Contents
Preface Ill
CHAPTER 1
Groups 1
1.1 Semigroups， Monoids and Groups 1
1.2 Subgroups 7
1.3 The Action of a Group on a Set 12
1.4 The Sylow Theorem 20
1.5 Homomorphisms 22
1.6 Direct Products and Direct Sums 30
1.7 Simple Groups 39
1.8 Nilpotent Groups and Solvable Groups 41
CHAPTER 2
Rings and Modules 47
2.1 Rings and Ring Homomorphisms 47
2.2 Modules， Indecomposable Modules and Free Modules 61
2.3 Projective Modules and Injective Modules 74
2.4 Homological Dimensions 82
2.5 Tensor Product and Weak Dimension 91
2.6 Localization 103
2.7 Noetherian Modules and UFD 113
2.8 Finitely Generated Modules Over a PID 124
CHAPTER 3
Fields and Galois Theory 135
3.1 Extensions of Fields 135
3.2 Splitting Fields and Normality 142
3.3 The Fundamental Theorem of Galois Theory 151
3.4 Radical Extensions 160
3.5 Construction with Straight-Edge and Compass 163
3.6 The Hilbert Nullstellensatz 166
CHAPTER 4
Introduction to Various Algebras 175
4.1 Associative Algebras 175
4.2 Coassociative Coalgebras and Hopf Algebras 188
4.3 Nonassociative Algebras 193
CHAPTER 5
Category 203
5.1 Category， Limit and Colimit 203
5.2 Functors and Natural Transformations 208
5.3 Abelian Categories and Homological Groups 216
Bibliography 227
Index 229