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Vietnam Journal of Mathematics 33:2 (2005) 214-221  When $M$-Cosingular Modules Are Projective Derya Keskin T\" ut\" unc\" u and Rachid Tribak Abstract.  Let $M$ be an $R$-module. Talebi and Vanaja investigate the category $\sigma[M]$ such that every $M$-cosingular module in $\sigma[M]$ is projective in $\sigma[M]$. In the light of this property we call $M$ a COSP-module if every $M$-cosingular module is projective in $\sigma[M]$. This note is devoted to the investigation of these classes of modules. We prove that every COSP-module is a coatomic module having a semisimple radical. We also characterise COSP-module when every injective module in $\sigma[M]$ is amply supplemented. Finally we obtain that a COSP-module is artinian if and only if every submodule has finite hollow dimension.

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