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On derivations in prime rings and Banach algebras


Author: J. Vukman
Journal: Proc. Amer. Math. Soc. 116 (1992), 877-884
MSC: Primary 46H40; Secondary 16N60, 16W25, 46H99
DOI: https://doi.org/10.1090/S0002-9939-1992-1072093-8
MathSciNet review: 1072093
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Abstract: Let $ R$ be a ring with center $ Z(R)$. A mapping $ F:R \to R$ is said to be centralizing on $ R$ if $ [F(x),x] \in Z(R)$ holds for all $ x \in R$. The main purpose of this paper is to prove the following result, which generalizes a classical result of Posner: Let $ R$ be a prime ring of characteristic not 2, 3, and 5. Suppose there exists a nonzero derivation $ D:R \to R$ , such that the mapping $ x \rightarrowtail [[D(x),x],x]$ is centralizing on $ R$ . In this case $ R$ is commutative. Combining this result with some well-known deep results of Sinclair and Johnson, we generalize Yood's noncommutative extension of the Singer-Wermer theorem.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1072093-8
Keywords: Prime ring, derivation, Jordan derivation, inner derivation, commuting mapping, centralizing mapping, Banach algebra
Article copyright: © Copyright 1992 American Mathematical Society

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