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 of integer-valued polynomials $\mathrm{Int}(S,V)=\{f\in K[X]\mid f(S)\subseteq V\}$ 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 $\mathrm{Int}(S,V)$ is Prüfer if and only if no element of the algebraic closure $\overline{K}$ of $K$ is a pseudo-limit of a pseudo-monotone sequence contained in $S$, with respect to some extension of $V$ to $\overline{K}$. This result expands a recent result by Loper and Werner.

to appear in J. Algebra. All comments are welcome. Keywords: Prüfer domain, pseudo-convergent sequence, pseudo-limit, residually transcendental extension, integer-valued polynomial

References in corpus (1)

• Transcendental extensions of a valuation domain of rank one

Cited by in corpus (6)

• Extending valuations to the field of rational functions using pseudo-monotone sequences
• The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences
• Polynomial Dedekind domains with finite residue fields of prime characteristic
• Stacked Pseudo-Convergent Sequences and Polynomial Dedekind Domains
• Nontriviality of rings of integral-valued polynomials
• Metrizability of spaces of valuation domains associated to pseudo-convergent sequences