About nilalgebras satisfying (xy)² = x²y²

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 ${(xy)}^2=x^2{y}^2$ 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 $\le 3.$

Keywords:

• Albert’s problem
• commutative algebras
• finite-dimensional algebras
• PI-algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A05
• 17A30

Additional information

Funding

The work was partially supported by the project 2018/23690-6, São Paulo Research Foundation (FAPESP).