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.