Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence

For $n\ge 2$ and fixed $k\ge 1$, we study when a square matrix $A$ over an arbitrary field $\mathbb{F}$ can be decomposed as $T+N$ where $T$ is a torsion matrix and $N$ is a nilpotent matrix with $N^k=0$. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of $A\in {\mathbb{M}}_n(\mathbb{F})$ is algebraic over its base field and the rank of $A$ is at least $\frac{n}{k}$, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for $k=2$ and nilpotent matrices over arbitrary fields (even over division rings). This somewhat continues our recent publications in Lin. & Multilin. Algebra (2023) and Internat. J. Algebra & Computat. (2022) as well as it strengthens results due to Calugareanu-Lam in J. Algebra & Appl. (2016).