Let R be a ring with identity.
The ain is to study some fundamental properties of a ring R when R is regular
or semi-potent and the radical Jacobson of R when R is semi-potent.
New results were obtained including necessary and sufficient condition
s of R
to be regular or semi-potent. New substructures of R are studied and their
relationship with the total of R.
It's considered that، the ring of linear operator of vector
space and stilled as a source of many mathematicians in general and
algebreians particularly in introducing a new concepts in algebra
and ring theory. In this subject I. Kaplansky proved
the following
theorem: "The ring of linear operators of finite dimension vector
space is regular".
The object of this paper is studying of ring of linear operator
of vector space in abstract algebra point of view.
The purpose of this paper is studying some properties of clean,
semi-clean and quasi-clean rings, and study the relationship between
these rings. A ring is called clean if each of its element is the sum of
an idempotent and a unit, a ring is calle
d semi-clean if each of its
element is the sum of an idempotent and a regular, a ring is called
quasi-clean if each of its element is the sum of an idempotent and an
pi .
