# 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."

Keywords » Kommutative Algebra

Related topics » Algebra

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S Title

CommutativeAlgebra, quantity I

© 1958, by way of D VAN NOSTRAND COMPANY

PREFACE

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

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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).