Some notes on Lie ideals in division rings

A Lie ideal of a division ring A is an additive subgroup L of A such that the Lie product [l, a] = la - al of any two elements l is an element of L, a is an element of A is in L or [l, a] is an element of L. The main concern of this paper is to present some properties of Lie ideals of A which may be interpreted as being dual to known properties of normal subgroups of A*. In particular, we prove that if A is a finite-dimensional division algebra with center F and charF not equal 2, then any finitely generated Z-module Lie ideal of A is central. We also show that the additive commutator subgroup [A, A] of A is not a finitely generated Z-module. Some other results about maximal additive subgroups of A and M-n(A) are also presented.