1st Edition

Simple Extensions with the Minimum Degree Relations of Integral Domains

By Susumu Oda, Ken-ichi Yoshida Copyright 2007
    296 Pages
    by Chapman & Hall

    296 Pages
    by Chapman & Hall

    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.

    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

    Biography

    Susumu Oda, Ken-ichi Yoshida

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

    EMS Newsletter, September 2008