On a theorem of Posner

Ahmad, Mansoor

doi:10.1090/S0002-9939-1977-0447348-X

On a theorem of Posner
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by Mansoor Ahmad

Proc. Amer. Math. Soc. 66 (1977), 13-16

DOI: https://doi.org/10.1090/S0002-9939-1977-0447348-X

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Abstract:

Posner [1] has proved the following theorem: Let R be a prime ring$^{1}$ and d a derivation of R such that, for all $a \in R,ada - daa \in Z$ (centre of R). Then, if d is not a zero derivation, R is commutative.

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
Ram Awtar, On a theorem of Posner, Proc. Cambridge Philos. Soc. 73 (1973), 25–27. MR 318236, DOI 10.1017/s0305004100047423
Ram Awtar, Lie and Jordan structure in prime rings with derivations, Proc. Amer. Math. Soc. 41 (1973), 67–74. MR 318233, DOI 10.1090/S0002-9939-1973-0318233-5
I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0271135