Envelopes. covers and semidualizing modules

Given an R-module C and a class of R-modules D over a commutative ring R, we investigate the relationship between the existence of D-envelopes (respectively, D-covers) and the existence of Hom(C, D)-envelopes or C circle times D-envelopes (respectively, Hom(C, D)-covers or C circle times D-covers) of modules. As a consequence, we characterize coherent rings, Noetherian rings, perfect, rings and Artinian rings in terms of envelopes and covers by C-projective, C-flat, C-injective and C-FP-injective modules, where C is a semidualizing R-module.