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

Tanay Kochrekar; A. S. Garge

paper

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

arXiv:2609.14064

Abstract

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 can be written as a sum of $p$ -th powers in $M_{n} (R)$ , for $n = 2, 3, 4$ and $p = 5$ and $n = 4, 5, 6$ for $p = 7$ . This paper extends the results of S. A. Katre and Kshipra Wadikar, $(p = 3)$ and Garge $(n \geq p), (p = 5, 7)$ , and S. A. Katre and Deepa Krishnamurthi $(n \geq p)$ .