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Simple Algebras of Type (1,1) are Associative

Published online by Cambridge University Press:  20 November 2018

Erwin Kleinfeld
Erwin Kleinfeld*
Affiliation:
Ohio State University
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In the classification of almost alternative algebras relative quasiequivalence by Albert two new classes of algebras of type (γ, δ) were introduced, namely those of type (1, 1) and ( —1, 0) (1, equations (34), (35), and Theorem 6). Since rings of type (1, 1) and ( — 1, 0) are anti-isomorphic it suffices to consider those of type (1, 1). They may be defined as rings satisfying

1

and

2

for all elements x, y, and z of the ring, where the associator (a, b, c) is given by (a, b, c) = (ab)c — a (bc).

Actually the identity

2'

together with (1) implies (2) whenever the characteristic of the ring is different from 2. This may readily be verified by linearizing (2’). Consequently we may use (1) and (2’) as the defining relations for a ring of type (1, 1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 129 - 148
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Albert, A. A., Almost alternative algebras, Portugal. Math., 8 (1949), 23–36.Google Scholar
2. Kleinfeld, E., Rings of (7, 5) type, Portugal. Math., 18 (1959), 107–110.Google Scholar
3. Kokoris, L. A., On a class of almost alternative algebras, Can. J. Math., 8 (1956), 250–255.Google Scholar
4. Kokoris, L. A., On rings of (T, 8) type, Proc. Amer. Math. Soc, 9 (1958), 897–904.Google Scholar
5. San Soucie, R. L., Right alternative division rings of characteristic 2, Proc. Amer. Math. Soc, 6 (1955), 291–296.Google Scholar