Integer-valued polynomials, Prüfer domains, and localization

Abstract:

Let $A$ be an integral domain with quotient field $K$ and let $\operatorname {Int} (A)$ be the ring of integer-valued polynomials on $A:\{ P \in K[X]|P(A) \subset A\}$. We study the rings $A$ such that $\operatorname {Int} (A)$ is a Prüfer domain; we know that $A$ must be an almost Dedekind domain with finite residue fields. First we state necessary conditions, which allow us to prove a negative answer to a question of Gilmer. On the other hand, it is enough that $\operatorname {Int} (A)$ behaves well under localization; i.e., for each maximal ideal $\mathfrak {m}$ of $A$, $\operatorname {Int} {(A)_\mathfrak {m}}$ is the ring $\operatorname {Int} ({A_\mathfrak {m}})$ of integer-valued polynomials on ${A_\mathfrak {m}}$. Thus we characterize this latter condition: it is equivalent to an "immediate subextension property" of the domain $A$. Finally, by considering domains $A$ with the immediate subextension property that are obtained as the integral closure of a Dedekind domain in an algebraic extension of its quotient field, we construct several examples such that $\operatorname {Int} (A)$ is Prüfer.