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

Abstract

In this paper, we investigate the notions of ‐projective, ‐injective, and ‐flat modules and give some characterizations of these modules, where is a class of left modules. We prove that the class of all ‐projective modules is Kaplansky. Further, if the class of all ‐injective R‐modules is contained in the class of all pure projective modules, we show the existence of ‐projective covers and ‐injective envelopes over a ‐hereditary ring. Further, we show that a ring R is Noetherian if and only if ‐injective R‐modules coincide with the injective R‐modules. Finally, we prove that if every module has a ‐injective precover over a coherent ring, where is the class of all pure projective R‐modules and is the class of all fp − Ω1‐modules.