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On Simple Alternative Rings

Published online by Cambridge University Press:  20 November 2018

A. A. Albert
A. A. Albert*
Affiliation:
The University of Chicago
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The only known simple alternative rings which are not associative are the Cayley algebras. Every such algebra has a scalar extension which is isomorphic over its center F to the algebra where . The elements e11 and e00 are orthogonal idempotents an , for every xij of . The multiplication table of C is then completed by the relations

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 129 - 135
Copyright
Copyright © Canadian Mathematical Society 1952

References

1 The multiplication table of a Cayley algebra was given in this form by M. Zorn, Theorie de alternativen Ringe, Abh. Math. Sem. Hamburgischen Univ., vol. 8 (1930), 123-147.

2 The structure of alternative division rings, Proc. Amer. Math. Soc, vol. 2 (1951), 878-890.

3 This seems to be one of the few places in our development where an assumption about the characteristic would make any difference.

4 If C has characteristic not two or three the property zy = yz implies that (z, x, y) = 0. However our proof is so arranged that (z, x, y) = 0 is quite trivial.