Rings and Their Modules

✍ Scribed by Paul E. Bland

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 466
• Category: Library

✦ Synopsis

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

✦ Table of Contents

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 <8> 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