Products of commutators in matrix rings

TL;DR

This paper investigates when elements in matrix rings over various rings can be expressed as products of commutators, revealing conditions under which such factorizations are possible or impossible.

Contribution

It provides new results on expressing matrix ring elements as products of commutators, especially under conditions like Bass stable rank one and division rings.

Findings

Not all elements in matrix rings over commutative rings are products of commutators.

Over division rings with infinite center, every matrix is a product of two commutators.

For fields, elements can be expressed as sums of specific commutator products depending on minimal polynomial degree.

Abstract

Let $R$ be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, 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 $2$.