This book is the child of an unborn parent. Some years ago the senior author began the preparation of a Colloquium volume on algebraic geometry, and he was then faced with the difficult task of incorporating in that volume the vast amount of purely algebraic material which is needed in abstract algebraic geometry. The original .plan was to insert, from time to time, algebraic digressions in which concepts and results from commutative algebra were to be developed in full as and when they were needed. However, it soon became apparent that such a parenthetical treatment of the purely algebraic topics, covering a wide range of commutative algebra, would impose artificial bounds on the manner, depth, and degree of generality with which these topics could be treated. As is well known, abstract algebraic geometry has been recently not only the main field of applications of commutative algebra but also the principal incentive of new research in commutative algebra.

CHAPTER Ⅰ.INTRODUCTORY CONCEPTS

1.Binary operations

2.Groups

3.Subgroups

4.Abelian groups

5.Rings

6.Rings with identity

7.Powers and multiples

8.Fields

9.Subrings and subfields

10.Transformations and mappings

11.Group homomorphisms

12.Ring homomorphisms

13.Identification of rings

14.Unique factorization domains

15.Euclidean domains

16.Polynomials in one indeterminate

17.Polynomial rings

18.Polynomials in several indeterminates

19.Quotient fields and total quotient rings

20.Quotient rings with respect to multiplicative systems

21.Vector spaces

Ⅱ.ELEMENTO OF FIELD THEORY

1.Field extensions

2.Algebraic quanities

3.Algebraic extensions

4.The characteristic of field

5.Separable and inseparable algebraic extensions

6.Splitting fields and normal extensions

7.The fundamental theorem of Galois theory

8.Galois fields

9.The theorem of the primitive element

……

Ⅲ.IDEALS AND MODULES

Ⅳ.NOETHERIAN RINGS

Ⅴ.DEDEKIND DOMAING.CLASSICAL IDEALS THEORY

INDEXL OF NOTATIONS

INDEX OF DEFINITIONS