The complete version of Kursov's theorem for matrices over division rings

Abstract

Let D be a division ring with center F, and let $n>1$ be an integer. A known result due to Kursov asserts that if D is finite-dimensional over F, then the commutator width of the general linear group ${\mathrm{GL}}_n\left(D\right)$ is at most one greater than that of ${\mathrm{GL}}_1\left(D\right)$. In the absence of the finite-dimensionality assumption, recent research has made significant progress, though the developments typically cease once F is infinite or D is algebraic over F. The purpose of this paper is to show that these restrictions are, in fact, unnecessary.