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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.
16D90 Module categories in associative algebras
16S50 Endomorphism rings; matrix rings
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