SOME STUDIES ON LIE IDEALS

In this note we study the structure of Lie ideals in associative algebras. In particular we prove that given any finite-dimensional division algebra A with the center F such that charF 6= 2, any finitely generated Z¡module Lie ideal of A is central. As a consequence we also prove that if A is a division algebra of finite dimension over its center F and charF 6= 2 then the additive commutator subgroup of A or [A;A] is not finitely generated Z¡module.