Proper C2-Modules and Structural Properties of Rings

In this paper, we study modules satisfy the condition any submodule that is isomorphic to a proper direct summand is itself a direct summand. It is shown that this class of modules is a proper extension of C2 modules. We give the structure of modules and rings via these modules. We prove that a ring Formula: see text is quasi-Frobenius iff Formula: see text is a proper C2 module for every Formula: see text and every Formula: see text-generated right Formula: see text-module embeds in a free module. It is shown that a proper C2 module is a C2 module over SI-rings.