\(w\)-divisorial domains (Q1770479)

Let \(R \) be an integral domain with quotient field \(K\). A star operation is a map \(I\to I^*\) from the set \(F(R)\) of nonzero fractional ideals of \(R\) to itself such that: (1) \(R^*=R\) and \((aI)^*=aI^*\), for all \(a\in K-\{0\}\); (2) \(I \subseteq I^*\) and \(I\subseteq J\Rightarrow I^* \subseteq J^*\); (3) \(I^{**}=I^*\). The identity is a star operation, called the \(d\)-operation. We set \(I_v=(R:(R: I))\) and \(I_t=\bigcup J_v\) with the union taken over all finitely generated ideals \(J\subseteq I\). Then they are star operations. A nonzero fractional ideal \(I\) is called a *-ideal if \(I=I^*\). If \(I=I_v\) we say that \(I\) is divisorial. A *-maximal ideal is an ideal that is maximal in the set of the proper *-ideals. The \(w\)-operation is the *-operation defined by setting \(I_w=\bigcap_{M\in t-\text{Max}(R)}IR_M\). The authors study the concept of \(w\)-divisorial domains, that is domains in which each \(w\)-ideal is divisorial. For example, they show that \(R\) is a \(w\)-divisorial domain if and only if \(R\) is a weakly Matlis domain (that is a domain with \(t\)-finite character such that each \(t\)-prime ideal is contained in a unique \(t\)-maximal ideal) and \(R_M\) is a divisorial domain, for each \(t\)-maximal ideal \(M\). They also show that \(R\) is an integrally closed \(w\)-divisorial domain if and only if \(R\) is a weakly Matlis PvMD and each \(t\)-maximal ideal is \(t\)-invertible.