Conditions on Lie ideals in rings

Abstract

We show that if [Formula: see text] is a non-central Lie ideal of a ring [Formula: see text] with Char[Formula: see text], such that all of its nonzero elements are invertible, then [Formula: see text] is a division ring. We prove that if [Formula: see text] is an [Formula: see text]-central algebra and [Formula: see text] is a Lie ideal without zero divisor such that the set of multiplicative cosets [Formula: see text] is of finite cardinality, then either [Formula: see text] is a field or [Formula: see text] is central. We show the only non-central Lie ideal without zero divisor of a non-commutative central [Formula: see text]-algebra [Formula: see text] with Char[Formula: see text] and radical over the center is [Formula: see text], the additive commutator subgroup of [Formula: see text] and in this case [Formula: see text] is a generalized quaternion algebra. Finally we prove that if [Formula: see text] is a Lie ideal without zero divisor in a central [Formula: see text]-algebra with characteristic not 2 and if [Formula: see text] is a finite residual group, then [Formula: see text] is central.