Simple Extensions with the Minimum Degree Relations of Integral Domains: 1st Edition (Paperback) book cover

Simple Extensions with the Minimum Degree Relations of Integral Domains

1st Edition

By Susumu Oda, Ken-ichi Yoshida

Chapman and Hall/CRC

296 pages

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pub: 2007-03-05
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Description

Although there are many types of ring extensions, simple extensions have yet to be thoroughly explored in one book. Covering an understudied aspect of commutative algebra, Simple Extensions with the Minimum Degree Relations of Integral Domains presents a comprehensive treatment of various simple extensions and their properties. In particular, it examines several properties of simple ring extensions of Noetherian integral domains.

As experts who have been studying this field for over a decade, the authors present many arguments that they have developed themselves, mainly exploring anti-integral, super-primitive, and ultra-primitive extensions. Within this framework, they study certain properties, such as flatness, integrality, and unramifiedness. Some of the topics discussed include Sharma polynomials, vanishing points, Noetherian domains, denominator ideals, unit groups, and polynomial rings.

Presenting a complete treatment of each topic, Simple Extensions with the Minimum Degree Relations of Integral Domains serves as an ideal resource for graduate students and researchers involved in the area of commutative algebra.

Reviews

"All topics … are developed in a clear way and illustrated by many examples."

EMS Newsletter, September 2008

Table of Contents

BIRATIONAL SIMPLE EXTENSIONS

The Ring R[a] n R[a-1]

Anti-Integral Extension and Flat Simple Extensions

The Ring R(Ia) and the Anti-Integrality of a

Strictly Closedness and Integral Extensions

Upper-Prime, Upper-Primary, or Upper-Quasi-Primary Ideals

Some Subsets of Spec(R) in the Birational Case

SIMPLE EXTENSIONS OF HIGH DEGREE

Sharma Polynomials

Anti-Integral Elements and Super-Primitive Elements

Integrality and Flatness of Anti-Integral Extensions

Anti-Integrality of a and a-1

Vanishing Points and Blowing-Up Points

SUBRINGS OF ANTI-INTEGRAL EXTENSIONS

Extensions R[a] n R[a-1] of Noetherian Domains R

The Integral Closedness of the Ring R[a] n R[a-1] (I)

The Integral Closedness of the Ring R[a] n R[a-1] (II)

Extensions of Type R[ß] n R[ß-1] with ß ? K(a)

DENOMINATOR IDEALS AND EXCELLENT ELEMENTS

Denominator Ideals and Flatness (I)

Excellent Elements of Anti-Integral Extensions

Flatness and LCM-Stableness

Some Subsets of Spec(R) in the High Degree Case

UNRAMIFIED EXTENSIONS

Unramifiedness and Etaleness of Super-Primitive Extensions

Differential Modules of Anti-Integral Extensions

Kernels of Derivations on Simple Extensions

THE UNIT GROUPS OF EXTENSIONS

The Unit-Groups of Anti-Integral Extensions

Invertible Elements of Super-Primitive Ring Extensions

EXCLUSIVE EXTENSIONS OF NOETHERIAN DOMAINS

Subring R[a] n K of Anti-Integral Extensions

Exclusive Extensions and Integral Extensions

An Exclusive Extension Generated by a Super-Primitive Element

Finite Generation of an Intersection R[a] n K over R

Pure Extensions

ULTRA-PRIMITIVE EXTENSIONS AND THEIR GENERATORS

Super-Primitive Elements and Ultra-Primitive Elements

Comparisons of Subrings of Type R[aa] n R[(aa)-1]

Subrings of Type R[Ha] n R[(Ha)-1]

A Linear Generator of an Ultra-Primitive Extension R[a]

Two Generators of Simple Extensions

FLATNESS AND CONTRACTIONS OF IDEALS

Flatness of a Birational Extension

Flatness of a Non-Birational Extension

Anti-Integral Elements and Coefficients of its Minimal Polynomial

Denominator Ideals and Flatness (II)

Contractions of Principal Ideals and Denominator Ideals

ANTI-INTEGRAL IDEALS AND SUPER-PRIMITIVE POLYNOMIALS

Anti-Integral Ideals and Super-Primitive Ideals

Super-Primitive Polynomials and Sharma Polynomials

Anti-Integral, Super-Primitive, or Flat Polynomials

SEMI ANTI-INTEGRAL AND PSEUDO-SIMPLE EXTENSIONS

Anti-Integral Extensions of Polynomial Rings

Subrings of R[a] Associated with Ideals of R

Semi Anti-Integral Elements

Pseudo-Simple Extensions

REFERENCES

INDEX

About the Series

Lecture Notes in Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General