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A ring $R$ is trinil clean if every element in $R$ is the sum of a tripotentand a nilpotent. If $R$ is a 2-primal strongly 2-nil-clean ring, we prove that$M_n(R)$ is trinil clean for all $n\in {\Bbb N}$. Furthermore, we show that thematrix 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 ofa tripotent and a nilpotent.

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