抽象代数讲义（第1卷 英文版）2013年版

抽象代数讲义（第1卷 英文版）
出版时间：2013年版
内容简介
　　The present volume is the first of three that will be published under the general title Lectures in Abstract Algebra.These vol- umes are based on lectures which the author has gi，ren during the past ten years at the University of North Carolina， at The Johns Hopkins University， and at Yale University.The general plan of the work is as follows： The present first volume gives an introduction to abstract algebra and gives an account of most of the important algebraic concepts. In a treatment of this type it is impossible to give a comprehensive account of the topics which are introduced.Nevertheless we have tried to go beyond the foundations and elementary properties of the algebraic sys- tems.This has necessitated a certain amount of selection and omission.We feel that even at the present stage a deeper under- standing of a few topics is to be preferred to a superficial under- standing of many.
目录
INTRODUCTION: CONCEPTS FROM SET THEORY THE SYSTEM OF NATURAL NUMBERS
SECTION
1. Operationsonsets
2. Product sets, mappings
3. Equivalencerelations
4. Thenaturalnumbers
5. Thesystemofintegers
6. The division process in I
CHAPTER I: SEMI-GROUPS AND GROUPS
1. Definition and examples ofsemi-groups
2. Non-associative binary compositions
3. Generalized associativelaw. Powers
4. Commutativity
5. Identities andinverses
6. Definition and examples of groups
7. Subgroups
8. Isomorphism
9. Transformation groups
10. Realization of a group as a transformation group
II. Cyclic groups. Order of an element
12. Elementary properties ofpermutations
13. Coset decompositions ofa group
14. Invariant subgroups and factor groups
15. Homomorphismofgroups
16. The fundamental theorem of homomorphism for groups
17. Endomorphisms, automorphisms, center of a group
18. Conjugatc classes
CHAPTER II: RINGS, INTEGRAL DOMAINS AND FIELDS
SECTION
1. Definition andexamples
2. Typesofrings
3. Quasi-regularity. The circle composition
4. Matrixrings
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 ofrings
10. Anti-isomorphism
11. Structure of the additive group of a ring. The charateristic ofaring
12. Algebra of subgroups of the additive group of a ring. Onr sidedideals
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. Polynomialrings
5. Structure of polynomial rings
6. Properties of the ring 2l[x]
7. Simple extensions ofa field
8. Structureofany field
9. The number of roots of a'polynomial in a field
10. Polynomials in several elements
11. Symmetric polynomials
12. Ringsoffunctions
CHAPTER IV: ELEMENTARY FACTORIZATlON THEORY
1. Factors, associates, irreducible elements
2. Gaussian semi-groups
3. Greatest common divisors
4. Principalidealdomains
……
CHAPTER V: GROUPS WITH OPERATORS
CHAPTER VI: MODULES AND IDEALS
CHAPTER VII: LATTICES
Index