Geodesic
On $\theta$-centralizers of semiprime rings~(II)
Algebra i analiz, Tome 21 (2009) no. 1, pp. 61-73.

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The following result is proved: Let $R$ be a 2-torsion free semiprime ring, and let $T\colon R\to R$ be an additive mapping, related to a surjective homomorphism $\theta\colon R\to R$, such that $2T(x^2)=T(x)\theta(x)+\theta(x)T(x)$ for all $x\in R$. Then $T$ is both a left and a right $\theta$-centralizer.
Keywords: prime ring, semiprime ring, left(right) centralizer, left(right) $\theta$-centralizer, left(right) Jordan $\theta$-centralizer, derivation, Jordan derivation. 
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M. N. Daif; M. S. Tammam El-Sayiad. On $\theta$-centralizers of semiprime rings~(II). Algebra i analiz, Tome 21 (2009) no. 1, pp. 61-73. http://geodesic.mathdoc.fr/item/AA_2009_21_1_a1/

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