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COMMUTATIVE ALGEBRAS SATISFYING IDENTITY $x^3y=0$
Keywords:
commutative, Jordan, power-associative, nilalgebra, nilpotent, solvableDOI:
https://doi.org/10.17654/0972555525003Abstract
It is known that commutative power-associative nilalgebras of nilindex 4 are not necessarily nilpotent. This was proved by Suttles' counterexample to a conjecture of Albert. This article is about commutative non-associative algebras of characteristic $\neq 2,3$ which satisfy the identity $x^3 y=0$. These algebras are nilpotent if they are finite dimensional. For dimension 3 or 4 , commutative nilalgebras of index 4 are such algebras. For dimension $\leq 5$, power-associativity implies that they are Jordan algebras. For dimension 6, if they are power-associative but not Jordan algebras, then they are nilpotent of index 5 and solvable of index 3.
Received: August 3, 2024
Accepted: November 23, 2024
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