On Bernstein algebras satisfying chain conditions II

Abstract

Following a previous work with Boudi, we continue to investigate Bernstein algebras satisfying chain conditions. First, it is shown that a Bernstein algebra $(A, ω)$ with ascending or descending chain condition on subalgebras is finite-dimensional. We also prove that $A$ is Nœtherian (Artinian) if and only if its barideal $N=\ker(ω)$ is. Next, as a generalization of Jordan and nuclear Bernstein algebras, we study whether a Nœtherian (Artinian) Bernstein algebra $A$ with a locally nilpotent barideal $N$ is finite-dimensional. The response is affirmative in the Nœtherian case, unlike in the Artinian case. This question is closely related to a result by Zhevlakov on general locally nilpotent nonassociative algebras that are Nœtherian, for which we give a new proof. In particular, we derive that a commutative nilalgebra of nilindex 3 which is Nœtherian or Artinian is finite-dimensional. Finally, we improve and extend some results of Micali and Ouattara to the Nœtherian and Artinian cases.