Products of commutators in matrix rings

Brešar, Matej ; Gardella, Eusebio ; Thiel, Hannes 

Canadian mathematical bulletin, Tome 68 (2025) no. 2, pp. 512-529

Résumé

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 $2$.

Détail

DOI : 10.4153/S0008439524000523

Mots-clés : Commutator, matrix ring, division ring, Bass stable rank, L’vov–Kaplansky conjecture