Quasi invo-clean rings

Abstract

An element v of an arbitrary ring R is called aninvolution if v2= 1 and a quasi-involution if either v or 1−vis an involution. We thereby define R to be quasi invo-cleanas the one whose elements are written in the form ofa 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 introducedby Breaz and Cˆımpean in Bull. Korean Math. Soc. (2018). We, more-over, prove the curious fact that the newly defined class of quasi invo-cleanrings actually coincides with the class of weakly invo-clean rings defined by Danchev in Afr. Mat. (2017).