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Cocentralizing and vanishing derivations on multilinear polynomials in prime rings

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Abstract

Let R be a prime ring of characteristic different from 2 and extended centroid C and let f(x1,..., x n ) be a multilinear polynomial over C not central-valued on R, while δ is a nonzero derivation of R. Suppose that d and g are derivations of R such that

$$ \delta (d(f(r_1 , \ldots ,r_n ))f(r_1 , \ldots ,r_n ) - f(r_1 , \ldots ,r_n )g(f(r_1 , \ldots ,r_n ))) = 0 $$

for all r1,..., r n ∈ R. Then d and g are both inner derivations on R and one of the following holds: (1) d = g = 0; (2) d = −g and f(x 1,..., x n )2 is central-valued on R.

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Authors and Affiliations

  1. University of Messina, Messina, Italy

    Vincenzo De Filippis

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  1. Vincenzo De Filippis
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Correspondence to Vincenzo De Filippis.

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Original Russian Text Copyright © 2009 De Filippis V.

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 4, pp. 806–817, July–August, 2009.

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De Filippis, V. Cocentralizing and vanishing derivations on multilinear polynomials in prime rings. Sib Math J 50, 637–646 (2009). https://doi.org/10.1007/s11202-009-0071-y

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