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A Theorem on Derivations of Prime Rings with Involution

Published online by Cambridge University Press:  20 November 2018

I. N. Herstein
I. N. Herstein*
Affiliation:
University of Chicago, Chicago, Illinois
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In a recent note [2] we showed that if R is a prime ring and d ≠ 0 a derivation of R such that d(x)d(y) = d(y)d(x) for all x, y ∈ R then, if R is not a characteristic 2, R must be commutative. (If char R = 2 we showed that R must be an order in a 4-dimensional simple algebra.)

In this paper we shall consider a similar problem, namely, that of a prime ring R with involution * where d(x)d(y) = d(y)d(x) not for all x, y ∈ R but merely for symmetric elements x* = x and y* = y. Although it is clear that some results can be obtained if R is of characteristic 2, we shall only be concerned with the case char R ≠ 2. Even in this case one cannot hope to extend the result cited in the first paragraph, that is, to show that R is commutative.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982 , pp. 356 - 369
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Bergen, J., Herstein, I. N. and Kerr, J. W., Lie ideals and derivations of prime rings, (to appear).Google Scholar
2. Herstein, I. N., A note on derivations, Canadian Math. Bull. 21 (1978), 369–370.Google Scholar
3. Herstein, I. N., Topics in ring theory (Univ. of Chicago Press, Chicago, 1969).Google Scholar
4. Herstein, I. N., Rings with involution (Univ. of Chicago Press, Chicago, 1976).Google Scholar
5. Herstein, I. N., A note on derivations II, Canadian Math. Bull. 22 (1979), 509–511.Google Scholar
6. Lin, J. S., On derivations of prime rings with involution, Ph.D. thesis, Univ. of Chicago (1981).Google Scholar
7. Miers, R. and Martindale, W., On the iterates of derivations of prime rings, (to appear).Google Scholar