We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and δ a nonzero σ-derivation of A, where σ is an epimorphism of A. For x, y∈ A, we set [x, y] = xy- yx. If [[…[[δ(xn0),xn1],xn2],…],xnk]=0 for all x∈ R, where n0, n1, … , nk are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) C≅ GF(2) , the Galois field of two elements; (3) there exist b∈ Q and λ∈ C such that δ(x) = σ(x) b- bx for all x∈ A, (b- λ) R= 0 and σ(R) = 0. The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).
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