Generalized W-Gorenstein Modules

In this paper, we introduce the notion of generalized W W -Gorenstein modules respect to some subclass W W , extending the classical notion of Gorenstein projective modules. By exploiting the correspondence between projective modules over the endomorphism ring EndR(C) End R ( C ) of a module C and elements of its additive closure W=AddR(C) W = Add R ( C ) , we establish a fundamental correspondence between Gorenstein projective EndR(C) End R ( C ) -modules and generalized AddR(C) Add R ( C ) -Gorenstein modules. This result refines existing relative homological settings and provides a natural extension of well-known results in Gorenstein homological algebra. We explore key properties, such as closure under direct summands and sums, and identify conditions under which the class of generalized W W -Gorenstein modules coincides with other classes of modules, like Gorenstein projective modules.