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Author Cahen, Paul-Jean
Source CiteSeerX
Content type Text
File Format PDF
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Integer-valued Polynomial ♦ Polynomial Closure Paul-jean Cahen ♦ Polynomial Closure ♦ Fractional Ideal ♦ Ring Int ♦ Integral Closure ♦ Krull Domain ♦ Divisorial Ideal ♦ Quotient Field ♦ Noetherian Domain ♦ Dedekind Domain ♦ Finite Residue Field ♦ Zariski Ring ♦ Topological Closure ♦ One-dimensional Noetherian Local Domain
Abstract Let D be a domain with quotient field K. The polynomial closure of a subset E of K is the largest subset F of K such that each polynomial (with coefficients in K), which maps E into D, maps also F into D. In this paper we show that the closure of a fractional ideal is a fractional ideal, that divisorial ideals are closed and that conversely closed ideals are divisorial for a Krull domain. If D is a Zariski ring, the polynomial closure of a subset is shown to contain its topological closure; the two closures are the same if D is a one-dimensional Noetherian local domain, with finite residue field, which is analytically irreducible. A subset of D is said to be polynomially dense in D if its polynomial closure is D itself. The characterization of such subsets is applied to determine the ring Rα formed by the values f(α) of the integer-valued polynomials on a Dedekind domain R (at some element α of an extension of R). It is also applied to generalize a characterization of the Noetherian domains D such that the ring Int(D) of integer-valued polynomials on D is contained in the ring Int(D′) of integer-valued polynomials on the integral closure D ′ of D.
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study
Learning Resource Type Article
Publisher Date 2005-01-01