Annihilators of power values of generalized skew derivations on lie ideals

Annihilator conditions with generalized skew derivations and Lie ideals of prime rings

“…Throughout this paper R always denotes a prime ring, Z(R) the center of R, Q r the right Martindale quotient ring of R and C = Z(Q r ), the center of Q r (C is usually called the extended centroid of R). We introduce on R an additive mapping d which satisfies the following rule: In [2], J.-C. Chang shows that if F is a generalized skew derivation of R, L is a non-commutative Lie ideal of R and n ≥ 1 a fixed integer such that F (x) n = 0, for all x ∈ L, then F (x) = 0, for all x ∈ R. Later, in [20], a generalization of the previous cited result involving an annihilator condition is given. More precisely, the main result in [20] proves that if F is a generalized skew derivation of R, L is a non-commutative Lie ideal of R, n ≥ 1 a fixed integer and a ∈ R is a fixed element such that aF (x) n = 0, for all x ∈ L, then aF (x) = 0, for all x ∈ R, unless R satisfies the standard identity s 4 .…”

“…proved that, if dim D V ≥ 3, both F and G are inner generalized derivations. The required conclusion then follows from Proposition 2.Proposition If R satisfies(20) then there exist a , c ∈ Q r such that F (x) = a x and G(x) = c x, for any x ∈ R, with pa = pc = 0, unless when R satisfies s 4 .…”