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.