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.
Kochi University, Japan Okayama University of Science, Japan
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