Skip to main content
SpringerLink
Log in

Biseparable extensions are not necessarily Frobenius

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We give necessary and sufficient conditions on an Ore extension \(A[x;\sigma ,\delta ]\), where A is a finite dimensional algebra over a field \({\mathbb {F}}\), for being a Frobenius extension of the ring of commutative polynomials \({\mathbb {F}}[x]\). As a consequence, as the title of this paper highlights, we provide a negative answer to a problem stated by Caenepeel and Kadison.

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. Caenepeel, S., Kadison, L.: Are biseparable extensions Frobenius? K-Theory 24(4), 361–383 (2001)

    Article  MathSciNet  Google Scholar 

  2. Eilenberg, S., Nakayama, T.: On the dimensions of modules and algebras, II. (Frobenius algebras and quasi-Frobenius rings). Nagoya Math. J. 9, 1–16 (1955)

    Article  MathSciNet  Google Scholar 

  3. Endo, S., Watanabe, Y.: On separable algebras over a commutative ring. Osaka J. Math. 4, 233–242 (1967)

    MathSciNet  MATH  Google Scholar 

  4. Gómez-Torrecillas, J., Lobillo, F.J., Navarro, G.: Computing separability elements for the sentence-ambient algebra of split ideal codes. J. Symb. Comput. 83, 211–227 (2017)

    Article  MathSciNet  Google Scholar 

  5. Gómez-Torrecillas, J., Lobillo, F.J., Navarro, G.: Ideal codes over separable ring extensions. IEEE Trans. Inf. Theory 63(5), 2702–2706 (2017)

    Article  MathSciNet  Google Scholar 

  6. Goodearl, K.R., Warfield Jr., R.B.: An Introduction to Noncommutative Noetherian Rings. London Mathematical Society Student Texts, vol. 61, 2nd edn. Cambridge University Press, Cambridge (2004)

  7. Hirata, K., Sugano, K.: On semisimple extensions and separable extensions over non commutative rings. J. Math. Soc. Jpn. 18(4), 360–373 (1966)

    Article  MathSciNet  Google Scholar 

  8. Kadison, L.: The Jones polynomial and certain separable Frobenius extensions. J. Algebra 186, 461–475 (1996)

    Article  MathSciNet  Google Scholar 

  9. Kadison, L.: New Examples of Frobenius Extensions. University Lecture Series 14. American Mathematical Society, Providence (1999)

  10. Kadison, L.: Separable equivalence of rings and symmetric algebras. Bull. Lond. Math. Soc. 51, 344–352 (2019)

    Article  MathSciNet  Google Scholar 

  11. Kasch, F.: Grundlagen einer theorie der Frobeniuserweiterungen. Math. Ann. 127, 453–474 (1954)

    Article  MathSciNet  Google Scholar 

  12. Kasch, F.: Projektive Frobenius–Erweiterungen, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, vol. 1960/1961/4. Springer, Berlin (1961)

  13. Lam, T.Y.: Lectures on Modules and Rings. Graduate Texts in Mathematics, vol. 189. Springer, Berlin (1999)

  14. Lam, T.Y., Leroy, A.: Vandermonde and Wronskian matrices over division rings. J. Algebra 119(2), 308–336 (1988)

    Article  MathSciNet  Google Scholar 

  15. Nakayama, T., Tsuzuku, T.: A remark on Frobenius extensions and endomorphism rings. Nagoya Math. J. 15, 9–16 (1959)

    Article  MathSciNet  Google Scholar 

  16. Nakayama, T., Tsuzuku, T.: On Frobenius extensions. I. Nagoya Math. J. 17, 89–110 (1960)

    Article  MathSciNet  Google Scholar 

  17. Ore, O.: Theory of non-commutative polynomials. Ann. Math. 34(3), 480–508 (1933)

    Article  MathSciNet  Google Scholar 

  18. Stenström, B.: Ring of Quotients, Grundlehren der mathematischen Wissenschaften, vol. 217. Springer, Berlin (1975)

    Google Scholar 

  19. Sugano, K.: Separable extensions and Frobenius extensions. Osaka J. Math. 7, 291–299 (1970)

    MathSciNet  MATH  Google Scholar 

  20. Sugano, K.: Note on separability of endomorphism rings. J. Fac. Sci. Hokkaido Univ. Ser. I Math. 21(3–4), 196–208 (1971)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Algebra, University of Granada, Granada, Spain

    José Gómez-Torrecillas & José Patricio Sánchez-Hernández

  2. Department of Algebra and CITIC, University of Granada, Granada, Spain

    F. J. Lobillo

  3. Department of Computer Sciences and AI, and CITIC, University of Granada, Granada, Spain

    Gabriel Navarro

Authors
  1. José Gómez-Torrecillas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. J. Lobillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gabriel Navarro
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. José Patricio Sánchez-Hernández
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. J. Lobillo.

Additional information

Publisher's Note

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

Research supported by the grants MTM2016-78364-P and PID2019-110525GB-I00 from Agencia Estatal de Investigación and Fondo Europeo de Desarrollo Regional (AEI/FEDER, UE). The fourth author was supported by The National Council of Science and Technology (CONACYT) of Mexico with a scholarship for a Postdoctoral Stay in the University of Granada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gómez-Torrecillas, J., Lobillo, F.J., Navarro, G. et al. Biseparable extensions are not necessarily Frobenius. Math. Z. 297, 517–533 (2021). https://doi.org/10.1007/s00209-020-02523-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-020-02523-7

Keywords

Mathematics Subject Classification

Search

Navigation