\(t\)-invertibility and Bazzoni-like statements
type: Item
instance of: scholarly article
title: \(t\)-invertibility and Bazzoni-like statements
zbMATH Open document ID: 1188.13010
author: Muhammad Zafrullah
published in: Journal of Pure and Applied Algebra
publication date: 2010-02-12T00:00:00Z
review text: \textit{S. Bazzoni} [Commun. Algebra 28, No. 11, 5157--5167 (2000; Zbl 0997.13005)] conjectured that if \(D\) is a Prüfer domain in which every nonzero locally principal ideal is invertible, then \(D\) has finite character. This was shown to be true by \textit{W. C. Holland, J. Martinez, W. Wm. McGovern} and \textit{M. Tesemma} [J. Algebra 320, No. 4, 1764--1768 (2008; Zbl 1151.13016)]. The paper under reviews gives an interesting alternative proof of this result and then some. It is shown that an integral domain \(D\) has the property that every nonzero locally principal ideal is invertible if and only if each proper principal ideal of \(D\) is contained in at most a finite number of proper mutually comaximal invertible ideals of \(D\). The author translates a result of Griffin to get that every nonzero nonunit of a PVMD\ belongs to only a finite number of maximal \(t\)-ideals if and only if every integral \(t\)-invertible \(t\)-ideal of \(D\) is contained in at most a finite number of mutually \(t\)-comaximal \(t\)-invertible \(t\)-ideals. An immediate corollary is that a Prüfer domain \(D\) has finite character if and only if each invertible integral ideal of \(D\) is contained in at most a finite number of mutually comaximal invertible ideals. This proves Bazzoni's conjecture. More generally, it is shown that for a PVMD or domain \(D\) in which every maximal \(t\)-ideal is \(t\)-invertible, \(D\) has finite \(t\)-character if and only if every \(t\)-locally principal \(t\)-ideal \(\;\)of \(D\) is \(t\)-invertible.
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 06F15
zbMATH DE Number: 5669134
zbMATH Keywords: Prüfer domain
zbMATH Keywords: PVMD
reviewed by: D. D. Anderson
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.07.004
OpenAlex ID: W2015780099
cites work: Groups in the class semigroup of a prüfer domain of finite character
cites work: Clifford regular domains
cites work: Bazzoni's conjecture
cites work: Clifford semigroups of ideals in monoids and domains
cites work: Some Results on v-Multiplication Rings
cites work: Putting \(t\)-invertibility to use
cites work: Q5661989
cites work: Q3283972
cites work: On Pruefer v-multiplication domains
cites work: Q4020938
cites work: Unruly Hilbert Domains
cites work: Characterizations of *-Multiplication Domains
cites work: Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains
cites work: Q4255424
DOI: 10.1016/J.JPAA.2009.07.004
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description: scientific article; zbMATH DE number 5669134