# On Bernstein algebras satisfying chain conditions II

## TL;DR

This paper investigates Bernstein algebras satisfying chain conditions, establishing their finite-dimensionality under certain conditions and exploring properties related to their barideals, with implications for related algebraic structures.

## Contribution

It proves finite-dimensionality of Bernstein algebras with chain conditions and explores their properties when their barideals are locally nilpotent, extending previous results in the field.

## Key findings

- Bernstein algebras with chain conditions are finite-dimensional.
- N ext{"o}therian Bernstein algebras have N ext{"o}}therian barideals.
- Commutative nilalgebras of nilindex 3 that are N ext{"o}therian or Artinian are finite-dimensional.

## Abstract

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