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tight; weakly tight; weakly injective; weakly projective; countably thick; locally q.f.d.; weakly semisimple
The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module $M$, there exists a module $K\in \sigma [M]$ such that $K\oplus N$ is weakly injective in $\sigma [M]$, for any $N\in \sigma [M]$. Similarly, if $M$ is projective and right perfect in $\sigma [M]$, then there exists a module $K\in \sigma [M]$ such that $K\oplus N$ is weakly projective in $\sigma [M]$, for any $N\in \sigma [M]$. Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. For some classes $\Cal M$ of modules in $\sigma [M]$, we study when direct sums of modules from $\Cal M$ satisfy property $\Bbb P$ in $\sigma [M]$. In particular, we get characterizations of locally countably thick modules, a generalization of locally q.f.d. modules.
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