On Bernstein algebras satisfying chain conditions

摘要

Our aim in this article is to study Noetherian and Artinian Bernstein algebras. We show that for Bernstein algebras which are either Jordan or nuclear, each of the Noetherian and Artinian conditions implies finite dimensionality. This result fails for general Noetherian or Artinian Bernstein algebras. We also investigate the relationships between the three finiteness conditions: Noetherian, Artinian, and. nitely generated. Especially, we prove that Noetherian Bernstein algebras are finitely generated.