Abstract
Let R be a prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C, and let \({f(x_1, \ldots, x_n)}\) be a multilinear polynomial over C, not central valued on R. Suppose that d is a derivation of R and G is a generalized derivation of R such that
for all \({r_1, \ldots, r_n \in R}\) . Then either d = 0 or G = 0, unless when d is an inner derivation of R, there exists \({\lambda \in C}\) such that G(x) = λ x, for all \({x \in R}\) and \({f(x_1, \ldots, x_n)^2}\) is central valued on R.
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Rania, F., Scudo, G. A Quadratic Differential Identity with Generalized Derivations on Multilinear Polynomials in Prime Rings. Mediterr. J. Math. 11, 273–285 (2014). https://doi.org/10.1007/s00009-013-0331-8
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DOI: https://doi.org/10.1007/s00009-013-0331-8