A note on generalized skew-derivations on multilinear polynomials in prime rings

Abstract

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid, f (X1, . , Xn) a multilinear polynomial over C that is noncentral-valued on R and F a generalized skew derivation of R. If for some 0 ≠ a ∈ R − C,a[F (f (r)), f (r)] − [F (f (r)), f (r)]a ∈ Z(R)for all r = (r1, . , rn) ∈ Rn then one of the following conditions holds:(1) there exists λ ∈ C such that F (x) = λx for all x ∈ R;(2) there exist b ∈ Qr and λ ∈ C such that F (x) = bx + xb + λx for all x ∈ R and f (X1, . , Xn) is central valued on R.As an application of this result, we investigate the commutator [S, T ] ∈ Z(R), where S = {[F (u), u]|u ∈ f (R)} and T = {[G(v), v]|v ∈ f (R)}, F and G two generalized skew derivations of R.