Skip to main content
SpringerLink
Log in

Primary decomposition in incomplete Noetherian algebras

  • Published:
Bollettino dell'Unione Matematica Italiana Aims and scope Submit manuscript

Abstract

We show that in a noetherian commutative unital topological algebra, the prime ideals associated with a closed ideal as well as its isolated primary components are closed. We obtain a version of the closed graph theorem. An example of a noetherian (even principal) commutative unital semi-simple and incomplete normed algebra whose each ideal is closed is also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allan, G.R.: Elements of finite closed descent in a Banach algebra. J. Lond. Math. Soc. 7, 462–466 (1973). (2)

    MathSciNet  MATH  Google Scholar 

  2. Choukri, R., El Kinani, A., Oudadess, M.: Algèbres topologiques à idéaux à gauche fermés. Stud. Math. 168(2), 159–164 (2005)

    Article  Google Scholar 

  3. Ferreira A. V., Tomassini, G.: Finiteness properties of topological algebras, Annali Scuola Normale Superiore-Pisa, Classe di Scienze, Serie IV-Vol. V, \(\text{n}^{\circ } \)3 (1978)

  4. Lafon, J. P.: Algèbre commutative, Paris, 1977

  5. Nanzetta, P., Plank, D.: Closed ideals in C(X). Proc. Am. Math. Soc. 35, 601–606 (1972)

    MathSciNet  MATH  Google Scholar 

  6. Sinclair, A.M., Tullo, A.W.: Noetherian Banach algebras are finite dimensional. Math. Ann. 211, 151–153 (1974)

    Article  MathSciNet  Google Scholar 

  7. Zelazko, W.: A characterization of commutative Fréchet algebras with all ideals closed. Stud. Math. 138(3), 293–300 (2000)

    Article  Google Scholar 

Download references

Acknowledgements

I would like to thank the referee for his/her constructive remarks and his/her valuable suggestions.

Author information

Authors and Affiliations

  1. Université Mohammed V de Rabat ,E.N.S de Rabat ,B. P. 5118, 10105, Rabat, Morocco

    Sara El Kinani

Authors
  1. Sara El Kinani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sara El Kinani.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El Kinani, S. Primary decomposition in incomplete Noetherian algebras. Boll Unione Mat Ital 14, 441–446 (2021). https://doi.org/10.1007/s40574-020-00274-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40574-020-00274-1

Keywords

Mathematics Subject Classification

Search

Navigation