Skolem properties, value-functions, and divisorial ideals
type: Item
instance of: scholarly article
title: Skolem properties, value-functions, and divisorial ideals
zbMATH Open document ID: 0931.13012
DOI: 10.1016/S0022-4049(97)00159-X
author: Paul-Jean Cahen
author: Jean-Luc Chabert
author: Thomas G. Lucas
author: Evan Houston
published in: Journal of Pure and Applied Algebra
publication date: 2000-03-08T00:00:00Z
review text: Let \(D\) be a domain with quotient field \(K\), and denote by \(\text{Int}(D)\) the ring of integer-valued polynomials on \(D\), that is, \(\text{Int}(D)=\{f\in K[X]\mid f(D)\subseteq D\}\). An ideal of \(\text{Int}(D)\) is called unitary if it contains nonzero constants. The authors prove that if \(D\) is a rank-one valuation domain with finite residue field, then the unitary ideals of \(\text{Int}(D)\) are entirely determined by their values on the completion of \(D\). Then, the authors globalize this result to any Dedekind domain \(D\) with finite residue fields. They prove that a prime ideal of \(\text{Int}(D)\) is invertible iff it is divisorial, and also, in the case where the characteristic of \(D\) is zero, iff it is upper to zero which is maximal. It follows that if \(D\) is the ring of integers of any algebraic number field, in particular if \(D=\mathbb{Z}\), then no prime ideal of \(\text{Int}(D)\) is divisorial.
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A05
zbMATH DE Number: 1308308
zbMATH Keywords: divisorial prime ideal
zbMATH Keywords: invertible prime ideal
zbMATH Keywords: ring of integer-valued polynomials
zbMATH Keywords: Dedekind domain
zbMATH Keywords: Skolem properties
reviewed by: Toma Albu
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00159-x
OpenAlex ID: W2013506183
description: scientific article; zbMATH DE number 1308308