Polynomial closure

[摘要] 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 coefficient 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 Notherian 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 is such subset is applied to determine the ring R, formed by the values f(alpha) of the integer-valued polynomials on a Dedekind domain R (at some element alpha 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. (C) 1996 Academic Press, Inc.

[发布日期] 1996-12-01