Matrices over non-commutative rings as sums of fifth and seventh powers

Tanay Kochrekar, A. S. Garge

Let $R$ be non-commutative ring with unity. In this paper, we prove that a given matrix in $M_n(R)$ is a sum of $p$-th powers in $M_n(R)$, where $n<p$ if and only if its trace ca…