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 Rmodule,
as a generalized rig
ht 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.
