Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The Wedderburn principal theorem for a generalization of alternative algebras


Author: Harry F. Smith
Journal: Trans. Amer. Math. Soc. 198 (1974), 139-154
MSC: Primary 17A30
MathSciNet review: 0352187
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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$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0352187-6
Keywords: Power-associative, noncommutative Jordan, generalized alternative rings I and II, alternative ring, finite-dimensional, nil, solvable, nilpotent, simple, degree one, algebraically closed field, nodal algebra, Albert decomposition, Peirce decomposition, Penico solvable, Wedderburn principal theorem, nil radical, separable
Article copyright: © Copyright 1974 American Mathematical Society