Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules

In this paper, we investigate the notions of X-1-projective, X-injective, and X-flat modules and give some characterizations of these modules, where X is a class of left modules. We prove that the class of all X-1-projective modules is Kaplansky. Further, if the class of all X-injective R-modules is contained in the class of all pure projective modules, we show the existence of X-1-projective covers and X-injective envelopes over a X1-hereditary ring. Further, we show that a ring R is Noetherian if and only if W-injective R-modules coincide with the injective R-modules. Finally, we prove that if W subset of S, every module has a W-injective precover over a coherent ring, where W is the class of all pure projective R-modules and S is the class of all fp -omega(1)-modules.