Retractable and Coretractable Modules

© 2016 Springer Science+Business Media New York 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 σ(M), every module is retractable. (2) In the category σ(M), every module is coretractable. (3) M is a CSL-module. (4) ExtR 1(S1, S2) = 0 for any two simple nonisomorphic modules S1, S2 ∈ σ(M). (5) M is a fully Kasch module