By Oscar Zariski

ISBN-10: 0387900896

ISBN-13: 9780387900896

From the Preface: "We have most popular to write down a self-contained ebook which can be utilized in a uncomplicated graduate process smooth algebra. it's also with an eye fixed to the coed that we've got attempted to offer complete and specific motives within the proofs... we've additionally attempted, this time with a watch to either the scholar and the mature mathematician, to offer a many-sided therapy of our subject matters, now not hesitating to provide a number of proofs of 1 and an identical outcome after we inspiration that whatever will be realized, as to tools, from all of the proofs."

Content point » Graduate

Keywords » Kommutative Algebra

Related topics » Algebra

Cover

Graduate Texts in arithmetic 28

S Title

CommutativeAlgebra, quantity I

Copyright

© 1958, by means of D VAN NOSTRAND COMPANY

PREFACE

TABLE OF CONTENTS

I. INTRODUCTORY CONCEPTS

§ 1. Binary operations.

§ 2. Groups

§ three. Subgroups.

§ four. Abelian groups

§ five. Rings

§ 6. jewelry with identity

§ 7. Powers and multiples

§ eight. Fields

§ nine. Subrings and subfields

§ 10. alterations and mappings

§ eleven. team homomorphisms

§ 12. Ring homomorphisms

§ thirteen. identity of rings

§ 14. exact factorization domains.

§ 15. Euclidean domains.

§ sixteen. Polynomials in a single indeterminate

§ 17. Polynomial rings.

§ 18. Polynomials in numerous indeterminates

§ 19. Quotient fields and overall quotient rngs

§ 20. Quotient earrings with admire to multiplicative systems

§ 21. Vector spaces

II. parts OF box THEORY

§ 1. box extensions

§ 2. Algebraic quantities

§ three. Algebraic extensions

§ four. The attribute of a field

§ five. Separable and inseparable algebraic extension

§ 6. Splitting fields and basic extensions

§ 7. the elemental theorem of Galois theory

§ eight. Galois fields

§ nine. the theory of the primitive element

§ 10. box polynomials. Norms and traces

§ eleven. The discriminant

§ 12. Transcendental extensions

§ thirteen. Separably generated fields of alebraic functions

§ 14. Algebrically closed fields

§ 15. Linear disjointness and separability

§ sixteen. Order of inseparability of a box of algebraic functions

§ 17. Derivations

III. beliefs AND MODULES

§ 1. beliefs and modules

§ 2. Operations on submodules

§ three. Operator homomorphisms and distinction modules

§ four. The isomorphism theorems

§ five. Ring homomorphisms and residue type rings.

§ 6. The order of a subset of a module

§ 7. Operations on ideals

§ eight. top and maximal ideals

§ nine. fundamental ideals

§ 10. Finiteness conditions

§ eleven. Composition series

§ 12. Direct sums

§ 12bis. endless direct sums

§ thirteen. Comaximal beliefs and direct sums of ideals

§ 14. Tensor items of rings

§ 15. loose joins of necessary domain names (or of fields).

IV. NOETHERIAN RINGS

§ 1. Definitions. The Hubert foundation theorem

§ 2. jewelry with descending chain condition

§ three. basic rngs

§ 3bis. replacement procedure for learning the earrings with d.c.c

§ four. The Lasker-Noether decomposition theorem

§ five. specialty theorems

§ 6. program to zero-divisors and nilpotent elements

§ 7. software to the intersection of the powers of an ideal.

§ eight. prolonged and gotten smaller ideals

§ nine. Quotient rings.

§ 10. kin among beliefs in R and beliefs in RM

§ eleven. Examples and purposes of quotient rings

§ 12. Symbolic powers

§ thirteen. size of an ideal

§ 14. top beliefs in noetherian rings

§ 15. valuable excellent rings.

§ sixteen. Irreducible ideals

V. DEDEKIND domain names. CLASSICAL perfect THEORY

§ 1. critical elements

§ 2. Integrally based rings

§ three. Integrally closed rings

§ four. Finiteness theorems

§ five. The conductor of an quintessential closure

§ 6. Characterizations of Dedekind domains

§ 7. extra homes of Dedekind domains

§ eight. Extensions of Dedekind domains

§ nine. Decomposition of best beliefs in extensions of Dedekind domains.

§ 10. Decomposition team, inertia crew, and ramification groups.

§ eleven. diverse and discriminant

§ 12. software to quadratic fields and cyclotomic fields.

INDEX OF NOTATIONS

INDEX OF DEFINITIQNS