Topological aspects of the ideal theory in rings of integer-valued polynomials

Abstract

Let D D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of I n t ( D ) Int(D) lying over a given maximal ideal m \mathfrak m of D D . We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the m \mathfrak m -adic completion of D D . Moreover, these results are naturally extended to overrings of I n t ( D ) Int(D) . As an application we provide explicit constructions of divisorial ideals of I n t ( D ) Int(D) which are not finitely generated.