conditions on lie ideals in rings

We show that if i is a non-central lie ideal of a ring r with char(r) 6= 2, such that all of its non-zero elements are invertible, then r is a division ring. we prove that if r is an f-central algebra and i is a lie ideal without zero divisor such that the set of multiplicative cosets faf j a 2 ig is of finite cardinality, then either r is a field or i is central. we show the only non-central lie ideal without zero divisor of a non-commutative central f-algebra r with char(r) 6= 2 and radical over the center is [r; r], the additive commutator subgroup of r.