Polynomial closure
We quote the author's words: ``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 \(R_\alpha\) 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 \(\text{Int}(D)\) of integer-valued polynomials on \(D\) is contained in the ring \(\text{Int}(D')\) of integer-valued polynomials on the integral closure \(D'\) of \(D\)''.
scientific article; zbMATH DE number 980267