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Lectures on modules and rings. (English) Zbl 0911.16001
Graduate Texts in Mathematics. 189. New York, NY: Springer. xxiii, 557 p. (1999).
The book under review is the third one written by the author in ring theory, the previous are “A first course in noncommutative rings” (abbreviation: FC) [1991; Zbl 0728.16001], and “Exercises in classical ring theory” [1994; Zbl 0823.16001]. The second book is tightly connected with FC containing solutions to all exercises from FC. The present work is in a way a continuation of FC including ring topics not considered in FC (or probably considered not enough). Comparing with FC, the book deals with much more vast and deep module theory; another topic not treated in FC is the theory of rings of quotients. Special attention is paid also to self-injective rings, particularly to QF-rings. The book ends with the Morita theory. A number of exercises helps the reader to study the book. It should be noted that by a ring the author means a ring with identity. Accordingly, a subring \(S\) of a ring \(R\) with an identity 1 means in particular \(1\in S\). A ring homomorphism from a ring \(R\) to a ring \(R'\) is supposed to take the identity of \(R\) to that of \(R'\).
From “Notes to the reader”. “Throughout the text, some familiarity with elementary ring theory is assumed, so that we can start our discussion at an “intermediate” level. Most (if not all) of the facts we need from commutative and noncommutative ring theory are available from standard first-year graduate algebra texts, such as those of Lang, Hungerford, and Isaacs, and certainly from the author’s FC”.
The book consists of 7 chapters, any chapter in its turn consists of sections (19 sections in total, numbered consecutively, independently of the chapters), and any section is divided into subsections (the total number of the subsections is 111). We give the titles of all chapters and sections only, thus omitting the titles of subsections.
Contents. Preface. Notes to the reader. Partial list of notations. Partial list of abbreviations. Chapter 1: Free modules, projective, and injective modules (1. Free modules, 2. Projective modules, 3. Injective modules). Chapter 2: Flat modules and homological dimensions (4. Flat and faithfully flat modules, 5. Homological dimensions). Chapter 3: More theory of modules (6. Uniform dimensions, complements, and CS modules, 7. Singular submodules and nonsingular rings, 8. Dense submodules and rational hulls). Chapter 4: Rings of quotients (9. Noncommutative localization, 10. Classical rings of quotients, 11. Right Goldie rings and Goldie’s theorems, 12. Artinian rings of quotients). Chapter 5: More rings of quotients (13. Maximal rings of quotients, 14. Martindale rings of quotients). Chapter 6: Frobenius and quasi-Frobenius rings (15. Quasi-Frobenius rings, 16. Frobenius rings and symmetric algebras). Chapter 7: Matrix rings, categories of modules, and Morita theory (17. Matrix rings, 18. Morita theory of category equivalences, 19. Morita duality theory). References. Name index. Subject index.

16-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras
16D10 General module theory in associative algebras
16E10 Homological dimension in associative algebras
16U20 Ore rings, multiplicative sets, Ore localization
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16D90 Module categories in associative algebras
16L60 Quasi-Frobenius rings
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)