Lie and Jordan ideals in prime rings with derivations

Ahmad, Mansoor

doi:10.1090/S0002-9939-1976-0399181-4

Lie and Jordan ideals in prime rings with derivations
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Proc. Amer. Math. Soc. 55 (1976), 275-278 Request permission

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 $u \in U$.

References

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
I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0271135

Additional Information

Journal: Proc. Amer. Math. Soc. 55 (1976), 275-278
DOI: https://doi.org/10.1090/S0002-9939-1976-0399181-4
MathSciNet review: 0399181