Table Of ContentFoundations of Module and Ring Theory
A Handbook for Study and Research
Robert Wisbauer
University of Du¨sseldorf
1991
Gordon and Breach Science Publishers, Reading
2
Contents
Preface ........................................................... v
Symbols ......................................................... vii
Chapter 1 Elementary properties of rings
1 Basic notions ................................................. 1
2 Special elements and ideals in rings ........................... 7
3 Special rings ................................................ 14
4 Chain conditions for rings ................................... 26
5 Algebras and group rings .................................... 30
Chapter 2 Module categories
6 Elementary properties of modules ........................... 36
7 The category of R-modules ................................. 43
8 Internal direct sum ......................................... 57
9 Product, coproduct and subdirect product .................. 64
10 Pullback and pushout ...................................... 73
11 Functors, Hom-functors ..................................... 80
12 Tensor product, tensor functor .............................. 90
Chapter 3 Modules characterized by the Hom-functor
13 Generators, trace .......................................... 105
14 Cogenerators, reject ....................................... 112
15 Subgenerators, the category σ[M] ......................... 118
16 Injective modules .......................................... 127
17 Essential extensions, injective hulls ........................ 137
18 Projective modules ........................................ 148
19 Superfluous epimorphisms, projective covers ............... 159
i
ii Contents
Chapter 4 Notions derived from simple modules
20 Semisimple modules and rings ............................. 165
21 Socle and radical of modules and rings ..................... 174
22 The radical of endomorphism rings ........................ 185
23 Co-semisimple and good modules and rings ................ 190
Chapter 5 Finiteness conditions in modules
24 The direct limit ........................................... 196
25 Finitely presented modules ................................ 207
26 Coherent modules and rings ............................... 214
27 Noetherian modules and rings ............................. 221
28 Annihilator conditions ..................................... 230
Chapter 6 Dual finiteness conditions
29 The inverse limit .......................................... 238
30 Finitely copresented modules .............................. 248
31 Artinian and co-noetherian modules ....................... 253
32 Modules of finite length ................................... 265
Chapter 7 Pure sequences and derived notions
33 P-pure sequences, pure projective modules ................ 274
34 Purity in σ[M], R-MOD and ZZ-MOD .................... 281
35 Absolutely pure modules .................................. 297
36 Flat modules .............................................. 304
37 Regular modules and rings ................................ 313
38 Copure sequences and derived notions ..................... 322
Chapter 8 Modules described by means of projectivity
39 (Semi)hereditary modules and rings ....................... 328
40 Semihereditary and hereditary domains .................... 341
41 Supplemented modules .................................... 348
42 Semiperfect modules and rings ............................ 371
43 Perfect modules and rings ................................. 382
Chapter 9 Relations between functors
44 Functorial morphisms ..................................... 393
45 Adjoint pairs of functors ................................... 399
46 Equivalences of categories ................................. 413
47 Dualities between categories ............................... 425
48 Quasi-Frobenius modules and rings ........................ 445
Contents iii
Chapter 10 Functor rings
49 Rings with local units ..................................... 464
50 Global dimensions of modules and rings ................... 476
51 The functor Hbom(V,−) .................................... 485
52 Functor rings of σ[M] and R-MOD ........................ 506
53 Pure semisimple modules and rings ........................ 521
54 Modules of finite representation type ...................... 531
55 Serial modules and rings ................................... 539
56 Homo-serial modules and rings ............................ 560
Bibliography ................................................... 576
Index ........................................................... 599
iv
Preface
On the one hand this book intends to provide an introduction to module
theoryandtherelatedpartofringtheory. Startingfromabasicunderstand-
ing of linear algebra the theory is presented with complete proofs. From the
beginning the approach is categorical.
On the other hand the presentation includes most recent results and
includes new ones. In this way the book will prove stimulating to those
doing research and serve as a useful work of reference.
Since the appearance of Cartan-Eilenberg’s Homological Algebra in the
1950smoduletheoryhasbecomeamostimportantpartofthetheoryofasso-
ciativeringswithunit. ThecategoryR-MODofunitalmodulesoveraringR
also served as a pattern for the investigation of more general Grothendieck
categories which are presented comprehensively in Gabriel’s work of 1962
(Bull.Soc.Math.France).
Whereas ring theory and category theory initially followed different di-
rections it turned out in the 1970s – e.g. in the work of Auslander – that
the study of functor categories also reveals new aspects for module theory.
In our presentation many of the results obtained this way are achieved by
purely module theoretic methods avoiding the detour via abstract category
theory (Chapter 10). The necessary extension of usual module theory to get
this is gained by an artifice.
From the very beginning the central point of our considerations is not
the entire category R-MOD but a full subcategory of it: for an R-module M
we construct the ’smallest’ subcategory of R-MOD which contains M and is
aGrothendieck category. Thisisthesubcategoryσ[M]whichissubgenerated
by M, i.e. its objects are submodules of M-generated modules.
The elaboration of module theoretic theorems in σ[M] is not more te-
dious than in R-MOD. However, the higher generality gained this way with-
out effort yields significant advantages.
The correlation of (internal) properties of the module M with properties
ofthecategoryσ[M]enablesahomological classificationofmodules. Among
other things, the Density Theorem has a new interpretation (in 15.8). All
in all the approach chosen here leads to a clear refinement of the customary
module theory and, for M = R, we obtain well-known results for the entire
module category over a ring with unit.
In addition the more general assertions also apply to rings without units
andcomprisethemoduletheoryfors-unitalringsandringswithlocalunits.
This will be especially helpful for our investigations of functor rings.
v
vi
For example, a new proof is obtained for the fact that a ring of left finite
(representation) type is also of right finite type (see 54.3). For serial rings
and artinian principal ideal rings we derive interesting characterizations in-
volving properties of the functor rings (see 55.15, 56.10).
Another special feature we could mention is the definition of linearly
compact modules through an exactness condition on the inverse limit (in
29.7). This permits more transparent proofs in studying dualities between
module categories (in section 47).
Let us indicate some more applications of our methods which are not
coveredinthebook. Categoriesofthetypeσ[M]arethestartingpointfora
richmoduletheoryovernon-associativeringsA. Forthis, Aisconsideredas
module over the (associative) multiplication algebra M(A) and the category
σ[A]isinvestigated. Alsotorsionmodulesoveratopologicalringandgraded
modules over a graded ring form categories of the type σ[M].
For orientation, at the beginning of every section the titles of the para-
graphs occurring in it are listed. At the end of the sections exercises are
includedtogivefurtherinsightintothetopicscoveredandtodrawattention
to related results in the literature. References can be found at the end of
the paragraphs. Only those articles are cited which appeared after 1970. In
citations of monographs the name of the author is printed in capital letters.
This book has evolved from lectures given at the Universities of Nantes
andDu¨sseldorffrom1978onwards. Theprintingwasmadepossiblethrough
thetechnicalassistanceoftheRechenzentrumoftheUniversityofDu¨sseldorf.
I wish to express my sincere thanks to all who helped to prepare and
complete the book.
Du¨sseldorf, Summer 1988
Besides several minor changes and improvements this English edition
contains a number of new results. In 48.16 cogenerator modules with com-
mutative endomorphism rings are characterized. In 51.13 we prove that a
category σ[M] which has a generator with right perfect endomorphism ring
also has a projective generator. In 52.7 and 52.8 the functor rings of regular
and semisimple modules are described. Three more theorems are added in
section 54.
Also a number of additional exercises as well as references are included.
I am very indebted to Patrick Smith and Toma Albu for their help in cor-
recting the text.
Du¨sseldorf, Spring 1991
Robert Wisbauer
Symbols
N(R) nil radical of R 11
Np(R) sum of the nilpotent ideals in R 11
P(R) prime radical of R 11
RG semigroup ring 32
An (M) annihilator of an R-module M 42
R
ENS category of sets 44
GRP category of groups 45
AB category of abelian groups 45
R-MOD category of left R-modules 45
R-mod category of finitely generated left R-modules 46
T morphism map for the functor T 81
A,B
Tr(U,L) trace of U in L 107
Re(L,U) reject of U in L 113
σ[M] subcategory of R-MOD subgenerated by M 118
QM N product of N in σ[M] 118
Λ λ λ
t(M) torsion submodule of a ZZ-module M 124
p(M) p-component of a ZZ-module M 124
ZZ Pru¨fer group 125
p∞
K EM K is an essential submodule of M 137
Nb M-injective hull of N 141
K (cid:28) M K is a superfluous submodule of M 159
SocM socle of M 174
RadM radical of M 176
JacR Jacobson radical of R 178
limM direct limit of modules M 197
−→ i i
An(K) annihilator of a submodule K ⊂ M 230
Ke(X) annihilator of a submodule X ⊂ Hom (N,M) 230
R
limN inverse limit of modules N 239
←− i i
lg(M) length of M 267
L∗, L∗∗ U-dual and U-double dual module of L 411
σ [M] submodules of finitely M-generated modules 426
f
Hbom(V,N) morphisms which are zero almost everywhere 485
Ebnd(V) endomorphisms which are zero almost everywhere 485
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