Commutative Algebra I by Oscar Zariski

By Oscar Zariski

From the Preface: "We have hottest to put in writing a self-contained e-book that may be utilized in a simple graduate process smooth algebra. it's also with a watch to the scholar that we have got attempted to provide complete and exact causes within the proofs... now we have additionally attempted, this time with an eye fixed to either the coed and the mature mathematician, to offer a many-sided remedy of our subject matters, now not hesitating to provide numerous proofs of 1 and an analogous end result once we concept that anything may be realized, as to equipment, 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 way 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. variations and mappings
§ eleven. workforce homomorphisms
§ 12. Ring homomorphisms
§ thirteen. identity of rings
§ 14. distinct factorization domains.
§ 15. Euclidean domains.
§ sixteen. Polynomials in a single indeterminate
§ 17. Polynomial rings.
§ 18. Polynomials in different indeterminates
§ 19. Quotient fields and overall quotient rngs
§ 20. Quotient earrings with appreciate to multiplicative systems
§ 21. Vector spaces

II. components 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 concept 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. basic 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 vital domain names (or of fields).

IV. NOETHERIAN RINGS
§ 1. Definitions. The Hubert foundation theorem
§ 2. earrings with descending chain condition
§ three. fundamental rngs
§ 3bis. substitute technique for learning the earrings with d.c.c
§ four. The Lasker-Noether decomposition theorem
§ five. area of expertise theorems
§ 6. software to zero-divisors and nilpotent elements
§ 7. program to the intersection of the powers of an ideal.
§ eight. prolonged and gotten smaller ideals
§ nine. Quotient rings.
§ 10. family members among beliefs in R and beliefs in RM
§ eleven. Examples and functions of quotient rings
§ 12. Symbolic powers
§ thirteen. size of an ideal
§ 14. best beliefs in noetherian rings
§ 15. critical excellent rings.
§ sixteen. Irreducible ideals

V. DEDEKIND domain names. CLASSICAL perfect THEORY
§ 1. necessary 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. additional homes of Dedekind domains
§ eight. Extensions of Dedekind domains
§ nine. Decomposition of top beliefs in extensions of Dedekind domains.
§ 10. Decomposition workforce, inertia crew, and ramification groups.
§ eleven. various and discriminant
§ 12. software to quadratic fields and cyclotomic fields.

INDEX OF NOTATIONS

INDEX OF DEFINITIQNS

Show description

Read Online or Download Commutative Algebra I PDF

Similar group theory books

Semigroup theory and evolution equations: the second international conference

Lawsuits of the second one foreign convention on traits in Semigroup idea and Evolution Equations held Sept. 1989, Delft collage of know-how, the Netherlands. Papers take care of contemporary advancements in semigroup thought (e. g. , optimistic, twin, integrated), and nonlinear evolution equations (e

Topics in Galois Theory

Written by means of one of many significant members to the sphere, this booklet is full of examples, workouts, and open difficulties for additional edification in this interesting subject.

Products of Finite Groups (De Gruyter Expositions in Mathematics)

The examine of finite teams factorised as a manufactured from or extra subgroups has develop into a topic of serious curiosity over the last years with functions not just in team thought, but in addition in different parts like cryptography and coding conception. It has skilled a huge impulse with the advent of a few permutability stipulations.

Automorphic Representation of Unitary Groups in Three Variables

The aim of this publication is to increase the solid hint formulation for unitary teams in 3 variables. The good hint formulation is then utilized to procure a category of automorphic representations. This paintings represents the 1st case during which the sturdy hint formulation has been labored out past the case of SL (2) and comparable teams.

Additional resources for Commutative Algebra I

Sample text

In this condition the elements a and b are automatically diffe"ent from zero, since the divisibility concepts introduced in the preceding section have been restricted to e1ements different from zero. INTRODUCTORY CONCEPTS 24 d. If is a unit, then Ch. I = p(l), and conversely. The direct statement follows from b. and the converse from c. THEOREM 5. A euclidean domain is a unique factorization domain. Weshall show that a euclidean domain E satisfies UF1 and UF3 (see § 14, Theorem 4). Then UF1 VERIFICATION OF UF1.

Identification of rings. As an application of the concept of § 13. isomorphism extension, we shall now discuss a certain standard procedure of ring identification which is frequently used in algebra. Given two rings R and S' we say that R can be imbedded in S' if there exists a ring S which contains the ring R as a subring 9) and which is isomorphic with 5'. It is clear that if R can be imbedded in 5', then 5' must contain a subring which is an isomorphic image of R. We shall prove now that this condition is also sufficient.

Moreover, if 0, then a = 0, aT0 = 0, and hence (a/b)T = 0. It follows, therefore, by Lemma 2, that T is a homomorphsm of F into F'. Since T0 is a mapping onto R' and F' is a total quotient ring of R', we a/b = conclude that T maps F onto F'. If b is regular in R and a is any element of R, then a = ab/b, so that aT = (ab)T0/bT0 = aT0. bT0/bT0 = aT0, so that T is an extension of T0. Finally, if (a/b)T = 0, then aT0/bT0 = 0, aT0 = 0, hence a = 0 (for T0 is an isomorphism), and a/b = 0; since only the zero of F maps into the zero of F', T is an sornorphism 11, Theorem 2).

Download PDF sample

Rated 4.73 of 5 – based on 31 votes