Retractable and coretractable modules

In this paper, we study mod-retractable modules, CSL-modules, fully Kasch modules, and their interrelations. Right fully Kasch rings are described. It is proved that for a module $M$ of finite length, the following conditions are equivalent. (1) In the category $\sigma(M)$, every module is retractable. (2) In the category $\sigma(M)$, every module is coretractable. (3) $M$ is a CSL-module. (4) $\mathrm{Ext}_R^1(S_1,S_2)=0$ for any two simple nonisomorphic modules $S_1,S_2\in\sigma(M)$. (5) $M$ is a fully Kasch module.