Ring Structure of Integer-Valued Rational Functions

$\text{DeclareMathOperator}\text{IntR}Int{}^{\text{R}}$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 $\text{IntR}(D)$ of $D$. For a valuation domain $V$, we characterize when $\text{IntR}(V)$ is a Prüfer domain and when $\text{IntR}(V)$ is a Bézout domain. We also extend the classification of when $\text{IntR}(D)$ is a Prüfer domain.