RING STRUCTURE OF INTEGER-VALUED RATIONAL FUNCTIONS

Abstract

Integer-valued rational functions are a natural generalization of integer-valued polynomials. Given a domain $D$, the collection of all integer-valued rational functions over $D$ forms a ring extension ${\text{Int}}^{\text{R}}(D)$ of $D$. For a valuation domain $V$, we characterize when ${\text{Int}}^{\text{R}}(V)$ is a Prüfer domain and when ${\text{Int}}^{\text{R}}(V)$ is a Bézout domain. We also extend the classification of when ${\text{Int}}^{\text{R}}(D)$ is a Prüfer domain.