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

Commutative Algebra