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Conditions on Lie ideals in rings

We show that if I is a non-central Lie ideal of a ring R with Char(R)≠2, such that all of its nonzero 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 {aF|a∈I} 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)≠2 and radical over the center is [R,R], the additive commutator subgroup of R and in this case R is a generalized quaternion algebra. Finally we prove that if I is a Lie ideal without zero divisor in a central F-algebra with characteristic not 2 and if (I+F F , +) is a finite residual group, then I is central.