Foundations of Module and Ring Theory A Handbook for Study and Research Robert Wisbauer University of D¨usseldorf 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 Homb (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 theory and the related part of ring theory. Starting from a basic understand- 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 1950s module theory has become a most important part of the theory of asso- ciative rings with unit. The category R-MOD of unital modules over a ring R 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 a Grothendieck category. This is the subcategory σ[M] which is subgenerated 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 of the category σ[M] enables a homological classification of modules. 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 and comprise the module theory for s-unital rings and rings with local units. 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 covered in the book. Categories of the type σ[M] are the starting point for a rich module theory over non-associative rings A. For this, A is considered as module over the (associative) multiplication algebra M(A) and the category σ[A] is investigated. Also torsion modules over a topological ring and graded 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 included to give further insight into the topics covered and to draw attention 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 and D¨usseldorf from 1978 onwards. The printing was made possible through the technical assistance of the Rechenzentrum of the University of D¨usseldorf. I wish to express my sincere thanks to all who helped to prepare and complete the book. D¨usseldorf, 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. D¨usseldorf, 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 AnR(M) annihilator of an R-module M 42 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 TA,B morphism map for the functor T 81 T r(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 Λ Nλ product of Nλ in σ[M] 118 t(M) torsion submodule of a ZZ-module M 124 p(M) p-component of a ZZ-module M 124 ZZp∞ Pr¨ufer group 125 K E MK is an essential submodule of M 137 NMb -injective hull of N 141 K MK is a superfluous submodule of M 159 Soc M socle of M 174 Rad M radical of M 176 Jac R Jacobson radical of R 178 lim M direct limit of modules M 197 −→ i i An(K) annihilator of a submodule K ⊂ M 230 Ke(X) annihilator of a submodule X ⊂ HomR(N, M) 230 lim N 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 σf [M] submodules of finitely M-generated modules 426 Homb (V, N) morphisms which are zero almost everywhere 485 Endb (V ) endomorphisms which are zero almost everywhere 485 vii 0 Chapter 1 Elementary properties of rings Before we deal with deeper results on the structure of rings with the help of module theory we want to provide elementary definitions and con- structions in this chapter. 1 Basic notions A ring is defined as a non-empty set R with two compositions + , · : R × R → R with the properties: (i) (R, +) is an abelian group (zero element 0); (ii) (R, · ) is a semigroup; (iii) for all a, b, c ∈ R the distributivity laws are valid: (a + b)c = ac + bc, a(b + c) = ab + ac.