Vector Space Generated by the Multiplicative Commutators of a Division Ring

Let D be a division ring with center F. An element of the form xyx(−1)y(−1) ∈ D is called a multiplicative commutator. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In this paper it is shown that if D is algebraic over F and Char(D) = 0, then D = T(D). We conjecture that it is true in general. Among other results it is shown that in characteristic zero if T(D) is algebraic over F, then D is algebraic over F.