\(t\)-invertibility and Bazzoni-like statements
\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.
scientific article; zbMATH DE number 5669134
scientific article