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On certain types of derivations

Published online by Cambridge University Press:  24 October 2008

E. M. Patterson
E. M. Patterson
Affiliation:
Department of MathematicsThe UniversityLeeds 2
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Extract

Let R be a ring, not necessarily associative. A derivation of R is a mapping D:R → R, not identically zero, such that

for all x, y ε R. If R is a linear algebra over a field F, then D also satisfies

for all x ε R and α ε F.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 3 , July 1958 , pp. 338 - 345
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

* We note that the distributive laws in R are not used in the proof of this theorem.

* I have obtained certain theorems concerning automorphisms analogous to those given in §§ 4 and 5 for derivations. For example, corresponding to Theorem 2, if an automorphism T of R is of degree 2 and leaves no non-zero element fixed then either R satisfies (xy)z = 0 = x(yz) or T 3 = E.