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 X ⊥ -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 ⊥ -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 ⊥ -projective covers and X -injective envelopes over a X ⊥ -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 ⊆ 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 f p − Ω 1 -modules.