On left nilalgebras of left nilindex four satisfying an identity of degree four

Abstract

We extend the concept of commutative nilalgebras to commutative algebras which are not power associative. We shall study commutative algebras A over fields of characteristic ≠ 2, 3 which satisfy the identities x(x(xx)) = 0 and β{x(y(xx)) - x(x(xy))} + γ{y(x(xx)) - x(x(xy))} = 0. In these algebras the multiplication operator was shown to be nilpotent by Correa, Hentzel and Labra [2]. In this paper we prove that for every x ∈ A we have A(A((xx)(xx))) = 0. We prove that there is an ideal I of A satisfying AI = IA = 0 and A/I is power associative. © World Scientific Publishing Company.