I 1 -RINGS

The objectiv 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 for it endomorphism ring of all free R-module, is a generalized right Baer ring, right I1-ring. Call a ring R is a generalized right Baer ring if any right annihilator contains a non-zero idempotent. We call a ring R is right I1-ring if the right annihilator of any element of R contains a non-zero idempotent. This text is showing that each right ideal of a ring R contains a projective right ideal if the endomorphism ring of any free R-module is a right I1-ring. And shown over a ring R, the endomorphism ring of any free R-module is a generalized right Baer ring if and only if endomorphism ring of any free R-module is an I1-ring.