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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Wedderburn theorem for alternative algebras with identity over commutative rings


Author: W. C. Brown
Journal: Trans. Amer. Math. Soc. 182 (1973), 145-158
MSC: Primary 17D05
MathSciNet review: 0325722
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Abstract: In this paper, we study alternative algebras $ {\mathbf{\Lambda }}$ over a commutative, associative ring R with identity. When $ {\mathbf{\Lambda }}$ is finitely generated as an R-module, we define the radical J of $ {\mathbf{\Lambda }}$. We show that matrix units and split Cayley algebras can be lifted from $ {\mathbf{\Lambda }}/J$ to $ {\mathbf{\Lambda }}$ when R is a Hensel ring. We also prove the following Wedderburn theorem: Let $ {\mathbf{\Lambda }}$ be an alternative algebra over a complete local ring R of equal characteristic. Suppose $ {\mathbf{\Lambda }}$ is finitely generated as an R-module, and $ {\mathbf{\Lambda }}/J$ is separable over $ \bar R$ ($ \bar R$ the residue class field of R). Then there exists an $ \bar R$-subalgebra S of $ {\mathbf{\Lambda }}$ such that $ S + J = {\mathbf{\Lambda }}$ and $ S \cap J = 0$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0325722-0
PII: S 0002-9947(1973)0325722-0
Keywords: Alternative algebra, Hensel ring, complete local ring
Article copyright: © Copyright 1973 American Mathematical Society