We show that if I is a non-central Lie ideal of a ring R with Char(R) not equal 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 vertical bar a is an element of 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) not equal 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.
展开▼
机译：我们证明，如果I是Char（R）不等于2的环R的非中心Lie理想，使得它的所有非零元素都是可逆的，则R是一个除法环。我们证明，如果R是F中心代数，而I是没有零除数的Lie理想，使得乘法陪集{aF竖线a是I的元素}的集合具有有限基数，则R是一个域或我是中心。我们显示了唯一的非中心Lie理想，即非交换性中心F代数R的零除数，且Char（R）不等于2且中心的根是[R，R]，R和R的加法交换子子群在这种情况下，R是广义四元数代数。最后我们证明，如果我是特征为2的中心F代数中没有零除数的Lie理想，并且如果（I + F / F，+）是有限的残基，那么我就是中心。
展开▼
