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Transactions of the American Mathematical Society

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



Alternator and associator ideal algebras

Authors: Irvin Roy Hentzel, Giulia Maria Piacentini Cattaneo and Denis Floyd
Journal: Trans. Amer. Math. Soc. 229 (1977), 87-109
MSC: Primary 17E05
MathSciNet review: 0447361
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Abstract: If I is the ideal generated by all associators, $ (a,b,c) = (ab)c - a(bc)$, it is well known that in any nonassociative algebra $ R,I \subseteq (R,R,R) + R(R,R,R)$. We examine nonassociative algebras where $ I \subseteq (R,R,R)$. Such algebras include $ ( - 1,1)$ algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (a, a, b), (a, b, a), (b, a, a). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where $ I = (R,R,R)$ as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form $ [x,(x,x,a)] = \gamma (x,x,[x,a])$ for fixed $ \gamma $. With two exceptions, if this algebra has an idempotent e such that $ (e,e,R) = 0$, then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.

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Keywords: Associator ideal, right alternative, associator dependent
Article copyright: © Copyright 1977 American Mathematical Society