On Kursov's theorem for matrices over division rings

Abstract

Let D be a division ring with center F and multiplicative group $D^{\times }$, where each element of the commutator subgroup of $D^{\times }$ can be expressed as a product of at most s commutators. A known theorem of Kursov states that if D is finite-dimensional over F, then every element of the commutator subgroup of the general linear group over D can be expressed as a product of at most $s+1$ commutators. We show that this result remains valid when F has a sufficiently large number of elements, without requiring D to be finite-dimensional. Our approach not only improves upon recent results on matrix decompositions over division rings but also provides a look at the Engel word map for matrices over arbitrary algebras.