A note on Dedekind non-\(D\)-rings
type: Item
instance of: scholarly article
title: A note on Dedekind non-\(D\)-rings
zbMATH Open document ID: 0840.13008
published in: Rocky Mountain Journal of Mathematics
publication date: 1994-03-14T00:00:00Z
review text: A non-\(D\)-ring is a commutative integral domain \(R\) for which there is a non-constant polynomial \(f\) over \(R\) with the property that \(f(a)\) is a unit of \(R\) for all \(a\) in \(R\). If also \(R\) is Dedekind, it is known that the class group \(\text{Cl} (R)\) of \(R\) is torsion, of exponent dividing the degree of \(f\). In this note, the author gives a method for constructing Dedekind non-\(D\)-rings as intersections of valuation rings. This method is applied to valuations in the rational function field \(\mathbb{Q} (X)\) to give some explicit examples of Dedekind non-\(D\)-rings and their class groups.
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 515197
zbMATH Keywords: non-\(D\)-ring
zbMATH Keywords: integral domain
zbMATH Keywords: Dedekind
zbMATH Keywords: intersections of valuation rings
author: K. Alan Loper
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072584
OpenAlex ID: W1977469693
cites work: On the Structure and Ideal Theory of Complete Local Rings
cites work: More Noneuclidian PID's and Dedekind Domains with Prescribed Class Group
cites work: On rings with a certain divisibility property
cites work: Q5603302
cites work: On rings without a certain divisibility property
DOI: 10.1216/RMJM/1181072584
description: scientific article; zbMATH DE number 515197