A classification of all $D$ such that $\mathrm \{Int\}(D)$ is a Prüfer domain

Abstract:

Let $D$ be an integral domain with quotient field $K$. The ring of integer-valued polynomials $\mathrm {Int}(D)$ over $D$ is defined by $\mathrm {Int}(D) = \{f(x) \in K[x] \mid f(D) \subseteq D\}$. It is known that if $\mathrm {Int}(D)$ is a Prüfer domain, then $D$ is an almost Dedekind domain with all residue fields finite. This condition is necessary and sufficient if $D$ is Noetherian, but has been shown to not be sufficient if $D$ is not Noetherian. Several authors have come close to a complete characterization by imposing bounds on orders of residue fields of $D$ and on normalized values of particular elements of $D$. In this note we give a double-boundedness condition which provides a complete charaterization of all integral domains $D$ such that $\mathrm {Int}(D)$ is a Prüfer domain.