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Subdirect representations of categories of modules. (English) Zbl 1196.16002

Dung, Nguyen Viet (ed.) et al., Rings, modules and representations. International conference on rings and things in honor of Carl Faith and Barbara Osofsky, Zanesville, OH, USA, June 15–17, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4370-3/pbk). Contemporary Mathematics 480, 139-151 (2009).
Summary: Let \(R\) be an associative ring with identity. We determine the minimal nonzero ideals of the category \(\text{Mod-}R\), and find a representation of \(\text{Mod-}R\) as a subdirect product of subdirectly irreducible preadditive categories. The category \(\text{Mod-}R\) turns out to be subdirectly irreducible if and only if \(R\) has a unique simple right module up to isomorphism.
For the entire collection see [Zbl 1158.16001].

MSC:

16D90 Module categories in associative algebras
18E05 Preadditive, additive categories
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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