Gorenstein Krull domains and their factor rings (Q6561438)

Let \(R\) an integral domain. If \(A\) is a \(w\)-ideal of \(R\) (where \(w\) is the \(w\)-operation on \(R\)), then the \(w\)-envelope \((R/A)_{w}\) of the \(R\)-module \(R/A\) has a natural ring structure which is called a \(w\)-factor ring of \(R\) modulo \(A\). If \(x\) is a nonzero nonunit of \(R\), then the \(w\)-factor ring \((R/Rx)_{w}\) of the principal ideal \(Rx\) is denoted by \(\overline{R}_{w}\). \N\NRecall that a Gorenstein Dedekind domain (G-Dedekind for short) is defined to be an integral domain with Gorenstein global dimension at most one (i. e. domain having all ideals Gorenstein projective), and Gorenstein Krull (G-Krull) is an integral domain \(R\) satisfying the following three conditions: \((1)\) \(R_{P}\) is G-Dedekind for each \(P\in X^{(1)}\) (where \(X^{(1)}\) is the set of all height-one prime ideals), \((2)\) \(R =\bigcap_{P\in{X^{(1)}}}R_{P}\), and \((3)\) Each nonzero element of \(R\) is contained in at most finitely many elements of \(X^{(1)}\). Using the \(w\)-operation, the author established a strong connection between a Gorenstein Krull domain and the Gorenstein global dimension of its \(w\)-factor ring. \N\NThe main results asserts that an integral domain \(R\) is a G-Krull domain if and only if \(\overline{R}_{w}\) is a \(QF\)-ring for any nonzero nonunit \(x\) of \(R\) if and only if the Gorenstein global dimension of \(\overline{R}_{w}\) is zero for any nonzero nonunit \(x\) of \(R\).