Simple Extensions with the Minimum Degree Relations of Integral Domains

Simple Extensions with the Minimum Degree Relations of Integral Domains

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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 more

Product details

  • Electronic book text | 296 pages
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • London, United Kingdom
  • 1584888520
  • 9781584888529

Table of contents

BIRATIONAL SIMPLE EXTENSIONSThe 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 DEGREESharma 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 PointsSUBRINGS OF ANTI-INTEGRAL EXTENSIONSExtensions 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 ExtensionsFlatness and LCM-Stableness Some Subsets of Spec(R) in the High Degree CaseUNRAMIFIED EXTENSIONS Unramifiedness and Etaleness of Super-Primitive ExtensionsDifferential Modules of Anti-Integral ExtensionsKernels 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 ElementFinite Generation of an Intersection R[a] n K over RPure ExtensionsULTRA-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 ExtensionsFLATNESS AND CONTRACTIONS OF IDEALS Flatness of a Birational Extension Flatness of a Non-Birational ExtensionAnti-Integral Elements and Coefficients of its Minimal PolynomialDenominator Ideals and Flatness (II) Contractions of Principal Ideals and Denominator IdealsANTI-INTEGRAL IDEALS AND SUPER-PRIMITIVE POLYNOMIALSAnti-Integral Ideals and Super-Primitive IdealsSuper-Primitive Polynomials and Sharma PolynomialsAnti-Integral, Super-Primitive, or Flat PolynomialsSEMI ANTI-INTEGRAL AND PSEUDO-SIMPLE EXTENSIONS Anti-Integral Extensions of Polynomial Rings Subrings of R[a] Associated with Ideals of RSemi Anti-Integral Elements Pseudo-Simple ExtensionsREFERENCESINDEXshow more