Hypercentralizing generalized skew derivations on left ideals in prime rings

Let R be a prime ring of characteristic different from 2, Qr the right Martindale quotient ring of R, C the extended centroid of R, I a nonzero left ideal of R, F a nonzero generalized skew derivation of R with associated automorphism α, and n, k ≥ 1 be fixed integers. If [F(rn),rn]k = 0 for all r ∈ I, then there exists λ ∈ C, such that F(x) = λx, for all x ∈ I. More precisely one of the following holds: (1) α is an X-inner automorphism of R and there exist b, c ∈ Qr and q invertible element of Qr, such that F(x) = bx - qxq-1c, for all x ∈Qr. Moreover there exists γ ∈ C such that I(q-1c-γ) = (0) and b - γq ∈ C; (2) α is an X-outer automorphism of R and there exist c ∈ Qr, λ ∈ C, such that F(x) = λx - α(x)c, for all x ∈ Qr, with α(I)c = 0. © 2013 Springer-Verlag Wien.