Rings of integer-valued rational functions
type: Item
instance of: scholarly article
title: Rings of integer-valued rational functions
zbMATH Open document ID: 0929.13013
DOI: 10.1016/S0022-4049(97)00120-5
author: Paul-Jean Cahen
author: K. Alan Loper
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-24T00:00:00Z
review text: For \(D\) an integral domain with quotient field \(K\) and \(E \subseteq D\), let \(\text{Int} (E, D)= \{f \in K[x]\mid f(E) \subseteq D\}\) and \(\text{Int}^R (E, D) = \{f \in K (x) \mid f(E) \subseteq D\}\). For a large class of rings, called \(D\)-rings (which includes \({\mathbb Z}\)), \(\text{Int}^R (D, D) = \text{Int} (D, D)\). This paper is devoted to studying \(\text{Int}^R(E, D)\) for non-\(D\)-rings. Conditions are given which insure that \(\text{Int}^R(E, D)\) is Prüfer or Bezout, and the value ideals and prime ideals are studied. Of particular interest is the case \(D = V\), a valuation domain. It is shown that if the maximal ideal, \(M\), of \(V\) is principal, then all prime ideals of \(\text{Int}^R(E, V)\) over \(M\) are maximal, while if \(M\) is not principal this fails for \(\text{Int}^R(V, V)\).
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 1308233
zbMATH Keywords: integer-valued polynomial
zbMATH Keywords: \(D\)-rings
zbMATH Keywords: Prüfer ring
zbMATH Keywords: Bezout ring
zbMATH Keywords: valuation domain
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308233