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Classes of modules related to Serre subcategories. (English) Zbl 1034.16012
Summary: Let $$R$$ be an associative ring with non-zero identity. For a Serre subcategory $$\mathcal C$$ of the category $$R$$-mod of left $$R$$-modules, we consider the class $${\mathcal A}_{\mathcal C}$$ of all modules that do not belong to $$\mathcal C$$, but all their proper submodules belong to $$\mathcal C$$. Alongside with basic properties of such associated classes of modules, we prove that every uniform module of $${\mathcal A}_{\mathcal C}$$ has a local endomorphism ring. Moreover, if $$R$$ is a commutative ring, then every torsionfree faithful $$R$$-module of $${\mathcal A}_{\mathcal C}$$ is isomorphic to the injective hull of $$R$$ and its endomorphism ring is a division ring.
##### MSC:
 16D90 Module categories in associative algebras 16S50 Endomorphism rings; matrix rings
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