Integer-valued polynomials, Prüfer domains, and localization

Abstract

Let A A be an integral domain with quotient field K K and let Int ⁡ ( A ) \operatorname {Int} (A) be the ring of integer-valued polynomials on A : { P ∈ K [ X ] | P ( A ) ⊂ A } A:\{ P \in K[X]|P(A) \subset A\} . We study the rings A A such that Int ⁡ ( A ) \operatorname {Int} (A) is a Prüfer domain; we know that A 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 Int ⁡ ( A ) \operatorname {Int} (A) behaves well under localization; i.e., for each maximal ideal m \mathfrak {m} of A A , Int ⁡ ( A ) m \operatorname {Int} {(A)_\mathfrak {m}} is the ring Int ⁡ ( A m ) \operatorname {Int} ({A_\mathfrak {m}}) of integer-valued polynomials on A m {A_\mathfrak {m}} . Thus we characterize this latter condition: it is equivalent to an "immediate subextension property" of the domain A A . Finally, by considering domains A 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 Int ⁡ ( A ) \operatorname {Int} (A) is Prüfer.