About nilalgebras satisfying (xy)2 = x 2 y 2

Abstract

A classical problem in nonassociative algebras involves the existence of simple finite-dimensional commutative nilalgebras. In this paper, we study the class Ω of nonassociative algebras satisfying the identity over a field of characteristic different from 2 and 3. We show that every unitary algebra in Ω is associative. Next, we prove that each prime algebra in Ω is either associative or its center vanishes. For nilalgebras, we obtain that every nilalgebra in Ω is an Engel algebra. Finally, we show that every commutative nilalgebra in Ω of nilindex 4 over a field of characteristic not 2, 3 and 5 is solvable of index