Introduction 1
Remarks on Notation and Terminology 3
Chapter 1 Basics 7
1.1 The Definitions 7
1.2 Fields and Vector Spaces 16
1.3 Matrices 27
1.4 Modules 31
1.5 The Language of Categories 43
Chapter 2 Linear Algebras and Artinian Rings 53
2.1 Linear Algebras 53
2.2 Chain Conditions 59
2.3 Artinian Rings：the Semisimple Case 66
2.4 Artinian Rings：the Radical 75
2.5 The Krull-Schmidt Theorem 80
2.6 Group Representations.Definitions and General Properties 84
2.7 Group Characters 94
Chapter 3 Noetherian Rings 103
3.1 Polynomial Rings 103
3.2 The Euclidean Algorithm 109
3.3 Factorization 115
3.4 Principal Ideal Domains 120
3.5 Modules over Principal Ideal Domains 125
3.6 Algebraic Integers 130
Chapter 4 Ring Constructions 135
4.1 The Direct Product of Rings 135
4.2 The Axiom of Choice and Zorn’s Lemma 140
4.3 Tensor Products of Modules and Algebras 143
4.4 Modules over General Rings 150
4.5 Projective Modules 156
4.6 Injective Modules 163
4.7 Invariant Basis Number and Projective-Free Rings 169
Chapter 5 General Rings 175
5.1 Rings of Fractions 175
5.2 Skew Polynomial Rings 181
5.3 Free Algebras and Tensor Rings 188
5.4 Free Ideal Rings 197
Outline Solutions 203
Notations and Symbols 219
Bibliography 223
Index 225