Nil Algebras Satisfying an Identity of Degree Three

Abstract

Let A be a nonassociative algebra over a field F with a function $g:A \times A \times A \to F$ such that $(xy)z = g(x,y,z)x(yz)$ for all x, y, and z in A. Algebras satisfying this identity have been studied by Michael Rich and the author. It is shown here that a finite-dimensional nil power-associative algebra satisfying the above identity is nilpotent.