MI the direct product of I copies of M IX
M(I) the direct sum of I copies ofM 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 ofall regular elements of A 101
Qcl(A) a classical right ring ofquotients of A 102
clQd(A) a classical two-sided ring ofquotients ofA 103
gs(M) the set ofall f E End(M) such that f(M) is superfluous in M 195
Qmax(A) the maximal right ring ofquotients of A 124
maxQ(A) the maximal left ring ofquotients of A 124
(a, b/A) the (generalized) quaternion algebraover 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 ofa polynomial f 302
B* the annihilator ofan ideal B ofa semiprime ring A 278
supp(X) the support ofa subset X of A[C] 318
(X) the subgroup ofC generated by X ~ C 318
IHI the order ofa subgroup H ofC 318
wH, WA[GIH the right ideal L:hEH(1 - h)A[C] of A[C] 318
