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 ⊥ -projective, id="M2"> -injective, and id="M3"> -flat modules give some characterizations these modules, where id="M4"> is a class left modules. We prove that all id="M5"> -projective Kaplansky. Further, if id="M6"> -injective id="M7"> R -modules contained in pure projective show existence id="M8"> covers id="M9"> envelopes over id="M10"> -hereditary ring. ring id="M11"> Noetherian only id="M12"> mathvariant="script">W id="M13"> coincide with injective id="M14"> -modules. Finally, id="M15"> ⊆ mathvariant="script">S , every module has id="M16"> precover coherent ring, id="M17"> id="M18"> id="M19"> id="M20"> f p − Ω 1