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Grundlagen der Modul- und Ringtheorie. Ein Handbuch für Studium und Forschung. (Foundations of module and ring theory. A handbook for study and research). (German) Zbl 0657.16001
München (FRG): Verlag Reinhard Fischer. vi, 596 S. DM 65.00 (1988).
Since the appearance of the book “Homological Algebra” by H. Cartan and S. Eilenberg (1956; Zbl 0075.243), the study of the category R-MOD of modules over a ring R has become an integral part of the theory of associative rings with unit element. The study of this category as a whole has lead to important insights concerning the internal structure of rings. The first systematic treatment along these lines is perhaps P. Gabriel’s fundamental paper “Des catégories abéliennes” [Bull. Soc. Math. Fr. 90, 323-448 (1962; Zbl 0201.356)]. Although the theory of certain categories, for example that of functor categories, offers new aspects for module theory, the abstract categorical machinery has often obscured the objectives, and the present book aims at obtaining the results derived from the purely category theoretical approach by purely module theoretical means. This is achieved by the following trick. Instead of studying R-MOD as a whole, the author considers a full subcategory: For an R-module M let \(\sigma\) [M] denote the smallest subcategory of R-MOD that contains M and that is a Grothendieck category. This is the subcategory whose objects are the modules subgenerated by M, that is, all submodules of homomorphic images of direct sums of copies of M. It turns out that the formulation of the module theoretical results in \(\sigma\) [M] is no more complicated than in R-MOD, but the greater generality has significant advantages. Thus the connection between internal properties of M and properties of the category \(\sigma\) [M] leads to a homological classification of modules, and well-known results, such as the Density Theorem get a new interpretation. Altogether, this approach provides a significant refinement of the usual module theory, yet for the special case \(M=R\) it yields all the known results about R-MOD. The following translation of the chapter headings may give a (very) rough idea of the contents of the book: Elementary properties of rings, module categories, modules characterizable by the Hom-functor, notions derived from simple modules, finiteness conditions in modules, dual finiteness conditions, pure sequences and derived notions, modules described by means of projectivity, relations between functors, functor rings.
This book is clearly written and very systematically put together, although the (perhaps inevitably) rather dry style will probably make it more useful as a reliable source of information for the expert than as an introduction for the newcomer to the field. The book contains a very extensive bibliography that leads right up to the current state of the subject, and each chapter ends with a number of problems, mostly designed to encourage the interested reader to consult the relevant research papers. All in all, this is a very useful handbook that should be on the shelf of anyone active in this area of mathematics.
Reviewer: G.Krause

16-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
18E15 Grothendieck categories (MSC2010)