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Commutative Non-Associative Algebras and Identities of Degree Four

Published online by Cambridge University Press:  20 November 2018

J. Marshall Osborn
J. Marshall Osborn*
Affiliation:
University of Wisconsin, Madison, Wisconsin ; Yale University, New Haven, Connecticut
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The main result of this paper is the following.

Theorem 1. Let A be a simple, commutative, finite-dimensional algebra containing an idempotent over a field of characteristic 0, and let the algebra A' obtained from A by adjoining a unity element satisfy an identity of degree ≦ 4 not implied by commutativity. Then either A is a Jordan algebra or A is two-dimensional over an appropriate field E.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 769 - 794
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

The research for this paper was supported by the National Science Foundation, Grant GP-3993, and by the Research Committee of the Graduate School of the University of Wisconsin from funds supplied by the Wisconsin Alumni Research Foundation.

References

1. Osborn, J. M., Identities on non-associative algebras, Can. J. Math. 17 (1965), 78–92.Google Scholar
2. Osborn, J. M., Commutative algebras satisfying an identity of degree four, Proc. Amer. Math. Soc. 16 (1965), 1114–1120.Google Scholar
3. Petersson, Holger, Ûber eine Verallgemeinerung von Jordan-Algebren, Dissertation, Univ. of Munich, 1965.Google Scholar
4. Schafer, R. D., An introduction to non-associative algebras (Academic Press, New York, 1966).Google Scholar