# Commutative Algebra I by Oscar Zariski By Oscar Zariski

From the Preface: "We have most popular to put in writing a self-contained publication that can be utilized in a simple graduate process glossy algebra. it's also with an eye fixed to the coed that we've got attempted to provide complete and unique causes within the proofs... we have now additionally attempted, this time with an eye fixed to either the coed and the mature mathematician, to provide a many-sided therapy of our issues, now not hesitating to provide numerous proofs of 1 and a similar consequence after we inspiration that whatever should be discovered, as to tools, from all of the proofs."

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Graduate Texts in arithmetic 28

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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. earrings with identity
§ 7. Powers and multiples
§ eight. Fields
§ nine. Subrings and subfields
§ 10. modifications and mappings
§ eleven. crew 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 jewelry with recognize 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 basic 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. best and maximal ideals
§ nine. basic ideals
§ 10. Finiteness conditions
§ eleven. Composition series
§ 12. Direct sums
§ 12bis. limitless direct sums
§ thirteen. Comaximal beliefs and direct sums of ideals
§ 14. Tensor items of rings
§ 15. loose joins of indispensable 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 approach for learning the jewelry with d.c.c
§ four. The Lasker-Noether decomposition theorem
§ five. area of expertise theorems
§ 6. program to zero-divisors and nilpotent elements
§ 7. program to the intersection of the powers of an ideal.
§ eight. prolonged and reduced in size ideals
§ nine. Quotient rings.
§ 10. kin among beliefs in R and beliefs in RM
§ eleven. Examples and functions of quotient rings
§ 12. Symbolic powers
§ thirteen. size of an ideal
§ 14. top beliefs in noetherian rings
§ 15. significant perfect rings.
§ sixteen. Irreducible ideals

V. DEDEKIND domain names. CLASSICAL perfect THEORY
§ 1. fundamental 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 houses of Dedekind domains
§ eight. Extensions of Dedekind domains
§ nine. Decomposition of major beliefs in extensions of Dedekind domains.
§ 10. Decomposition workforce, inertia workforce, and ramification groups.
§ eleven. assorted and discriminant
§ 12. program to quadratic fields and cyclotomic fields.

INDEX OF NOTATIONS

INDEX OF DEFINITIQNS

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Extra resources for Commutative Algebra I

Example text

Let us make the additional requirement that all the elements of M are regular in R. Thus R contains reguiar elements and hence has the set a total quotient ring F. Since M is closed under of all quotients a/m, where a e R, m e M, is a subring of F containing R. will be denoted by RM and will be called the quotient ring of R with respect to the system M. Note the following extreme cases. m1m2 E (1) R has an identity, and M is the set of all units of R. In this case RM = R. (2) M is the set of all regular elements of R.

With this definition of the ring operations in S it follows directly from Lemma 1 of § 12 that S is a ring and that T is an isomorphism of S onto Since T0 is an isomorphism of R onto RF and T coincides with T0 on R, it follows from the very definition of the ring operations in S that if a, b e R, then a b = a + b and a 0 b = b, where + and refer to the ring operations in R. Hence the ring R is a subring of S. Moreover, T is, by definition, an extension of T0. This completes the proof if R and 5F are disjoint.

The factor c is cafled the content of f(x) and s denoted by c(f). We observe thatf(x) is primitive if and only if c(f) is a unit in R. We can now prove that every element of RIxI factors into irreducible ones. It is clear that an element of R s irreducible (or a unit) in if and only if it is irreducible (or a unit) in R. From this it follows (since R is a UFD) that every polynomial of Rrxl of degree zero factors into Supposef(x) has positive degree n and that factoriza- ton has been proved for 'olynomias of lower degree.