Modules and rings. (English) Zbl 0817.16001
Cambridge: Cambridge University Press. xviii, 442 p. (1994).
This book is very useful for any graduated student interested in the modern theory of rings and modules. The first part of the book (Chapters 1-8) deals with free, projective and injective modules, simple modules, primitive rings, the Jacobson radical, subdirect products. The more advanced topics are considered in Chapters 9-18: primes and semiprimes, the prime radical, hereditary rings and more detailed theory of projective modules, categories and functors, module categories, purity. Every chapter ends with a number of exercises which sometimes contain important concepts not considered in the main text (e.g. some basic concepts of homological algebra are given in Exercises for Chapter 10). To call a statement or a paragraph the author uses a large scale of terms: theorem, proposition, construction, lemma, corollary, consequence, observation, remark (the terms are given here, according to the author’s meaning, in the descending order of importance). Only the most important results are called “theorems”; the author notes in the preface: “In other texts the reader has to decide which theorems are important and which are not, but not in this book. There just are no unimportant theorems”.
Contents: Preface; Note to the reader; Chapter 1 Modules: Introduction, 1-1 Definitions, 1-2 Direct products and sums, 1-3 Adjunction of 1 to \(R\), 1-4 Sequences of modules, 1-5 Exercises; Chapter 2 Free Modules: Introduction, 2-1 Definition of free modules, 2-2 Bases of free modules, Exercises; Chapter 3 Injective Modules: Introduction, 3-1 Properties of injectives, 3-2 Divisibility, 3-3 Embeddings in injectives, 3-4 Injective hulls, 3-5 Noetherian rings, 3-6 Examples, 3-7 Exercises; Chapter 4 Tensor Products: Introduction, 4-1 Tensor products of modules, 4-2 Definitions for algebras, 4-3 Tensor products of algebras, 4-4 Exercises; Chapter 5 Certain Important Algebras: 5-1 Free and tensor algebras, 5-2 Exterior algebras, 5-3 Exercises; Chapter 6 Simple Modules and Primitive Rings: Introduction, 6-1 Preliminaries, 6-2 Cyclic modules, 6-3 Simple modules, 6-4 Examples, 6-5 Density, 6-6 More on density and simples, 6-7 Examples, 6-8 Exercises; Chapter 7 The Jacobson Radical: Introduction, 7- 1 Characterizations, 7-2 Radicals of related rings, 7-3 Local rings, 7-4 Examples, 7-5 Exercises; Chapter 8 Subdirect Product Decompositions: Introduction, 8-1 Subdirect products, 8-2 Dense subdirect products, 8-3 Exercises; Chapter 9 Primes and Semiprimes: Introduction, 9-1 Prime ideals, 9-2 Semiprime ideals and the prime radical, 9-3 Nil radicals, 9-4 Primes and semiprimes in derived rings, 9-5 Exercises; Chapter 10 Projective Modules and more on Wedderburn Theorems: Introduction, 10-1 Projective modules, 10-2 Projective dimension, 10-3 Minimal right ideals, 10-4 Main theorems, 10-5 Direct proofs, 10-6 Uniqueness, 10-7 Rings with d.c.c. and idempotents, 10-8 Exercises; Chapter 11 Direct sum Decompositions: Introduction, 11-1 Completely reducible modules, 11-2 Radical of a module, 11-3 Artinian and Noetherian modules, 11-4 Direct sums of indecomposables, 11-5 Singular submodule, 11-6 Exercises; Chapter 12 Simple Algebras: Introduction, 12-1 Algebra modules, 12-2 Multiplication algebra, 12-3 Tensor products of simple rings, 12-4 Centralizers, 12-5 Double centralizers, 12-6 Exercises; Chapter 13 Hereditary Rings, Free and Projective Modules: Introduction, 13-1 Hereditary rings, 13-2 Injectivity and projectivity, 13-3 Finitely generated modules, 13-4 Examples, 13-5 Exercises; Chapter 14 Module Constructions: Introduction, 14-1 Pullbacks, 14-2 Pushouts, 14-3 Pushout application, 14-4 Exercises; Chapter 15 Categories and Functors: Introduction, 15-1 Basics of categories, 15-2 Objects, 15-3 Pre-additive categories, 15-4 Adjoint functors, 15-5 Exercises; Chapter 16 Module Categories: Introduction, 16-1 Generators and cogenerators, 16-2 Hom functor, 16-3 Tensor product functor, 16-4 Adjoint associativity, 16-5 Elements of tensor products, 16-6 Direct and inverse limits, 16-7 Exercises, 16-8 Exercises on direct and inverse limits; Chapter 17 Flat Modules: Introduction, 17-1 Character modules, 17-2 Flat module basics, 17-3 Exercises; Chapter 18 Purity: Introduction, 18-1 Systems of equations in modules, 18-2 Pure projectives and pure exact sequences, 18- 3 Direct limits, 18-4 Pure injectives, 18-5 Pure injective hull, 18-6 Exercises; Appendix A Basics: Introduction, A-1 Sets, A-2 Background review, A-3 Exercises; Appendix B Certain Important Algebras: Introduction, B-2 Exterior algebras, B-3 A unified approach; List of Symbols and Notation; Bibliography; Subject Index; Author Index.
Contents: Preface; Note to the reader; Chapter 1 Modules: Introduction, 1-1 Definitions, 1-2 Direct products and sums, 1-3 Adjunction of 1 to \(R\), 1-4 Sequences of modules, 1-5 Exercises; Chapter 2 Free Modules: Introduction, 2-1 Definition of free modules, 2-2 Bases of free modules, Exercises; Chapter 3 Injective Modules: Introduction, 3-1 Properties of injectives, 3-2 Divisibility, 3-3 Embeddings in injectives, 3-4 Injective hulls, 3-5 Noetherian rings, 3-6 Examples, 3-7 Exercises; Chapter 4 Tensor Products: Introduction, 4-1 Tensor products of modules, 4-2 Definitions for algebras, 4-3 Tensor products of algebras, 4-4 Exercises; Chapter 5 Certain Important Algebras: 5-1 Free and tensor algebras, 5-2 Exterior algebras, 5-3 Exercises; Chapter 6 Simple Modules and Primitive Rings: Introduction, 6-1 Preliminaries, 6-2 Cyclic modules, 6-3 Simple modules, 6-4 Examples, 6-5 Density, 6-6 More on density and simples, 6-7 Examples, 6-8 Exercises; Chapter 7 The Jacobson Radical: Introduction, 7- 1 Characterizations, 7-2 Radicals of related rings, 7-3 Local rings, 7-4 Examples, 7-5 Exercises; Chapter 8 Subdirect Product Decompositions: Introduction, 8-1 Subdirect products, 8-2 Dense subdirect products, 8-3 Exercises; Chapter 9 Primes and Semiprimes: Introduction, 9-1 Prime ideals, 9-2 Semiprime ideals and the prime radical, 9-3 Nil radicals, 9-4 Primes and semiprimes in derived rings, 9-5 Exercises; Chapter 10 Projective Modules and more on Wedderburn Theorems: Introduction, 10-1 Projective modules, 10-2 Projective dimension, 10-3 Minimal right ideals, 10-4 Main theorems, 10-5 Direct proofs, 10-6 Uniqueness, 10-7 Rings with d.c.c. and idempotents, 10-8 Exercises; Chapter 11 Direct sum Decompositions: Introduction, 11-1 Completely reducible modules, 11-2 Radical of a module, 11-3 Artinian and Noetherian modules, 11-4 Direct sums of indecomposables, 11-5 Singular submodule, 11-6 Exercises; Chapter 12 Simple Algebras: Introduction, 12-1 Algebra modules, 12-2 Multiplication algebra, 12-3 Tensor products of simple rings, 12-4 Centralizers, 12-5 Double centralizers, 12-6 Exercises; Chapter 13 Hereditary Rings, Free and Projective Modules: Introduction, 13-1 Hereditary rings, 13-2 Injectivity and projectivity, 13-3 Finitely generated modules, 13-4 Examples, 13-5 Exercises; Chapter 14 Module Constructions: Introduction, 14-1 Pullbacks, 14-2 Pushouts, 14-3 Pushout application, 14-4 Exercises; Chapter 15 Categories and Functors: Introduction, 15-1 Basics of categories, 15-2 Objects, 15-3 Pre-additive categories, 15-4 Adjoint functors, 15-5 Exercises; Chapter 16 Module Categories: Introduction, 16-1 Generators and cogenerators, 16-2 Hom functor, 16-3 Tensor product functor, 16-4 Adjoint associativity, 16-5 Elements of tensor products, 16-6 Direct and inverse limits, 16-7 Exercises, 16-8 Exercises on direct and inverse limits; Chapter 17 Flat Modules: Introduction, 17-1 Character modules, 17-2 Flat module basics, 17-3 Exercises; Chapter 18 Purity: Introduction, 18-1 Systems of equations in modules, 18-2 Pure projectives and pure exact sequences, 18- 3 Direct limits, 18-4 Pure injectives, 18-5 Pure injective hull, 18-6 Exercises; Appendix A Basics: Introduction, A-1 Sets, A-2 Background review, A-3 Exercises; Appendix B Certain Important Algebras: Introduction, B-2 Exterior algebras, B-3 A unified approach; List of Symbols and Notation; Bibliography; Subject Index; Author Index.
Reviewer: J.Ponizovskij (St.Peterburg)
MSC:
16-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras |
16D90 | Module categories in associative algebras |
16D40 | Free, projective, and flat modules and ideals in associative algebras |
16D25 | Ideals in associative algebras |
16D50 | Injective modules, self-injective associative rings |
16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
16N60 | Prime and semiprime associative rings |
16N80 | General radicals and associative rings |