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Relative finiteness in module theory. (English) Zbl 0556.16001
The aim of this monograph is a unified presentation of certain relativizations of notions in ring and module theory, such as injectivity, projectivity and especially chain conditions. The main attention of the authors is directed to the study of the lattice \(C_ F(M)\); here F denotes a Gabriel filter over a ring R, M an R-module and \(C_ F(M)\) the set of all submodules U of M such that M/U is torsion- free relative to the torsion theory (\({\mathcal T},{\mathcal F})\) defined by F. In order to prove results on \(C_ F(M)\), the authors systematically use the fact that this lattice is isomorphic to the lattice of all subobjects of T(M), where T denotes the canonical functor \(T:Mod R\to Mod R/{\mathcal T}.\)
The core of the book consists in a transparent proof of a theorem due to Miller and Teply [R. W. Miller and M. L. Teply; Pac. J. Math. 83, 207-219 (1979; Zbl 0444.16017)] stating that if \(C_ F(R)\) and \(C_ F(M)\) are artinian, then \(C_ F(M)\) is noetherian. It generalizes the classical theorem of Hopkins and Levitzki contending that (onesided) artinian rings with unit are noetherian. In subsequent sections the authors demonstrate that many ”artinian\(\Rightarrow noetherian''\) results in the literature, especially such concerning injective and projective modules, can be derived from Miller’s and Teply’s theorem. A number of further statements on relative chain conditions are loosely grouped around these observations.
We give a short summary of the contents. In Chapter I, elementary facts on M-injective and M-projective modules are compiled. Chapter II contains a systematic treatment on chain conditions of the module \(Hom_ R(M,N)\) considered as a bimodule over the endomorphism rings of N and of M. The central Chapter III begins with a review of hereditary torsion theories and quotient categories and contains the proof as well as applications of the Miller-Teply theorem. In the concluding Chapter IV, \(\Sigma\)- and \(\Delta\)-injective modules are analyzed, and some applications are given, for instance to the theory of quotient rings.
At the end of each section a large number of very useful exercises are included. The text is written in a clear and precise style and may serve as a source of reference for parts of ring and module theory.
Reviewer: W.Zimmermann

MSC:
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16P50 Localization and associative Noetherian rings
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16P40 Noetherian rings and modules (associative rings and algebras)
16P20 Artinian rings and modules (associative rings and algebras)