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.