n-Torsion Clean and Almost n-Torsion Clean Matrix Rings

We (completely) determine those natural numbers n for which the full matrix ring $\mathbb{M}_n(\mathbb{F}_2)$ and the triangular matrix ring $\mathbb{T}_n(\mathbb{F}_2)$ over the two elements field $\mathbb{F}_2$ are either n-torsion clean or are almost n-torsion clean, respectively. These results somewhat address and settle a question, recently posed by Danchev–Matczuk in Contemp. Math. (2019) as well as they supply in a more precise aspect the nil-cleanness property of the full matrix $n\times n$ ring $\mathbb{M}_n(\mathbb{F}_2)$ for all naturals $n\geq 1$ , established in Linear Algebra & Appl. (2013) by Breaz–Călugăreanu–Danchev–Micu and again in Linear Algebra & Appl. (2018) by Ster as well as in Indag. Math. (2019) by Shitov.