Strongly 2-nil-clean rings

A ring R is strongly 2-nil-clean if every element in R is the sum of two idempotents 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 is an element of R, a - a(3) is an element of R is nilpotent, if and only if for all a is an element of R, a(2) is an element of R is strongly nil-clean, if and only if every element in R is the sum of a tripotent and a nilpotent that commute. Furthermore, we prove that a ring R is strongly 2-nil-clean if and only if R/J(R) is tripotent and J(R) is nil, if and only if R congruent to R-1, R-2 or R-1 x 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. Strongly 2-nil-clean group algebras are investigated as well.