SOME STUDIES ON LIE IDEALS

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 $l in L$, $a in A$ is in $L$ or $[l,a] in L$. 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 fi_x000C_nite-dimensional division algebra with center $F$ and $char F eq 2$,then any _x000C_finitely generated $Bbb{Z}-$module Lie ideal of $A$ is central. We also show that the additive commutator subgroup $[A,A]$ of $A$ is not a finitely generated $Bbb{Z}-$module.