## Nilpotency of derivations in prime rings

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- by David W. Jensen
- Proc. Amer. Math. Soc.
**123**(1995), 2633-2636 - DOI: https://doi.org/10.1090/S0002-9939-1995-1291775-9
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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.

## References

- L. O. Chung and Jiang Luh,
*Nilpotency of derivations*, Canad. Math. Bull.**26**(1983), no. 3, 341–346. MR**703409**, DOI 10.4153/CMB-1983-057-5 - Irving Kaplansky,
*Lie algebras and locally compact groups*, University of Chicago Press, Chicago, Ill.-London, 1971. MR**0276398** - Edward C. Posner,
*Derivations in prime rings*, Proc. Amer. Math. Soc.**8**(1957), 1093–1100. MR**95863**, DOI 10.1090/S0002-9939-1957-0095863-0

## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- 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