Simple Extensions with the Minimum Degree Relations of Integral Domains
This monograph is devoted to a study of structure and properties of a simple algebraic extension of a Noetherian integral domain. It generalizes classical and naturally motivated questions in commutative algebra about algebraic field extensions but the techniques of study are completely different. The most fruitful tool for solving the problem is an investigation of flatness of an extension as a module over an original ring or over another suitable extension. The authors focus mainly on the case of anti-integral extension and naturally defined and rather common classes of extensions that satisfy stronger conditions (such as super-primitive and ultra-primitive extensions). Suppose that R is a Noetherian domain, K is its quotient field and α is an algebraic element over K . Let I[α] = ∩(R:Rηi), where ηi are coefficients of the monic minimal polynomial ϕα of α over K, then α is called an anti-integral element of degree d over R if I[α] ϕα R[X] is precisely the kernel of the natural projection R[X] onto R[α]. Anti-integral elements and the corresponding anti-integral extensions are the central notions of the first two chapters, where their basic properties are proved. Extensions R < α > = R[α] ∩ R[α -1] for an anti-integral element α and their generalizations are studied in the large third chapter (note R < α > = R provided α is anti-integral of degree 1), and connections between flatness and excellent elements are shown in the fourth chapter.
The next chapter treats unramified extensions, i.e. extensions S of R for which every prime ideal of S is unramified over R, and with their differential modules. Chapter 6 is an investigation of subgroups of units in a simple extension. The following two chapters contain descriptions of particular classes of extensions such as super-primitive, exclusive, pure and ultra-primitive. The ninth chapter deepens and extends some previous results on flatness and the last two chapters are devoted to a study of (simple birational) extensions of polynomial rings over Noetherian domains. The monograph sums up results that the authors and their collaborators have obtained during more than ten years of intensive research and publication activity. All topics of the book, which is almost absolutely self-contained, are developed in a clear way and illustrated by many examples. The monograph might be useful for researchers and students interested in methods of modern commutative algebra, particularly for those who need to become familiar with birational ring extensions.