A characterization of large Dedekind domains
type: Item
instance of: scholarly article
title: A characterization of large Dedekind domains
zbMATH Open document ID: 1445.13017
published in: Archiv der Mathematik
publication date: 2020-07-03T00:00:00Z
review text: A commutative domain \(D\) (with identity) is said to be \textit{ideal upper finite} if every nonzero ideal of \(D\) is contained in but finitely many ideals of \(D\). The author proves that being a Dedekind domain is a sufficient but not necessary condition for being an ideal upper finite domain. However, if \(D\) has cardinality \(\kappa >2^{\aleph_0}\), \(D\) is a Dedekind domain if and only if \(D\) is ideal upper finite. In ZFC, the cardinal \(2^{\aleph_0}\) is the best possible, since if \(\kappa\) is a cardinal \(\aleph_0 \leq \kappa \leq 2^{\aleph_0}\), there exists an ideal upper finite domain \(D\) of cardinality \(\kappa\) which is not Dedekind.
reviewed by: Laura Cossu
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 7217591
zbMATH Keywords: Dedekind domain
zbMATH Keywords: field of fractions
zbMATH Keywords: Krull's intersection theorem
zbMATH Keywords: principal ideal ring
zbMATH Keywords: (discrete) valuation (over) ring
author: Greg Oman
MaRDI profile type: Publication
full work available at URL: s00013-020-01443-6
OpenAlex ID: W3012346982
cites work: Residually Finite Rings
cites work: Q3137901
cites work: Q2770573
cites work: Commutative Artinian Principal Ideal Rings
cites work: Jónsson Modules over Noetherian Rings
cites work: SMALL AND LARGE IDEALS OF AN ASSOCIATIVE RING
DOI: 10.1007/S00013-020-01443-6
description: scientific article; zbMATH DE number 7217591