The element $q$ of a ring $R$ is called quasi-idempotent element if $q^2=uq$ for some central unit $u$ of $R$, or equivalently $q=ue$, where $u$ is a central unit and $e$ is an idempotent of $R$. In this paper, we define that the ring $R$ is almost quasi-clean if each element of $R$ is the sum of a regular element and a quasi-idempotent element. Several properties of almost-quasi clean rings are investigated. We prove that every quasi-continuous and nonsingular ring is almost quasi-clean. Finally, it is determined that the conditions under which the idealization of an $R$-module $M$ is almost quasi clean.
Quasi-idempotent, almost quasi-clean ring
"Almost quasi clean rings,"
Turkish Journal of Mathematics: Vol. 45:
2, Article 24.
Available at: https://journals.tubitak.gov.tr/math/vol45/iss2/24