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Proceedings of the American Mathematical Society

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

 
 

 

Lie and Jordan ideals in prime rings with derivations


Author: Mansoor Ahmad
Journal: Proc. Amer. Math. Soc. 55 (1976), 275-278
DOI: https://doi.org/10.1090/S0002-9939-1976-0399181-4
MathSciNet review: 0399181
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Abstract: In this paper derivations on Lie and Jordan ideals of a prime ring $ R$ are studied. The following results are proved. (i) Let $ R$ be a prime ring of characteristic not $ 2$, and let $ U$ be a Lie or Jordan ideal of $ R$. If $ d$ is a derivation defined on $ U$, and if $ a$ is an element of the subring $ T(U)$, generated by $ U$, or $ a$ is an element of $ R$, according as $ U$ is a Lie or Jordan ideal of $ R$, such that $ adu = 0$, for all $ u \in U$, then either $ a = 0$ or $ du = 0$. (ii) Let $ {d_1},{d_2}$ be derivations defined for all $ u \in U$, and also for $ {u^2}$ and $ {u^3}$ if $ U$ is a Lie ideal of $ R$, such that the iterate $ {d_1}{d_2}$ is also a derivation, satisfying the same conditions as $ {d_1},{d_2}$. Let $ {d_1}(u) \in U$, whether $ U$ is a Lie or Jordan ideal of $ R$. Then, at least, one of $ {d_1}(u)$ and $ {d_2}(u)$ is zero, for all $ u \in U$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0399181-4
Keywords: Prime rings, Lie and Jordan ideals, derivation
Article copyright: © Copyright 1976 American Mathematical Society

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