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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Alternator and associator ideal algebras
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by Irvin Roy Hentzel, Giulia Maria Piacentini Cattaneo and Denis Floyd
Trans. Amer. Math. Soc. 229 (1977), 87-109
DOI: https://doi.org/10.1090/S0002-9947-1977-0447361-7

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.
References
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Bibliographic Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 229 (1977), 87-109
  • MSC: Primary 17E05
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0447361-7
  • MathSciNet review: 0447361