## The Wedderburn principal theorem for a generalization of alternative algebras

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- by Harry F. Smith
- Trans. Amer. Math. Soc.
**198**(1974), 139-154 - DOI: https://doi.org/10.1090/S0002-9947-1974-0352187-6
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## Abstract:

A generalized alternative ring I is a nonassociative ring $R$ in which the identities $(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z$; and $(x,x,x)$ are identically zero. It is here demonstrated that if $A$ is a finite-dimensional algebra of this type over a field $F$ of characteristic # 2, 3, then $A$ a nilalgebra implies $A$ is nilpotent. A generalized alternative ring II is a nonassociative ring $R$ in which the identities $(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x$ and $(x,y,x)$ are identically zero. Let $A$ be a finite-dimensional algebra of this type over a field $F$ of characteristic # 2. Then it is here established that (1) $A$ a nilalgebra implies $A$ is nilpotent; (2) $A$ simple with no nonzero idempotent other than 1 and $F$ algebraically closed imply $A$ itself is a field; and (3) the standard Wedderburn principal theorem is valid for $A$.## References

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## Bibliographic Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**198**(1974), 139-154 - MSC: Primary 17A30
- DOI: https://doi.org/10.1090/S0002-9947-1974-0352187-6
- MathSciNet review: 0352187