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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Scalar dependent algebras


Authors: Raymond Coughlin, Erwin Kleinfeld and Michael Rich
Journal: Proc. Amer. Math. Soc. 39 (1973), 69-72
MSC: Primary 17A30
DOI: https://doi.org/10.1090/S0002-9939-1973-0311728-X
MathSciNet review: 0311728
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Abstract: An algebra $ A$ over a field $ F$ will be called scalar dependent in case for every $ x,y,z$ in $ A$ there exists a function $ g(x,y,z)$ in $ F$ such that $ (xy)z = g(x,y,z)x(yz)$. The main result of this paper is that any scalar dependent algebra which contains a nonzero idempotent must always be associative. Since there are known to exist scalar dependent algebras which are not associative, the hypothesis regarding the existence of an idempotent is actually necessary.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0311728-X
Keywords: Associator, scalar dependent algebra
Article copyright: © Copyright 1973 American Mathematical Society