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 finitedimensional. We also prove that A is Noetherian (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 Noetherian (Artinian) Bernstein algebra A with a locally nilpotent barideal N is finite-dimensional. The response is affirmative in the Noetherian case, unlike in the Artinian case. This question is closely related to a result by Zhevlakov on general locally nilpotent nonassociative algebras that are Noetherian, for which we give a new proof. In particular, we derive that a commutative nilalgebra of nilindex 3 which is Noetherian or Artinian is finite-dimensional. Finally, we improve and extend some results of Micali and Ouattara to the Noetherian and Artinian cases.

Bernstein algebras that are algebraic and the Kurosh problem

“…Peresi [40] and Krapivin [30] have shown independently that the barideal of a finitely generated nuclear algebra is nilpotent and so finite-dimensional [43]. Later, Boudi and Zitan [12,53] undertook a systematic study of Bernstein algebras satisfying chain conditions on ideals.…”