Characterizing domains of finite \(*\)-character
An integral domain \(R\) is said to be of finite \(*\)-character, for a given star operation of finite type \(*\) on \(R\), if every nonzero nonunit element of \(R\) is contained in at most a finite number of maximal \(*\)-ideal. In the particular case where \(*=d\) is the trivial star operation, one obtains the so-called domain of finite character, that is, every nonzero nonunit element of \(R\) is contained in at most a finite number of maximal ideal. In this paper, the authors characterize integral domains with finite \(*\)-character. The main result asserts that for an integral domain \(R\), let \(\mathcal{T}\) be a set of proper \(*\)-ideals of finite type of \(R\) such that every proper \(*\)-finite \(*\)-ideal of D is contained in some member of \(\mathcal{T}\) and let \(I\) be a nonzero finitely generated ideal of \(R\) with \(I^{*}\not= R\). Then \(I\) is contained in an infinite number of maximal \(*\)-ideals if and only if there exists an infinite family of mutually \(*\)-comaximal ideals in \(\mathcal{T}\) containing \(I\). As an immediate consquence, a Prüfer domain \(R\) is of finite character if and only if every invertible ideal of \(R\) is contained in at most a finite number of mutually comaximal invertible ideals, and a Noetherian domain \(R\) is of finite character if and only if every proper nonzero ideal of \(D\) is contained in at most a finite number of mutually comaximal proper ideals. The paper closes with other applications to domains with finite \(t\)-character. In particular, an integral domain \(R\) with PSP property is of finite \(t\)-character if and only if every nonzero nonunit of \(R\) is divisible by at most a finite number of mutually coprime nonunits.
scientific article; zbMATH DE number 5726614