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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Nilpotency of derivations in prime rings


Author: David W. Jensen
Journal: Proc. Amer. Math. Soc. 123 (1995), 2633-2636
MSC: Primary 16W25; Secondary 16N60
DOI: https://doi.org/10.1090/S0002-9939-1995-1291775-9
MathSciNet review: 1291775
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Abstract: In 1957, E. C. Posner proved that if $ \lambda $ and $ \delta $ are derivations of a prime ring R, characteristic $ R \ne 2$, then $ \lambda \delta = 0$ implies either $ \lambda = 0$ or $ \delta = 0$. We extend this well-known result by showing that, without any characteristic restriction, $ \lambda {\delta ^m} = 0$ implies either $ \lambda = 0$ or $ {\delta ^{4m - 1}} = 0$. We also prove that $ {\lambda ^n}\delta = 0$ implies either $ {\delta ^2} = 0$ or $ {\lambda ^{12n - 9}} = 0$. In the case where $ {\lambda ^n}{\delta ^m} = 0$, we show that if $ \lambda $ and $ \delta $ commute, then at least one of the derivations must be nilpotent.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1291775-9
Keywords: Derivations, nilpotency, prime rings
Article copyright: © Copyright 1995 American Mathematical Society