Prüfer intersection of valuation domains of a field of rational functions

Abstract

Let V be a rank one valuation domain with quotient field K. We characterize the subsets S of V for which the ring Int(S,V)={f∈K[X]|f(S)⊆V} of integer-valued polynomials over S is a Prüfer domain. The characterization is obtained by means of the notion of pseudo-monotone sequence and pseudo-limit in the sense of Chabert, which generalize the classical notions of pseudo-convergent sequence and pseudo-limit by Ostrowski and Kaplansky, respectively. We show that Int(S,V) is Prüfer if and only if no element of the algebraic closure K‾ of K is a pseudo-limit of a pseudo-monotone sequence contained in S, with respect to some extension of V to K‾. This result expands a recent result by Loper and Werner.