Idempotents and zero divisors in commutative algebras satisfying an identity of degree four

Year 2024, Volume: 36 Issue: 36, 194 - 205, 12.07.2024

Manuel Arenas Ivan Correa Irvin Roy Hentzel , Alicia Labra

https://doi.org/10.24330/ieja.1438748

https://izlik.org/JA47ZF66WF

Abstract

We study commutative algebras satisfying the identity
$ ((wx)y)z+((wy)z)x+((wz)x)y-((wy)x)z- ((wx)z)y-((wz)y)x = 0. $ We assume
characteristic of the field $\neq 2,3.$ We prove that given any $\lambda \in F,$ there exists a commutative algebra with idempotent $e,$ which satisfies the identity, and has $\lambda $ as an eigen value of the multiplication operator $L_e$. For algebras with idempotent, the containment relations for the product of the eigen spaces are not as precise as they are for Jordan or power-associative algebras. A great part of this paper is calculating these containment relations.

Keywords

Commutative algebra , degree four identity , idempotent element , Peirce decomposition

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