Remarks on vector space generated by the multiplicative commutators of a division ring

Abstract

Let $D$ be a division ring with centre $F$. An element of the form $xyx^{-1}y^{-1}\in D$ is called a multiplicative commutator. Let $T(D)$ be the vector space over $F$ generated by all multiplicative commutators in $D$. In M. Aghabali et al., J. Algebra Appl. 12 (2013), no. 8, art. 1350043, the authors have conjectured that every division ring is generated as a vector space over its centre by all of its multiplicative commutators. In this note it is shown that if $D$ is centrally finite, then the conjecture holds.