INTEGER-VALUED POLYNOMIALS, PRUFER DOMAINS, AND LOCALIZATION

Let A be an integral domain with quotient field K and let Int(A) be the ring of integer-valued polynomials on A : {P is-an-element-of K[X]\P(A) subset-of A}. We study the rings A such that Int(A) is a Prufer 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 Int(A) behaves well under localization; i.e., for each maximal ideal m of A , Int(A)m is the ring Int(A(m)) of integer-valued polynomials on A(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 Int(A) is Prufer.