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 ${\mathcal{X}}^{\perp }$ -projective, $\mathcal{X}$ -injective, and $\mathcal{X}$ -flat modules and give some characterizations of these modules, where $\mathcal{X}$ is a class of left modules. We prove that the class of all ${\mathcal{X}}^{\perp }$ -projective modules is Kaplansky. Further, if the class of all $\mathcal{X}$ -injective $R$ -modules is contained in the class of all pure projective modules, we show the existence of ${\mathcal{X}}^{\perp }$ -projective covers and $\mathcal{X}$ -injective envelopes over a ${\mathcal{X}}^{\perp }$ -hereditary ring. Further, we show that a ring $R$ is Noetherian if and only if $\mathcal{W}$ -injective $R$ -modules coincide with the injective $R$ -modules. Finally, we prove that if $\mathcal{W}\subseteq \mathcal{S},$ every module has a $\mathcal{W}$ -injective precover over a coherent ring, where $\mathcal{W}$ is the class of all pure projective $R$ -modules and $\mathcal{S}$ is the class of all $fp-{\Omega }^1$ -modules.