Generalized Gorenstein Modules

We introduce a generalization of the Gorenstein injective modules: the Gorenstein FPn-injective modules (denoted by GIn). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-), with A any FPn-injective module. Thus, GI0 is the class of classical Gorenstein injective modules, and GI1 is the class of Ding injective modules. We prove that over any ring R, for any n≥2, the class GIn is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For n=1 we show that GI1 (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring R is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein FPn-projectives (denoted by GIn). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for anyn≥2 the class GPn is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.