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Some Division Algebras

Published online by Cambridge University Press:  20 November 2018

Joseph L. Zemmer*
Affiliation:
University of Missouri
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Let K* be an associative algebra over a field F with identity u, and let u, e1, e2, … , be a basis for K*. Denote by K the linear space, over F, spanned by the ei,i = 1, 2, … . Then for x, y in K, xy = αu + a, where a ∈ K. Define h(x, y) = α and x.y = a. With respect to the operation thus defined, K becomes an algebra over F satisfying

1

Further, the bilinear form h(x, y) is associative on K. Any algebra, over a field F, which possesses an associative bilinear form h(x, y) and satisfies (1) will be called a algebra. It is not difficult to show that any algebra K can be obtained from a unique associative algebra K* with identity by the process described above. The algebra K* will be called the associated associative algebra of K.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Albert, A. A., On nonassociative division algebras, Trans. Amer. Math. Soc, 72 (1952), 296309.Google Scholar
2. Bruck, R. H., Some results in the theory of linear nonassociative algebras, Trans. Amer. Math. Soc, 56 (1944), 141199.Google Scholar
3. Dickson, L. E., Linear algebras in which division is always uniquely possible, Trans. Amer. Math. Soc, 7 (1906), 370390.Google Scholar