Decomposition of matrices into a sum of invertible matrices and matrices of fixed index

Abstract: For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix $N$ with $N^k=0$ over an arbitrary field $\mathbb{F}$.