On the Existence of Relative Injective Covers

Abstract

Let $\sigma = ({\mathcal{T}},{\mathcal{F}})$ be a hereditary torsion theory for the category ${\mathcal{R}}$-mod of unital left ${\mathcal{R}}$-modules over an associative ring ${\mathcal{R}}$ with an identity element. The purpose of this note is to prove that if the associated Gabriel filter ${\mathcal{L}}$ consists of finitely presented left ideals, then every module has a $\sigma$-injective cover and if ${\mathcal{L}}$ contains a cofinal subset of finitely presented left ideals, then every module has a $\sigma$-torsionfree $\sigma$-injective cover. The methods used working with pure submodules contained in ``large" submodules also allow to unify the proofs of some previously known results.