Abstract
Let R be a non-commutative prime of characteristic different from 2, C the extended centroid of R, \(f(x_1,\ldots ,x_n)\) a non-central polynomial over C and \(S=\{f(r_1,\ldots ,r_n) : r_1,\ldots ,r_n \in R\}\). If F and G are generalized derivations of R such that \(F(x)G(y)+G(y)F(x)=xy+yx\), for any \(x\in S\), then there exist \(\lambda , \mu \in C\) such that \(\lambda \mu =1\) and \(F(x)=\lambda x\), \(G(x)=\mu x\), for all \(x\in R\).
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Rania, F. Generalized derivations with skew-commutativity conditions on polynomials in prime rings. Beitr Algebra Geom 60, 513–525 (2019). https://doi.org/10.1007/s13366-018-0424-4
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DOI: https://doi.org/10.1007/s13366-018-0424-4