Quasi invo-clean rings

Abstract

An element v of an arbitrary ring R is called an involution if v2 = 1 and a quasi-involution if either v or 1 - v is an involution. We thereby define R to be quasi invo-clean as the one whose elements are written in the form of a sum of an idempotent and a quasi-involution. This considerably extends the class of invo-clean rings introduced by the present author in Commun. Korean Math. Soc. (2017) and the class of weakly tripotent rings introduced by Breaz and Cîmpean in Bull. Korean Math. Soc. (2018). We, moreover, prove the curious fact that the newly defined class of quasi invo-clean rings actually coincides with the class of weakly invo-clean rings defined by Danchev in Afr. Mat. (2017).