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On the existence of relative injective covers. (English) Zbl 1006.16006
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.

##### MSC:
 16D50 Injective modules, self-injective associative rings 16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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