Rings and Their Modules

Description

This book is an introduction to the theory of rings and modules that goes beyond what one normally obtains in a graduate course in abstract algebra. The theme of the text is the interplay between rings and modules. At times rings are investigated by considering given sets of conditions on the modules they admit and at other times rings of a certain type are considered to see what structure is forced on their modules. Standard topics in ring and module theory such as chain conditions on rings and modules, injective and projective modules and semisimple rings are included as well as subjects like category theory and homological algebra. The text also contains presentations on topics such as flat modules and coherent rings, injective envelopes, projective covers and perfect rings, reflexive modules and quasi-Frobenius rings, and graded rings and modules. The book is a self-contained volume written in a very systematic style: all proofs are clear and easy for the reader to understand and all arguments are based on materials contained in the book. A problem sets follow each section.  It is assumed that the reader is familiar with concepts such as Zorn's lemma, commutative diagrams and ordinal and cardinal numbers. It is also assumed that the reader has a basic knowledge of rings and their homomorphisms. The text is suitable for graduate and PhD students who have chosen ring theory for their research subject. Suitable for graduate students and researchers Includes lots of exercises with tips Preface About the Text Acknowledgements Contents 0 Preliminaries 0.1 Classes, Sets and Functions Partial Orders and Equivalence Relations Zorn’s Lemma and Well-Ordering 0.2 Ordinal and Cardinal Numbers 0.3 Commutative Diagrams 0.4 Notation and Terminology Problem Set 1 Basic Properties of Rings and Modules 1.1 Rings Problem Set 1.1 1.2 Left and Right Ideals Factor Rings Problem Set 1.2 1.3 Ring Homomorphisms Problem Set 1.3 1.4 Modules Factor Modules Problem Set 1.4 1.5 Module Homomorphisms Problem Set 1.5 2 Fundamental Constructions 2.1 Direct Products and Direct Sums Direct Products External Direct Sums Internal Direct Sums Problem Set 2.1 2.2 Free Modules Rings with Invariant Basis Number Problem Set 2.2 2.3 Tensor Products of Modules Problem Set 2.3 3 Categories 3.1 Categories Functors Properties of Morphisms Problem Set 3.1 3.2 Exact Sequences in ModR Split Short Exact Sequences Problem Set 3.2 3.3 Hom and as Functors Properties of Hom Properties of Tensor Products Problem Set 3.3 3.4 Equivalent Categories and Adjoint Functors Adjoints Problem Set 3.4 4 Chain Conditions 4.1 Generating and Cogenerating Classes Problem Set 4.1 4.2 Noetherian and Artinian Modules Problem Set 4.2 4.3 Modules over Principal Ideal Domains Free Modules over a PID Finitely Generated Modules over a PID Problem Set 4.3 5 Injective, Projective, and Flat Modules 5.1 Injective Modules Injective Modules and the Functor HomR(—,M) Problem Set 5.1 5.2 Projective Modules Projective Modules and the Functor HomR (M, —) Hereditary Rings Semihereditary Rings Problem Set 5.2 5.3 Flat Modules Flat Modules and the Functor M R — Coherent Rings Regular Rings and Flat Modules Problem Set 5.3 5.4 Quasi-Injective and Quasi-Projective Modules Problem Set 5.4 6 Classical Ring Theory 6.1 The Jacobson Radical Problem Set 6.1 6.2 The Prime Radical Prime Rings Semiprime Rings Problem Set 6.2 6.3 Radicals and Chain Conditions Problem Set 6.3 6.4 Wedderburn–Artin Theory Problem Set 6.4 6.5 Primitive Rings and Density Problem Set 6.5 6.6 Rings that Are Semisimple Problem Set 6.6 7 Envelopes and Covers 7.1 Injective Envelopes Problem Set 7.1 7.2 Projective Covers The Radical of a Projective Module Semiperfect Rings Perfect Rings Problem Set 7.2 7.3 QI-Envelopes and QP-Covers Quasi-Injective Envelopes Quasi-Projective Covers Problem Set 7.3 8 Rings and Modules of Quotients 8.1 Rings of Quotients The Noncommutative Case The Commutative Case Problem Set 8.1 8.2 Modules of Quotients Problem Set 8.2 8.3 Goldie’s Theorem Problem Set 8.3 8.4 The Maximal Ring of Quotients Problem Set 8.4 9 Graded Rings and Modules 9.1 Graded Rings and Modules Graded Rings Graded Modules Problem Set 9.1 9.2 Fundamental Concepts Graded Direct Products and Sums Graded Tensor Products Graded Free Modules Problem Set 9.2 9.3 Graded Projective, Graded Injective and Graded Flat Modules Graded Projective and Graded Injective Modules Graded Flat Modules Problem Set 9.3 9.4 Graded Modules with Chain Conditions Graded Noetherian and Graded Artinian Modules Problem Set 9.4 9.5 More on Graded Rings The Graded Jacobson Radical Graded Wedderburn–Artin Theory Problem Set 9.5 10 More on Rings and Modules 10.1 Reflexivity and Vector Spaces Problem Set 10.1 10.2 Reflexivity and R-modules Self-injective Rings Kasch Rings and Injective Cogenerators Semiprimary Rings Quasi-Frobenius Rings Problem Set 10.2 11 Introduction to Homological Algebra 11.1 Chain and Cochain Complexes Homology and Cohomology Sequences Problem Set 11.1 11.2 Projective and Injective Resolutions Problem Set 11.2 11.3 Derived Functors Problem Set 11.3 11.4 Extension Functors Right Derived Functors of HomR(—, X) Right Derived Functors of HomR(X, —) Problem Set 11.4 11.5 Torsion Functors Left Derived Functors of — r X and X R — Problem Set 11.5 12 Homological Methods 12.1 Projective and Injective Dimension Problem Set 12.1 12.2 Flat Dimension Problem Set 12.2 12.3 Dimension of Polynomial Rings Problem Set 12.3 12.4 Dimension of Matrix Rings Problem Set 12.4 12.5 Quasi-Frobenius Rings Revisited More on Reflexive Modules Problem Set 12.5 A Ordinal and Cardinal Numbers Ordinal Numbers Cardinal Numbers Problem Set Bibliography List of Symbols Index