On Strongly Quasipolar Rings

Abstract

Idempotent elements, invertible elements and quasinilpotents are
some important tools to study structures of rings. By using these
kinds of elements, we study a class of rings, called strongly
quasipolar rings, which is a subclass of that of quasipolar rings.
Let $R$ be a ring with identity. An element $a\in R$ is said to be
strongly quasipolar if there exists $p^2=p\in comm^{2}(a)$ such
that $a+p$ is invertible and $a^{2}p$ is quasinilpotent. The ring
$R$ is called strongly quasipolar in case each of its elements is
strongly quasipolar. Some basic properties of the strongly
quasipolar rings are obtained. The class of strongly quasipolar
rings lies properly between the classes of pseudopolar rings and
quasipolar rings. We determine the conditions under which a
quasipolar ring is strongly quasipolar. We also show that strongly
quasipolarity is a generalization of uniquely cleanness. When we
consider this concept in terms of generalized inverses, we get
that every pseudo Drazin invertible element is strongly
quasipolar, and every strongly quasipolar element is generalized
Drazin invertible.