On the existence of relative injective covers (Q1857399)

Let \(R\) be an associative ring with identity and \(\mathcal G\) an abstract class of \(R\)-modules (i.e. \(\mathcal G\) is closed under isomorphic copies). A homomorphism \(\varphi\colon G\to M\) with \(G\in{\mathcal G}\) is called a \(\mathcal G\)-precover of \(M\) if for each homomorphism \(\psi\colon F\to M\) with \(F\in{\mathcal G}\) there is a homomorphism \(f\colon F\to G\) such that \(\varphi f=\psi\). If every endomorphism \(g\) of \(G\) for which \(\varphi g=\varphi\) is an automorphism of \(G\) then the precover is called a \(\mathcal G\)-cover of \(M\). The purpose of this paper is to investigate sufficient conditions for the existence of covers with respect to the class of modules which are injective relative to a hereditary torsion theory. The main result is: Let \(\sigma=(\mathcal{T,F})\) be a hereditary torsion theory for the category \(R\)-mod with associated Gabriel filter \(\mathcal L\). Then every module has a \(\sigma\)-torsion free \(\sigma\)-injective cover provided one of the following conditions hold: (a) \(\mathcal L\) contains a cofinal subset \({\mathcal L}'\) of finitely presented left ideals; (b) \(R\) is left semihereditary and \(\sigma\) is of finite type; (c) \(R\) is a left Noetherian ring.