Some characterizations of Krull domains (Q856353)

Let \(*\) be a star operation on an integral domain \(R\). A proper integral \(*\)-ideal \(I\) of \(R\) has \(*\)-finite length if the supremum of the lengths of all finite chains of \(*\)-ideals between \(I\) and \(R\) is finite. In this case, this supremum length is called the \(*\)-length of \(I\) and denoted by \(L_{*}(I)\). Then \(R\) is called a \(*\)-FLD (i.e., \(*\)-finite length domain) if every proper integral \(*\)-ideal of R has \(*\)-finite length. The main result of this paper asserts that \(R\) is a Krull domain if and only if \(R\) is a \(v\)-FLD and \(L_{v}((IJ)_{v}) = L_{v}(I) + L_{v}(J)\) for any two proper integral \(v\)-ideals \(I\) and \(J\) of \(R\). In the third section, the authors give Euclidean-like characterizations of factorial, Krull, and \(\pi\)-domains. In the last section, they define the notion of quasi \(*\)-invertibility and show that \(R\) is a Krull domain if and only if every proper \(t\)-prime ideal of \(R\) is quasi \(t\)-invertible.