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A note on weak injectivity. (English) Zbl 1063.16004
For any module \(M\) in Mod-\(R\), let \(\sigma[M]\) denote the full subcategory of Mod-\(R\) whose objects are submodules of \(M\)-generated modules. Let the injective hull of \(X\in\text{Mod-}R\) in \(\sigma[M]\) be denoted by \(\widehat X\). Given any two modules \(Q\) and \(N\) in \(\sigma[M]\), \(Q\) is called weakly \(N\)-injective in \(\sigma[M]\) if for any homomorphism \(\psi\colon N\to\widehat Q\) there exists a homomorphism \(\widehat\psi\colon N\to Q\) and a monomorphism \(\sigma\colon Q\to\widehat Q\) such that \(\psi=\sigma\widehat\psi\). If \(Q\) is weakly \(N\)-injective for all finitely generated modules \(N\) in \(\sigma[M]\), \(Q\) is said to be weakly injective in \(\sigma[M]\). A module \(M_R\) is called weakly semisimple iff every module \(N\in\sigma[M]\) is weakly injective in \(\sigma[M]\). If every finitely generated module \(N\in\sigma[M]\) has finite uniform dimension, then the module \(M_R\) is said to be locally q.f.d.
A number of characterizations for a ring \(R\) or for an \(R\)-module \(M_R\) to be weakly semisimple are found in terms of the local q.f.d. property and weak injectivity.
16D50 Injective modules, self-injective associative rings
16D90 Module categories in associative algebras