Applications of Epi-Retractable and Co-Epi-Retractable Modules

‎A module $M$ is called epi-retractable if every submodule of $M$‎ ‎is a homomorphic image of $M$‎. ‎Dually‎, ‎a module $M$ is called‎ ‎co-epi-retractable if it contains a copy of each of its factor‎ ‎modules‎. ‎In special case‎, ‎a ring $R$ is called co-pli (respectively, ‎co-pri) if $_{R}R$ (respectively, ‎$R_{R}$) is co-epi-retractable‎. ‎It is‎ ‎proved that if $R$ is a left principal right duo ring‎, ‎then every‎ ‎left ideal of $R$ is an epi-retractable $R$-module‎. ‎A co-pli‎ ‎strongly prime ring $R$ is a simple ring‎. ‎A left self-injective‎ ‎co-pli ring $R$ is left Noetherian if and only if $R$ is a left‎ ‎perfect ring‎. ‎It is shown that a cogenerator ring $R$ is a pli‎ ‎ring if and only if it is a co-pri ring‎. ‎Moreover‎, ‎if $R$ is a‎ ‎left perfect ring such that every projective $R$-module is‎ ‎co-epi-retractable‎, ‎then $R$ is a quasi-Frobenius ring‎.