Locally factorial integral domains
type: Item
instance of: scholarly article
title: Locally factorial integral domains
zbMATH Open document ID: 0545.13007
DOI: 10.1016/0021-8693(84)90214-X
author: David F. Anderson
author: D. D. Anderson
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: An integral domain R is locally factorial if \(R_ f=R[1/f]\) is factorial for each nonzero nonunit f of R. A non quasi-local locally factorial domain is a Krull domain. Claborn gave examples of locally factorial Dedekind domains with cyclic divisor class groups (and with the further property that every proper overring is a PID). We show that if R is a locally factorial Krull domain, then \(Cl(R)={\mathbb{Z}}^ n\oplus {\mathbb{Z}}/m{\mathbb{Z}}\) where \(n\geq 0\), \(m\geq 1\). Conversely, for each such n and m we construct a locally factorial Dedekind domain R with \(Cl(R)={\mathbb{Z}}^ n\oplus {\mathbb{Z}}/m{\mathbb{Z}}.\) We also give an example of a locally factorial Dedekind domain that has an overring which is not locally factorial. Among the many other results, we show that if R is a Krull domain with finitely generated nonzero divisor class group, then some localization of R is locally factorial but not factorial. Quasi- local locally factorial domains that are not Krull domains are also investigated.
Mathematics Subject Classification ID: 13F15
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13C99
zbMATH DE Number: 3867471
zbMATH Keywords: locally factorial Krull domain
zbMATH Keywords: locally factorial Dedekind domain
zbMATH Keywords: finitely generated nonzero divisor class group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3867471