Generalized derivations acting on multilinear polynomials in prime rings

Abstract

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F , G and H be three generalized derivations of R, I an ideal of R and f (x 1 , . . ., xn) a multilinear polynomial over C which is not central valued on R. Iffor all r = (r 1 , . . ., rn) ∈ I n , then one of the following conditions holds:(1) there exist a ∈ C and b ∈ U such that F (x) = ax, G(x) = xb and H(x) = xab for all x ∈ R; (2) there exist a, b ∈ U such that F (x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C; (3) there exist b ∈ C and a ∈ U such that F (x) = ax, G(x) = bx and H(x) = abx for all x ∈ R; (4) f (x 1 , . . ., xn) 2 is central valued on R and one of the following conditions holds: (a) there exist a, b, p, p ′ ∈ U such that F (x) = ax, G(x) = xb and H(x) = px + xp ′ for all x ∈ R, with ab = p + p ′ ; (b) there exist a, b, p, p ′ ∈ U such that F (x) = xa, G(x) = bx and H(x) = px + xp ′ for all x ∈ R, with p + p ′ = ab ∈ C.