Skip to main content
SpringerLink
Log in

On the separability of the partial skew groupoid ring

  • Published:
São Paulo Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Given a partial (resp. a global) action \(\alpha \) of a connected finite groupoid \(\mathscr {G}\) on a ring \(\mathcal {A},\) we determine necessary and sufficient conditions for the partial (resp. the global) skew groupoid ring \(\mathcal {A}\star _\alpha \mathscr {G}\) to be a separable extension of \(\mathcal {A}\).

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. Alfaro, R., Szeto, G.: The Centralizer on H-Separable Skew Group Rings. Rings, extension and cohomology, vol. 159. Marcel Dekker, New York (1994)

    MATH  Google Scholar 

  2. Alfaro, R., Szeto, G.: Skew group rings which are Azumaya. Commun. Alg. 23, 2255–2261 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bagio, D., Flôres, D., Paques, A.: Partial actions of ordered groupoids on rings. J. Algebra Appl. 9, 501–517 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bagio, D., Lazzarin, J.R., Paques, A.: Crossed products by twisted partial actions: separability, semisimplicity and Frobenius properties. Commun. Alg. 38, 496–508 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bagio, D., Paques, A.: Partial Groupoid Actions: globalization, morita theory, and Galois theory. Commun. Alg. 40, 3658–3678 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Castaño, F., Gómez Torrecillas, J., Nǎstǎsescu, C.: Separable functors in graded rings. J. Pure Appl. Algebra 127, 219–230 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Caenepeel, S., Ion, B., Militaru, G.: The structure of Frobenius algebras and separable algebras. K-Theory 19, 365–402 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. DeMeyer, F.M., Ingraham, E.: Separable Algebras Over Commutative Rings, Springer Lecture Notes, 181 (1971)

  9. Dokuchaev, M., Ferrero, M., Paques, A.: Partial actions and Galois theory. J. Pure Appl. Algebra 208, 77–87 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Ikehata, S.: Note on Azumaya algebras and H-separable extensions. Math. J. Okayama Univ. 23, 17–18 (1981)

    MATH  MathSciNet  Google Scholar 

  12. Lundström, P.: Crossed product algebras defined by separable extensions. J. Algebra 283, 723–737 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lundström, P.: Separable groupoid rings. Commun. Algebra 34(9), 3029–3041 (2005)

    MATH  MathSciNet  Google Scholar 

  14. Nystedt, P., Öinert, J., Pinedo, H.: Epsilon-Strongly Graded Rings, Separability and Semisimplicity. arXiv:1606.07592

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Santa Maria, Santa Maria, RS, 97105-900, Brazil

    Dirceu Bagio

  2. Escuela de Matemáticas, Universidad Industrial de Santander, Cra. 27 Calle 9 UIS Edificio 45, Bucaramanga, Colombia

    Hector Pinedo

Authors
  1. Dirceu Bagio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hector Pinedo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dirceu Bagio.

Additional information

To Antonio Paques, on his 70th birthday.

D. Bagio was partially supported by CAPES. H. Pinedo was supported by FAPESP.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bagio, D., Pinedo, H. On the separability of the partial skew groupoid ring. São Paulo J. Math. Sci. 11, 370–384 (2017). https://doi.org/10.1007/s40863-017-0068-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40863-017-0068-6

Keywords

Mathematics Subject Classification

Search

Navigation