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

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.