The objective of this paper is to continue our study for a right 1 I - rings and
to generalize the concept of 1 I - rings to modules. We call a ring R a right
1 I - ring if every right annihilator for any element of R contains a nonzero
idempotent
.
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
generali
zed 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.
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.
