Decompositions of matrices over division algebras into products of commutators

Abstract

Let D be a division algebra of degree m. The first aim of this paper is to show that if D is tame and totally ramifield and if the center of D is Henselian, then there exists a positive d depending on m such that every element in the commutator subgroup D′ of the unit group D⁎=D∖{0} is a product at most d commutators, which answers a problem of P. Draxl ([5], Problem 1, Page 102) for tame and totally ramifield division algebras whose centers are Henselian. The second goal is to prove that if D is infinite and every element in D′ is a product at most α commutators in D⁎, then every matrix in the special linear group SLn(D) of degree n>1 is a product of at most 2+6α commutators of involutions.