Integer-valued rational functions over globalized pseudovaluation domains

TL;DR

This paper characterizes when the ring of integer-valued rational functions over certain domains forms a globalized pseudovaluation domain, extending previous work on integer-valued polynomials.

Contribution

It provides a complete characterization of when $ ext{Int}^ ext{R}(D)$ is a GPVD for domains $D$ that are GPVDs, especially pseudosingular and non-valuation pseudovaluation domains.

Findings

If $D$ is a pseudosingular GPVD, then $ ext{Int}^ ext{R}(D)$ is a GPVD.

Complete characterization for $ ext{Int}^ ext{R}(D)$ being a GPVD when $D$ is a non-valuation pseudovaluation domain.

Abstract

$\DeclareMathOperator{\IntR}{Int{}^\text{R}}$$\DeclareMathOperator{\Int}{Int}$Let $D$ be a domain. Park determined the necessary and sufficient conditions for which the ring of integer-valued polynomials $\Int(D)$ is a globalized pseudovaluation domain (GPVD). In this work, we investigate the ring of integer-valued rational functions $\IntR(D)$. Since it is necessary that $D$ be a GPVD for $\IntR(D)$ to be a GPVD, we consider $\IntR(D)$, where $D$ is a GPVD. We determine that if $D$ is a pseudosingular GPVD, then $\IntR(D)$ is a GPVD. We also completely characterize when $\IntR(D)$ is a GPVD if $D$ is a pseudovaluation domain that is not a valuation domain.