抽象代数讲义

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

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.