Relative FP-injective modules and relative IF rings

Let R be a commutative ring and w be the so-called w-operation on R. Then we introduce and study two concepts of w-FP-injective modules and w-IF rings. To do so, we use two main methods, of which one is to localize at maximal w-ideals of R and the other is to utilize w-Nagata modules over w-Nagata rings. As an application, we characterize Prufer v-multiplication domains in terms of w-FP-injective modules. Finally we provide an example of a w-IF ring, but not an IF ring using a trivial extension.