Abstract
Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x 1, …, x n ) a noncentral multilinear polynomial over C. If δ(G(f(r 1, …, r n ))f(r 1, …, r n )) = 0 for all r 1, …, r n ∈ R, then f(x 1, …, x n )2 is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.
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Original Russian Text Copyright © 2012 Carini L. and De Filippis V.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 6, pp. 1310–1320, November–December, 2012.
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Carini, L., De Filippis, V. Centralizers of generalized derivations on multilinear polynomials in prime rings. Sib Math J 53, 1051–1060 (2012). https://doi.org/10.1134/S0037446612060092
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DOI: https://doi.org/10.1134/S0037446612060092