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A Theorem on Derivations of Prime Rings with Involution

  • I. N. Herstein (a1)

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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, yR 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, yR 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.

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References

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1. Bergen, J., Herstein, I. N. and Kerr, J. W., Lie ideals and derivations of prime rings, (to appear).
2. Herstein, I. N., A note on derivations, Canadian Math. Bull. 21 (1978), 369370.
3. Herstein, I. N., Topics in ring theory (Univ. of Chicago Press, Chicago, 1969).
4. Herstein, I. N., Rings with involution (Univ. of Chicago Press, Chicago, 1976).
5. Herstein, I. N., A note on derivations II, Canadian Math. Bull. 22 (1979), 509511.
6. Lin, J. S., On derivations of prime rings with involution, Ph.D. thesis, Univ. of Chicago (1981).
7. Miers, R. and Martindale, W., On the iterates of derivations of prime rings, (to appear).
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A Theorem on Derivations of Prime Rings with Involution

  • I. N. Herstein (a1)

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