The Skolem closure as a semistar operation

Abstract

Skolem properties are usually defined on rings of integer-valued polynomials, and are thought to be about finitely generated ideals. We contend that Skolem properties are more naturally stated in terms of $\star$-ideals. By this method, we characterize the classes of ideals on which $\text{ Int}\left(\mathbb{Z}\right)$ has the Skolem and strong Skolem properties. We extend the definitions of Skolem properties to rings comprising rational functions. We further define semistar operations which generalize the Skolem closure in this broader context. Finally we give some conditions under which a ring has Skolem properties.