On Bernstein Algebras Satisfying Chain Conditions

Abstract

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 finitely generated. Especially, we prove that Noetherian Bernstein algebras are finitely generated.

Key Words:

• Artinian algebras
• Artinian modules
• Bernstein algebras
• Noetherian algebras

Mathematics Subject Classification:

• 17D92

ACKNOWLEDGMENTS

We thank Professor A. Facchini for helpful discussions concerning Theorem 3.14 and Remark 3.16. We also would like to thank the referee for some useful suggestions, especially for simplifying the proof of Theorem 2.2 and for bringing to our attention references (Shirshov, 1957; Zel'manov and Skosyrskii, 1983).

Nadia Boudi was partially Supported by the spanish Junta de Andalucia (Proyecto de Cooperacion Interuniversitaria con Marruecos titulado Estudio Analitico-Algebraico de Sistemas Triples y de Pares en diferentes clases de Estructuras no Asociativas).

Notes

Communicated by I. P. Shestakov.