The Skolem property in rings of integer-valued rational functions

TL;DR

This paper investigates the Skolem property in rings of integer-valued rational functions, establishing conditions for its validity and generalizing the concept using star operations.

Contribution

It characterizes the Skolem property in $ ext{Int}^R(D)$ and extends the notion through star operations, providing new equivalences and insights.

Findings

Skolem property in $ ext{Int}^R(D)$ linked to maximal spectrum conditions

Obstructions like unit-valued polynomials are absent in $ ext{Int}^R(D)$

Generalization of Skolem property via star operations and new equivalences

Abstract

$\DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\IntR}{Int{}^\text{R}} \newcommand{\Z}{{\mathbb Z}}$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.