Rings over which every matrix is the sum of a tripotent and a nilpotent

Abstract

A ring R is trinil clean if every element in R is the sum of a tripotent and a nilpotent. If R is a 2-primal strongly 2-nil-clean ring, we prove that M n (R) is trinil clean for all n ∈ N. Furthermore, we show that the matrix ring over a strongly 2-nil-clean ring of bounded index is trinil clean. We thereby provide various type of rings over which every matrix is the sum of a tripotent and a nilpotent.