\(\ast \)-finite ideals contained in infinitely many maximal \(\ast _{s}\)-ideals (Q658933)

The authors give a new characterization of \(*\)-finite ideals that are contained in infinitely many \(*_{f}\)-maximal ideals. That is, let \(R\) be an integral domain, \(*\) a star operation on \(R\), \(*_{f}\) the star operation of finite type associated to \(*\) and \(\Gamma\) a set of proper \(*\)-ideals of finite type of \(R\) such that every proper \(*\)-finite \(*\) ideal of \(R\) is contained in some element of \(\Gamma\). Let \(A\) be a nonzero finitely generated ideal of \(R\) with \(A^{*}\not =R\). Then \(A\) is contained in an infinite number of \(*_{f}\)-maximal ideals of \(R\) if and only if there exists an infinite family of mutually \(*_{f}\)-comaximal ideals in \(\Gamma\) containing \(A\).