The Skolem property in rings of integer-valued rational functions

Abstract

Let D be a domain and let Int(D) and IntR(D) be the ring of integer-valued polynomials and the ring of integer-valued rational functions, respectively. Skolem proved that if I is a finitely-generated ideal of Int(Z) with all the value ideals of I not being proper, then I=Int(Z). This is known as the Skolem property, which does not hold in Z[x]. One obstruction to Int(D) having the Skolem property is the existence of unit-valued polynomials. This is no longer an obstruction when we consider the Skolem property on IntR(D). We determine that the Skolem property on IntR(D) is equivalent to the maximal spectrum being contained in the ultrafilter closure of the set of maximal pointed ideals. We generalize the Skolem property using star operations and determine an analogous equivalence under this generalized notion.