Strongly 2-nil-clean rings with involutions

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 , a 2 ∈ 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 ≅ R 1 , R 2 or R 1 × R 2 , where R 1 / J ( R 1 ) is a *-Boolean ring and J ( R 1 ) is nil, R 2 / J ( R 2 ) is a *-Yaqub ring and J ( R 2 ) is nil. The uniqueness of projections of such rings are thereby investigated.