On a generalization of $C_2$-modules

Abstract

A module $M$ is called a $C_{21}$-module if, whenever $A$ and $B$ are submodules of $M$ with $A \cong B$, $A$ is nonsingular and $B$ is a direct summand of $M$, then $A$ is a direct summand of $M$. Various examples of $C_{21}$-modules are presented. Some basic properties of these modules are investigated. It is shown that the class of rings $R$ over which every $C_{21}$-module is a $C_2$-module is exactly that of right SI-rings. Also, we prove that for a ring $R$, every $R$-module has $(C_{21})$ if and only if $R$ is a right t-semisimple ring.