Identities with Engel conditions on derivations

摘要

Let R be a semiprime ring with a derivation D. The focus is on the two identities with Engel condition on ${D: [x^m, D(x^{n_1}),ldots,D(x^{n_s})]_s=0}$ for all ${xin R}$ and ${[x^m, D(x)^{n_1},ldots,D(x)^{n_s}]_s=0}$ for all ${xin R}$ , where s, m, n 1, . . . , n s are fixed positive integers. Our results are natural generalizations of Posner’s theorem on centralizing derivations, Herstein’s theorem on derivations with power-central values and a recent result by A. Fošner, M. Fošner and Vukman.