Transactions of the American Mathematical Society

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



Generalization of right alternative rings

Authors: Irvin Roy Hentzel and Giulia Maria Piacentini Cattaneo
Journal: Trans. Amer. Math. Soc. 207 (1975), 143-161
MSC: Primary 17A30
MathSciNet review: 0369451
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Abstract: We study nonassociative rings $ R$ satisfying the conditions (1) $ (ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b$ for all $ a,b,c,d \in R$, and (2) $ (x,x,x) = 0$ for all $ x \in R$. We furthermore assume weakly characteristic not 2 and weakly characteristic not 3. As both (1) and (2) are consequences of the right alternative law, our rings are generalizations of right alternative rings. We show that rings satisfying (1) and (2) which are simple and have an idempotent $ \ne 0, \ne 1$, are right alternative rings.

We show by example that $ (x,e,e)$ may be nonzero. In general, $ R = R' + (R,e,e)$ (additive direct sum) where $ R'$ is a subring and $ (R,e,e)$ is a nilpotent ideal which commutes elementwise with $ R$.

We examine $ R'$ under the added assumption of Lie admissibility: (3) $ (a,b,c) + (b,c,a) + (c,a,b) = 0$ for all $ a,b,c \in R$. We generate the Peirce decomposition. If $ R'$ has no trivial ideals contained in its center, the table for the multiplication of the summands is associative, and the nucleus of $ R'$ contains $ {R'_{10}} + {R'_{01}}$. Without the assumption on ideals, the table for the multiplication need not be associative; however, if the multiplication is defined in the most obvious way to force an associative table, the new ring will still satisfy (1), (2), (3).

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Keywords: Right alternative, Peirce decomposition
Article copyright: © Copyright 1975 American Mathematical Society