A Gorenstein homological characterization of Krull domains (Q6589632)

This paper provides a novel Gorenstein homological characterization of Krull domains through the lens of \(w\)-operations. The authors establish that an integral domain \(R\) is Krull if and only if every nonzero proper \(w\)-ideal \(I\) satisfies \(\mathrm{G\text{-}gl.dim}((R/I)_w) = 0\), where \((R/I)_w\) is the \(w\)-factor ring of \(R\) modulo \(I\) (which was first introduced in [Zbl 1474.13039]). This result extends and complements existing characterizations of Krull domains, particularly those involving \(2\)-SG-semisimple rings. Additionally, they demonstrate that an integral domain \(R\) is Dedekind if and only if \(R/I\) is a QF-ring for every nonzero proper ideal \(I\). These results unify and expand upon various existing results in the literature, offering a deeper understanding of the relationship between Gorenstein homological dimensions and the structure of Krull and Dedekind domains.