Preserving and reflecting covers by functors. Applications to graded modules Original Research Article

We study image-covers in the context of Grothendieck categories. Namely, we analyse when a functor between two Grothendieck categories preserves or reflects image-covers. We apply our general study to the category of graded modules over a graded ring, by showing that relative injective covers with respect to a torsion theory are preserved and reflected, in some cases, among the categories R-gr, R1-Mod and R-Mod.