On w-universal injective modules and their applications in commutative rings

summary:Let R be a commutative ring and w be the w-operation on R. We introduce the concept of w-universal injective modules and establish their fundamental properties. It is shown that the product of E(R/ m), where m ranges over maximal w-ideals of R, is a w-universal injective w-module over R, albeit not necessarily a universal injective R-module. As applications, we characterize w-IF rings and w-coherent rings using w-universal injective modules. Specifically, we demonstrate that R is a w-IF ring if and only if R is w-coherent and E(R/ m) is a flat R-module for every m ∈ w -Max(R). These results extend existing results and provide deeper insights into the structure of w-modules.