Skolem properties, value-functions, and divisorial ideals
Let \(D\) be a domain with quotient field \(K\), and denote by \(\text{Int}(D)\) the ring of integer-valued polynomials on \(D\), that is, \(\text{Int}(D)=\{f\in K[X]\mid f(D)\subseteq D\}\). An ideal of \(\text{Int}(D)\) is called unitary if it contains nonzero constants. The authors prove that if \(D\) is a rank-one valuation domain with finite residue field, then the unitary ideals of \(\text{Int}(D)\) are entirely determined by their values on the completion of \(D\). Then, the authors globalize this result to any Dedekind domain \(D\) with finite residue fields. They prove that a prime ideal of \(\text{Int}(D)\) is invertible iff it is divisorial, and also, in the case where the characteristic of \(D\) is zero, iff it is upper to zero which is maximal. It follows that if \(D\) is the ring of integers of any algebraic number field, in particular if \(D=\mathbb{Z}\), then no prime ideal of \(\text{Int}(D)\) is divisorial.
scientific article; zbMATH DE number 1308308