## Mohammad Saleh

*On weakly projective and weakly injective modules *

Comment.Math.Univ.Carolinae 45,3 (2004) 389-402. **Abstract:**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.

**Keywords:** tight, weakly tight, weakly injective, weakly projective, countably thick, locally q.f.d., weakly semisimple

**AMS Subject Classification:** 16D50, 16D60, 16D70, 16P40

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