Generalized Derivations with Engel Conditions on Polynomials

Abstract

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X 1,…, X t ) a nonzero polynomial in noncommutative indeterminates X 1,…, X t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X 1,…, X t ) are characterized if the Engel identity is satisfied: [D(f(x 1,…, x t )), f(x 1,…, x t )] k = 0 for all x 1,…, x t ∈ R.