English

Abstract

A *-ring R is strongly 2-nil-*-clean if every element in R is the sum of two projections and a nilpotent that commute. Fundamental properties of such *-rings are obtained. We prove that a *-ring R is strongly 2-nil-*-clean if and only if for all a ∈ R, a2 ∈ R is strongly nil-*-clean, if and only if for any a ∈ R there exists a *-tripotent e ∈ R such that a − e ∈ R is nilpotent and ea = ae, if and only if R is a strongly *-clean SN ring, if and only if R is abelian, J(R) is nil and R/J(R) is *-tripotent. Furthermore, we explore the structure of such rings and prove that a *-ring R is strongly 2-nil-*-clean if and only if R is abelian and R ≅ R1,R2 or R1 × R2, where R1/J(R1) is a *-Boolean ring and J(R1) is nil, R2/J(R2) is a *-Yaqub ring and J(R2) is nil. The uniqueness of projections of such rings are thereby investigated.