Semicommutativity of rings by the way of idempotents

Abstract

In this paper, we focus on the semicommutative property of rings via idempotent elements.
In this direction, we introduce a class of rings, so-called right
$e$-semicommutative rings. The notion of right $e$-semicommutative
rings generalizes those of semicommutative rings, $e$-symmetric
rings and right $e$-reduced rings. We present examples of right
$e$-semicommutative rings that are neither semicommutative nor
$e$-symmetric nor right $e$-reduced. Some extensions of rings such
as Dorroh extensions and some subrings of matrix rings are
investigated in terms of right $e$-semicommutativity. As an
application, we give an answer to the question ``If $R$ is a clean
ring and $e^2 = e \in R$, is the ring $eRe$ clean?" in the case of
$e$-semicommutative rings.