On a class of rings that generalize Dedekind rings
type: Item
instance of: scholarly article
title: On a class of rings that generalize Dedekind rings
zbMATH Open document ID: 0844.13012
author: Marco Fontana
author: N. Popescu
published in: Journal of Algebra
publication date: 1995-07-25T00:00:00Z
review text: Let \(A\) be a commutative integral domain and let \({\mathcal F}\) be a localizing system of ideals of \(A\). One can then construct a localization \(A_{\mathcal F}\) of \(A\), which is a subring of the field of fractions \(K\) of \(A\). The ring \(A\) is said to be a generalized Dedekind domain if conversely any ring \(B\) with \(A \subseteq B \subseteq K\) can be realized in the form \(A_{\mathcal F}\) for some such \({\mathcal F}\). In this paper, the authors first give a characterization of Prüfer rings in terms of their localizations, followed by many alternative descriptions of generalized Dedekind domains. These in turn lead to a number of characterizations of standard Dedekind domains. They then consider extensions of a generalized Dedekind domain in an algebraic extension of \(K\), not necessarily finite, and give criteria for such an extension to be itself generalized Dedekind. The results are illustrated with a number of explicit examples.
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 763934
zbMATH Keywords: localization
zbMATH Keywords: generalized Dedekind domain
zbMATH Keywords: Prüfer rings
MaRDI profile type: Publication
full work available at URL: jabr.1995.1077
OpenAlex ID: W2093285336
Wikidata QID: Q105439730
DOI: 10.1006/JABR.1995.1077
description: scientific article; zbMATH DE number 763934