Simple Extensions with the Minimum Degree Relations of Integral Domains  book cover
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Simple Extensions with the Minimum Degree Relations of Integral Domains





ISBN 9781584888512
Published March 5, 2007 by Chapman and Hall/CRC
296 Pages

 
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Book 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.

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

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Reviews

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

EMS Newsletter, September 2008