Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2001; 38(5): 915-935

Printed September 1, 2001

Copyright © The Korean Mathematical Society.

High dimension Pr\"ufer domains of integer-valued polynomials

Paul-Jean Cahen, Jean-Luc Chabert, and K. Alan Loper

Universit´e d’Aix-Marseille III, Universit´e de Picardie, Ohio State University

Abstract

Let $V$ be any valuation domain and let $E$ be a subset of the quotient field $K$ of $V$. We study the ring of integer-valued polynomials on $E$, that is, Int$(E,V)=\{f\in K[X]\mid f(E)\subseteq V\}$. We show that, if $E$ is precompact, then Int$(E,V)$ has many properties similar to those of the classical ring Int$(\mathbb Z)$. In particular, Int$(E,V)$ is dense in the ring of continuous functions $\mathcal C(\hat{E},\hat{V})$; each finitely generated ideal of Int$(E,V)$ may be generated by two elements; and finally, Int$(E,V)$ is a Pr\"ufer domain.

Keywords: integer-valued polynomial, valuation domain, Pr\"ufer domain, Stone-Weierstrass theorem, Skolem property, two-generators property

MSC numbers: 13F20, 12J25

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