Ring structure of integer-valued rational functions
type: Item
instance of: scholarly article
title: Ring structure of integer-valued rational functions
zbMATH Open document ID: 1545.13018
author: Baian Liu
published in: Journal of Commutative Algebra
publication date: 2024-09-03T00:00:00Z
review text: Let \(D\) be an integral domain, \(K\) its field of quotients and \(\mathrm{Int}^R(D) = \{f\in K(x):\ f(D)\subset D\}\) its ring of integer-valued rational functions. It has been shown by \textit{A. Loper} and \textit{P. J. Cahen} [J. Pure Appl. Algebra 131, 179--193 (1998; Zbl 0929.13013)] that if \(D\) is a monic or singular Prüfer ring, then \(\mathrm{Int}^R(D)\) is a Prüfer ring. The author presents sufficient conditions for the remaining Prüfer rings to have \(\mathrm{Int}^R(D)\) Prüfer, resp. non-Prüfer. He gives also necessary and sufficient conditions for valuation domains whose ring of integer-valued rational functions is a Prüfer ring, resp. a Bézout ring.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 7905766
zbMATH Keywords: Prüfer domain
zbMATH Keywords: valuation domain
zbMATH Keywords: Bézout domain
zbMATH Keywords: integer-valued rational functions
MaRDI profile type: Publication
cites work: Integer-valued polynomials
cites work: Interpolation domains
cites work: Integer-valued polynomials and Prüfer \(v\)-multiplication domains
cites work: Q3137901
cites work: On rings with a certain divisibility property
cites work: An Essential Ring Which is Not A v-Multiplication Ring
cites work: Prüfer v-multiplication domains and the ring \(R[X]_{N_ v}\)
cites work: Q5654135
cites work: On rings without a certain divisibility property
cites work: On Prüfer non-\(D\)-rings
cites work: Sequence domains and integer-valued polynomials
cites work: A classification of all 𝐷 such that {𝐼𝑛𝑡}(𝐷) is a Prüfer domain
cites work: Rings of integer-valued rational functions
cites work: Q3436774
cites work: Prüfer intersection of valuation domains of a field of rational functions
DOI: 10.1216/JCA.2024.16.305
description: scientific article; zbMATH DE number 7905766