Almost quasi clean rings

The element q of a ring R is called quasi-idempotent element if q2 = 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. © 2021. TÜBİTAK.