Nathan Jacobson Collected Mathematical Papers pp 173-178 | Cite as

# A Kronecker Factorization Theorem for Cayley Algebras and the Exceptional Simple Jordan Algebra

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## Abstract

A well known theorem, due to Wedderburn, states that if \( \mathfrak{B} \) is an arbitrary associative algebra with an identity 1 and \( \mathfrak{A} \) is a finite dimensional central simple subalgebra containing 1, then \( \mathfrak{B} \) is the Kronecker product \( \mathfrak{B} = \mathfrak{A} \otimes \mathfrak{U} \) where \( \mathfrak{U} \) is the subalgebra of elements which commute with every \( a\,\varepsilon \,\mathfrak{A} \). Recently Kaplansky indicated that the same result holds for \( \mathfrak{B} \) any alternative algebra and \( \mathfrak{A} \) an algebra of Cayley numbers.^{2} It is the purpose of the present note to give a new proof of this result and to show that an analogous result holds for \( \mathfrak{B} \) any Jordan algebra and \( \mathfrak{A} \) an exceptional simple Jordan algebra of finite dimensions.

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## References

- [1]A. A. Albert, “On simple alternative rings,”
*Canadian Journal of Mathematics*, vol. 2 (1952), pp. 129–135.CrossRefGoogle Scholar - [2]N. Jacobson, “Structure of alternative and Jordan bimodules,” forthcoming.Google Scholar
- [3]I. Kaplansky, “Semi-simple alternative rings,”
*Portugaliae Mathematica*, vol. 10 (1951), pp. 37–50.Google Scholar - [4]R. D. Schafer, “The exceptional simple Jordan algebras,”
*American Journal of Mathematics*, vol. 70 (1948), pp. 82–84.CrossRefGoogle Scholar - [5]T. Nakayama, “Wedderburn’s theorem, weakly normal rings,” Journal of the Mathematical Society of Japan, vol. 5 (1953), pp. 154–170.CrossRefGoogle Scholar