Revisiting G-Dedekind domains

Abstract Let R be an integral domain with $qf(R)=K$ , and let $F(R)$ the set of nonzero fractional ideals . Call a dually compact (DCD) if, for each $I\in F(R)$ ideal $I_{v}=(I^{-1})^{-1}$ is finite intersection principal ideals. We characterize DCDs show that class properly contains various classes domains, such as Noetherian, Mori, Krull domains. In addition, we Schreier DCD greatest common divisor (GCD) (Greatest Common Divisor) property that, $A\in $A_{v}$ principal. G-Dedekind (i.e., has invertible ) if only satisfying $\ast :$ For all pairs subsets $\{a_{1},\ldots ,a_{m}\},\{b_{1},\ldots ,b_{n}\}\subseteq K\backslash \{0\}, (\cap _{i=1}^{m}(a_{i})(\cap _{j=1}^{n}(b_{j}))=\cap _{i,j=1}^{m,n}(a_{i}b_{j})$ discuss what appropriate names domains related notions should be. also make some observations about how behave under localizations polynomial ring extensions.