《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。

Chapter 1
　　Groups
　　The concept of a group historically emerged from the study of solutions of equations such as aia; for variable x. It is of fundamental importance in algebra and other subjects. We say that two groups are the same if they are isomorphic. Just as classifications of finite-dimensional vector spaces over a number field， one of the fundamental questions in group theory is to classify all groups up to isomorphism， which means finding a necessary and sufficient condition for two groups to be isomorphic. This is a very complicated question. However， a larger amount of miscellaneous information on the structure of a group has been explored in this chapter.
　　1.1 Semigroups， Monoids and Groups
　　A group is a nonempty set with a binary operation satisfying some axioms. Let us first recall some known binary operations. Given two nx n matrices A， B over the complex number field C， we have binary operations of A and B， such as and AB. For any two maps f : X Y and g : Y Z， the composition of and g is defined by g of ：. Abstractly， a binary operation on a nonempty set S is a map. It is customary to write a b， simply ab， instead of f(a， b). With this notation， the binary operation of S is usually called the product， or multiplication of S. A binary operation of S is commutative if ab ba for any a，b. When a binary operation f is commutative， f(a， b) is sometimes denote by a+b and called the addition of S.
　　Definition 1.1.1. A binary operation on a set S is to be associative if. A semigroup is a nonempty set S together with an associative binary operation. A commutative semigroup is a semigroup whose binary operation is commutative.
　　For any ai， an in a semigroup， where n， define an inductively. In particular， if all， then an is denoted by an. In a commutative semigroup with a binary operation ， and a a is denoted by na.
　　Example 1.1.1. Let N be the set of all natural numbers. Then are semigroups with ordinary addition and multiplication of numbers respectively. Both and are commutative semigroups.
　　For any n 1，the set Z of all integers can be divided into disjoint subsets，where r. For any m G Z， we still use m to denote the subset. It is obvious that there is a unique 1 such that m for any m G Z and that mi = m2 if and only if m m2 is divisible by n. Let. Define and. It is easy to check that r and rs. It is obvious that both are semigroups. Further， let Affc(Zra) be the set of all k x k matrices with entries in Zn. Define the addition of matrices by and product of matrices by， where k ditbtj. Then both are semigroups. If fe then is not a commutative semigroup.
　　It is well known that h f for any maps X Y Z W. For any nonempty set X，let map from. Then Xx is a semigroup with composition of maps.
　　Let Q be an open subset of and Q be a fixed point. A loop with a fixed point xo in ￡1 is a continuous map 9 : Q such that cp. Let L be the set of all loops with a fixed point xq in Q.
　　Example 1.1.2. Let 5(^4) be the set of all finite sequences (or strings) of elements from a nonempty set A. The elements in 5(^4) are also called words over A， or words with alphabets in A. Then becomes a semigroup with the string concatenation. The number of letters in a word is called the length of word.
　　Definition 1.1.2. An element e in a semigroup S is called an identity of S provided that ea ae a for all S. A monoid is a semigroup with an identity. A commutative monoid is a commutative semigroup with identity. In a commutative monoid S with binary operation “+”，the identity is denoted by 0，which is called a zero element of S.
　　Let ei， e2 be identities of a monoid M. Then as 62 is an identity. Similarly， 62 eie2- Hence e Thus a monoid has a unique identity. We denote the unique identity of a monoid by e in this chapter unless otherwise specified. We also write e，M for the identity of a monoid M instead of e if necessary to avoid confusion.
　　Example 1.1.3. The semigroups is a monoid with identity 0 (resp. 1). The set 2 of all even numbers is not a monoid with ordinary multiplication of numbers. It is only a semigroup. Moreover， is a monoid with identity，and is a monoid with identity，where is the Kronecker9s symbol， which is equal to 1 when i j and zero otherwise.
　　Example 1.1.4. Let 5 be a semigroup and choose an element e S. Define a binary operation on as follows. If a， 6 G S， then ab is the product of a and b in S. Otherwise a for any a. It is easy to check that S+ is a monoid with identity e. In particular， for any given nonempty set Ay is a monoid，where 5 is the semigroup defined in Example 1.1.2. The identity of M(A) is also called an empty word. We define the length of empty word by zero.
　　Definition 1.1.3. Let M be a monoid with identity e. An element a 6 M is said to be invertible in M if there is an element b ￡ M such that ab， the element b， denoted by a-1， is called the inverse of a.
　　IS M is a commutative monoid with binary operation “ + ”，then t

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