Generalized Right Bear Rings

The object of this paper is to study the relationship between certain ring R and endomorphism rings of free modules over R. Specifically, the basic problem is to describe ring R, which is endomorphism ring of all free R- module, as a generalized right Bear ring. Call a ring R a generalized right Bear ring if any right annihilator contains a nonzero idempotent. A structure theorem is obtained: endomorphism ring of a free module F is a generalized right Bear ring if and only if every closed submodule of F contains a direct summand of F. It is shown that every torsionless R-module contains a projective R-module if endomorphism ring of any free R-module is a generalized right Bear ring.