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 ringR is almost quasi-clean if each element of R is the sum of a regular element and a quasi-idempotent element. Severalproperties of almost-quasi clean rings are investigated. We prove that every quasi-continuous and nonsingular ring isalmost quasi-clean. Finally, it is determined that the conditions under which the idealization of an R-module M isalmost quasi clean.

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