Simple Extensions with the Minimum Degree Relations of Integral Domains 1st Edition by Susumu Oda ISBN 978-1584888512 1584888512

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Product details: 

ISBN 10:  1584888512

ISBN 13: 978-1584888512

Author: Susumu Oda

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 $R[a]\cap R[a-1]$

• Anti-Integral Extension and Flat Simple Extensions

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

• Strictly Closedness and Integral Extensions

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

• Some Subsets of Spec(RRR) 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 aaa and a−1a^{-1}a−1

• Vanishing Points and Blowing-Up Points

SUBRINGS OF ANTI-INTEGRAL EXTENSIONS

• Extensions $R[a]\cap R[a-1]$ of Noetherian Domains RRR

• The Integral Closedness of the Ring $R[a]\cap R[a-1]$ (I)

• The Integral Closedness of the Ring $R[a]\cap R[a-1]$ (II)

• Extensions of Type $R[\beta ]\cap R[\beta -1]$ with $\beta \in K(a)$

DENOMINATOR IDEALS AND EXCELLENT ELEMENTS

• Denominator Ideals and Flatness (I)

• Excellent Elements of Anti-Integral Extensions

• Flatness and LCM-Stableness

• Some Subsets of Spec(RRR) 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 $R[a]\cap K$ of Anti-Integral Extensions

• Exclusive Extensions and Integral Extensions

• An Exclusive Extension Generated by a Super-Primitive Element

• Finite Generation of an Intersection $R[a]\cap K$ over RRR

• Pure Extensions

ULTRA-PRIMITIVE EXTENSIONS AND THEIR GENERATORS

• Super-Primitive Elements and Ultra-Primitive Elements

• Comparisons of Subrings of Type R[aa]∩R[(aa)−1]R[aa] cap R[(aa)^{-1}]R[aa]∩R[(aa)−1]

• Subrings of Type R[Ha]∩R[(Ha)−1]R[Ha] cap R[(Ha)^{-1}]R[Ha]∩R[(Ha)−1]

• A Linear Generator of an Ultra-Primitive Extension R[a]R[a]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]R[a]R[a] Associated with Ideals of RRR

• Semi Anti-Integral Elements

• Pseudo-Simple Extensions

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Tags: Susumu Oda, Simple Extensions, the Minimum Degree, Integral Domains