抽象代數講義：第1卷(英文)（簡體書）

作者：（德國）雅格布斯（Jacobson N.）

《抽象代數講義》是一套久負盛名的三卷集教材，是作者雅格布斯根據他在霍普金斯大學和耶魯大學講課時的講義編寫而成的，后又成為作者《基本代數學》一書的藍本。《抽象代數講義（第1卷）》介紹了群、環、域、同構等抽象代數的重要的基本概念和抽象代數的基本性質。

INTRODUCTION： CONCEPTS FROM SET THEORY
THE SYSTEM OF NATURAL NUMBERS
1. Operations on sets
2， Product sets， mappings
3. Equivalence relations
4. The natural numbers
5. The system of integers
6. The division process in I
CHAPTER I： SEMI—CROUPS AND GROUPS
1. Definition and examples of semi—groups
2. Non—associative binary compositions
3. Generalized associative law. Powers
4. Commutativity
5. Identities and inverses
6. Definition and examples of groups
7. Subgroups "
8. Isomorphism
9. Transformation groups
10. Realization of a group as a transformation group
11. Cyclic groups. Order of an element
12. Elementary properties of permutations
13. Coset decompositions of a group
14. Invariant subgroups and factor groups
15. Homomorphism of groups
16. The fundamental theorem of homomorphism for groups
17. Endomorphisms， automorphisms， center of a group
18. Conjugate classes
CHAPTER II： RINGS， INTEGRAL DOMAINS AND FIELDS
SECTION
1. Definition and examples
2. Types of rings
3. Quasi—regularity. The circle composition
4. Matrix rings
5. Quaternions
6. Subrings generated by a set of elements. Center
7. Ideals， difference rings
8. Ideals and difference rings for the ring of integers
9. Homomorphism of rings
10. Anti—isomorphism
11. Structure of the additive group of a ring. The charateristic
of a ring
12. Algebra of subgroups of the additive group of a ring. Onesided ideals
13. The ring of endomorphisms of a commutative group
14. The multiplications of a ring
CHAPTER III： EXTENSIONS OF RINGS AND FIELDS
1. Imbedding of a ring in a ring with an identity
2. Field of fractions of a commutative integral domain
3. Uniqueness of the field of fractions
4. Polynomial rings
5. Structure of polynomial rings
6. Properties of the ring
7. Simple extensions of a field
8. Structure of any field
9. The number of roots of a polynomial in a field
10. Polynomials in several elements
11. Symmetric polynomials
12. Rings of functions
CHAPTER IV： ELEMENTARY FACTORIZATION THEORY
1. Factors， associates， irreducible elements
2. Gaussian semi—groups
3. Greatest common divisors
4. Principal ideal domains
SECTION
5. Euclidean domains
6. Polynomial extensions of Gaussian domains
CHAPTER V： GROUPS WITH OPERATORS
1. Definition and examples of gr.oups with operators
2. M—subgroups， M—factor groups and M—homomorphisms
3. The fundamental theorem of homomorphism for M—groups
4. The correspondence between M—subgroups determined by a
homomorphism
5. The isomorphism theorems for M—groups
6. Schreiers theorem
7. Simple groups and the Jordan—H61der theorem
8. The chain conditions
9. Direct products
10. Direct products of subgroups
11. Projections
12. Decomposition into indecomposable groups
13. The Krull—Schmidt theorem
14. Infinite direct products
CHAPTER VI： MODULES AND IDEALS
1. Definitions
2. Fundamental concepts
3. Generators. Unitary modules
4. The chain conditions
5. The Hilbert basis theorem
6. Noetherian rings. Prime and primary ideals
7. Representation of an ideal as intersection of primary ideals
8. Uniqueness theorems
9. Integral dependence
10. Integers of quadratic fields
CHAPTER VII： LATTICES
1. Partially ordered sets
2. Lattices
3. Modular lattices
4. Schreiers theorem. The chain conditions
5. Decomposition theory for latticeswith ascending chain condition
6. Independence
7. Complemented modular lattices
8. Boolean algebras
Index