Let R be a domain with its field Q of quotients. An R-module M is said to be weak w-projective if Undefined control sequence Ext for all Undefined control sequence dag, where Undefined control sequence dag denotes the class of Undefined control sequence GV-torsionfree R-modules N with the property that Undefined control sequence Ext for all w-projective R-modules M and for all integers k≥1. In this paper, we define a domain R to be w-Matlis if the weak w-projective dimension of the R-module Q is ≤1. To characterize w-Matlis domains, we introduce the concept of w-Matlis cotorsion modules and study some basic properties of w-Matlis modules. Using these concepts, we show that R is a w-Matlis domain if and only if Undefined control sequence Ext for any Undefined control sequence dag-divisible R-module D and any integer k≥1, if and only if every Undefined control sequence dag-divisible module is w-Matlis cotorsion, if and only if w.w-Undefined control sequence pd.
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