Generalized derivations with Engel condition on multilinear polynomials

Abstract

Let R be a prime ring with extended centroid C, δ a nonzero generalized derivation of R, f(x 1, ..., x n ) a nonzero multilinear polynomial over C, I a nonzero right ideal of R and k ≥ a fixed integer. If [δ(f(r 1, ..., r n )), f(r 1, ..., r n )] k = 0, for all r 1, ..., r n ∈ I, then either δ(x) = ax, with (a-γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds (1) if char(R) = 0 then f(x 1, ..., x n ) is central valued in eRCe (2) if char(R) = p > 0 then $$ f(x_1 , \ldots ,x_n )^{p^s } $$ is central valued in eRCe, for a suitable s ≥ 0, unless when char(R) = 2 and eRCe satisfies the standard identity s 4 (3) δ(x) = ax−xb, where (a+b+α)e = 0, for α ∈ C, and f(x 1, ..., x n )2 is central valued in eRCe.