Generalized Gorenstein Modules

We introduce a generalization of the Gorenstein injective modules: the Gorenstein [Formula: see text]-injective modules (denoted by [Formula: see text]). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor [Formula: see text], with [Formula: see text] any [Formula: see text]-injective module. Thus, [Formula: see text] is the class of classical Gorenstein injective modules, and [Formula: see text] is the class of Ding injective modules. We prove that over any ring [Formula: see text], for any [Formula: see text], the class [Formula: see text] is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For [Formula: see text] we show that [Formula: see text] (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring [Formula: see text] is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein [Formula: see text]-projectives (denoted by [Formula: see text]). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for any[Formula: see text] the class [Formula: see text] is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.