ON A MODULE CONTAINING A COPY OF ITS FACTOR BY A $t$-CLOSED SUBMODULE

In this paper, we introduce the notion of $t$-co-epi-retractable as generalization of $c$-co-epi-retractable. An $R$-module $M$ is called $t$-co-epi-retractable if contains a copy of its factor modules by a $t$-closed submodule. The ring $R$ is called $t$-co-pri if $R_R$ is $t$-co-epi-retractable. Conditions are found under which, a $t$-co-epi-retractable module is $t$-extending, $t$-continuous, $t$-quasi-continuous, $t$-semisimple, $t$-co-compressible and other modules. Finally, $t$-semisimple rings and some well-known rings are characterized in terms of $t$-co-epi-retractable.