Abstract 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.
Bennis et al. (Mon,) studied this question.