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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.
##### MSC:
 16D50 Injective modules, self-injective associative rings 16D90 Module categories in associative algebras