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.