Products of commutators in matrix rings

Matej Brešar; Eusebio Gardella; Hannes Thiel

paper

Products of commutators in matrix rings

arXiv:2404.18116

Abstract

Let $R$ be a ring and let $n \geq 2$ . We discuss the question of whether every element in the matrix ring $M_{n} (R)$ is a product of (additive) commutators $[x, y] = x y - y x$ , for $x, y \in M_{n} (R)$ . An example showing that this does not always hold, even when $R$ is commutative, is provided. If, however, $R$ has Bass stable rank one, then under various additional conditions every element in $M_{n} (R)$ is a product of three commutators. Further, if $R$ is a division ring with infinite center, then every element in $M_{n} (R)$ is a product of two commutators. If $R$ is a field and $a \in M_{n} (R)$ , then every element in $M_{n} (R)$ is a sum of elements of the form $[a, x] [a, y]$ with $x, y \in M_{n} (R)$ if and only if the degree of the minimal polynomial of $a$ is greater than $2$ .

15 pages