描述
开 本: 16开纸 张: 胶版纸包 装: 平装是否套装: 否国际标准书号ISBN: 9787030823267
内容简介
《Algebra Basic Concepts of Abstract Algebra（代数——抽象代数基本概念）（第二版）》是作者在2014年在科学出版社出版的《algebra》基础之上，结合近几年的教学实践进行了大量篇幅修改而成的，主要将前面教材中一些较难的习题改成了讲解内容，添加了一些章节的，简化了一些证明，尤其是主理想整环上有限生成模的结构定理的证明，添加了一些新的习题，完善了*后一章*后一节有关同调的内容。《Algebra Basic Concepts of Abstract Algebra（代数——抽象代数基本概念）（第二版）》短小精悍，所有定理的证明都经过精心处理，故以较小的篇幅涵盖了代数学与同调代数以及表示理论的内容；联系了高等代数或线性代数的内容，使得在更高层次理解高等代数，并能够更加容易理解抽象的内容；对焦代数学前沿，这是其他代数书没有的。
目 录
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
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
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