n-torsion clean and almost n-torsion clean matrix rings

TL;DR

This paper classifies when matrix rings over F2 are n-torsion clean or almost n-torsion clean, resolving a recent open question and refining understanding of nil-cleanness in these rings.

Contribution

It completely determines the natural numbers n for which matrix rings over F2 are n-torsion clean or almost n-torsion clean, addressing an open problem.

Findings

Identifies all n for which M_n(F_2) is n-torsion clean.

Identifies all n for which T_n(F_2) is almost n-torsion clean.

Provides a more precise understanding of nil-cleanness in matrix rings over F2.

Abstract

We completely determine those natural numbers $n$ for which the full matrix ring $M_n(F_2)$ and the triangular matrix ring $T_n(F_2)$ over the two elements field $F_2$ 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 $M_n(F_2)$ for all naturals $n \geq 1$, established in Linear Algebra and Appl. (2013) by Breaz-Calugareanu-Danchev-Micu and again in Linear Algebra & Appl. (2018) by Ster as well as in Indag. Math. (2020) by Shitov.