Generalized Lie ideals in ★-prime rings

Abstract

Let R be a ★-prime ring with skew elements K, extended centroid C, and central closure RC. For U, W ⊂- R we define U n ( W) inductively: U (1)( W) = [ U, W], U ( n + 1) ( W) = [ U, U ( n) ( W)]. An additive subgroup V of K is called a generalized Lie ideal (GLI) of K of index ⩽ n if V ( n) ( K) ⊆ V. The notion of a GLI includes that of a Lie ideal of K, a Lie inner ideal of K, and an additive subgroup of K which generates a nilpotent subring of R. Theorem. If char R = 0 and V ⊆ K such that V ( n) ( K) = 0, then V ⊆ C + B, where B is a nilpotent subring of RC. Theorem. Suppose ★ is an involution of the first kind, V is a GLI of K of index ⩽n, and T = {t ϵ K¦[V,[t, K]] ⊆ V}. If char R ≠ 2 and [T, T] ≠ 0 then [I ∩ K, K] ⊆ V for some nonzero ★-ideal I of R. If char R = 0 and [ T, T] = 0 then V 6 n − 7 = 0. A similar result is obtained when ★ is of the second kind.