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\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document} a nonzero σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-derivation of A, where σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} is an epimorphism of A. For x,y∈A\documentclass[12pt]{minimal}
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\begin{document}$$x,y\in A$$\end{document}, we set [x,y]=xy-yx\documentclass[12pt]{minimal}
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\begin{document}$$[x,y] = xy - yx$$\end{document}. If [[…[[δ(xn0),xn1],xn2],…],xnk]=0\documentclass[12pt]{minimal}
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\begin{document}$$[[\ldots [[\delta (x^{n_0}),x^{n_1}],x^{n_{2}}],\ldots ],x^{n_k}]=0$$\end{document} for all x∈R\documentclass[12pt]{minimal}
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\begin{document}$$x\in R$$\end{document}, where n0,n1,…,nk\documentclass[12pt]{minimal}
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\begin{document}$$n_{0},n_{1},\ldots ,n_{k}$$\end{document} are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) C≅GF(2)\documentclass[12pt]{minimal}
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\begin{document}$$C\cong GF(2)$$\end{document}, the Galois field of two elements; (3) there exist b∈Q\documentclass[12pt]{minimal}
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\begin{document}$$b\in Q$$\end{document} and λ∈C\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in C$$\end{document} such that δ(x)=σ(x)b-bx\documentclass[12pt]{minimal}
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\begin{document}$$\delta (x)=\sigma (x)b-bx$$\end{document} for all x∈A\documentclass[12pt]{minimal}
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\begin{document}$$x\in A$$\end{document}, (b-λ)R=0\documentclass[12pt]{minimal}
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\begin{document}$$(b-\lambda )R=0$$\end{document} and σ(R)=0\documentclass[12pt]{minimal}
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\begin{document}$$\sigma (R)=0$$\end{document}. 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).
