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. To approach the underlying algebra only in a strictly utilitarian, auxiliary, and parenthetical manner, to stop short of going further afield where the applications of algebra to algebraic geometry stop and the general algebraic theories inspired By geometry begin, impressed us increasingly as being a program scientifically too narrow and psychologically frustrating, not to mention the distracting effect that repeated algebraic digressions would inevitably have had on the reader, vis-h-vis the central algebro-geometrlc theme. Thus the idea of a separate book on commutative algebra was born, and the present book--of which this is the first of two volumes--is a realization of this idea, come to fruition at a time when its parent--a treatise on abstract algebraic geometry-has still to see the light of the day.
目录
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
