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On regularity and injectivity of the ring of real-continuous functions on a topoframe

Abstract

A frame is a complete lattice in which the meet distributes over arbitrary joins. Let \(\tau \) be a subframe of a frame L such that every element of \(\tau \) has a complement in L, then \((L, \tau )\), briefly \(L_{ \tau }\), is said to be a topoframe. Let \({\mathcal {R}}L_\tau \) be the ring of real-continuous functions on a topoframe \(L_{ \tau }\). We define P-topoframes and show that \(L_{\tau }\) is a P-topoframe if and only if \({\mathcal {R}}L_{\tau }\) is a regular ring if and only if it is a \(\aleph _0\)-self-injective ring. We define extremally disconnected topoframes and show that \(L_{\tau }\) is an extremally disconnected topoframe if and only if \(\tau \) is an extremally disconnected frame. For a completely regular topoframe \(L_\tau \), it is shown that \(L_\tau \) is an extremally disconnected topoframe if and only if \({\mathcal {R}}L_\tau \) is a Baer ring if and only if it is a CS-ring. Finally, we prove that a completely regular topoframe \(L_\tau \) is an extremally disconnected P-topoframe if and only if \({\mathcal {R}}L_\tau \) is a self-injective ring.

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References

  1. 1.

    Banaschewski, B.: The real numbers in pointfree topology. Textos de Mathemática (Séries B), No. 12. Departamento de Mathemática da Universidade de Coimbra, Coimbra (1997)

  2. 2.

    Berberian, S.: Baer *-rings. Springer, New York (1972)

    Book  Google Scholar 

  3. 3.

    Dube, T.: Concerning \(P\)-frames, essential \(P\)-frames, and strongly zero-dimensional frames. Algebra Universalis 61, 115–138 (2009)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Dube, T.: Notes on pointfree disconnectivity with a ring-theoretic slant. Appl. Categr. Struct. 18, 55–72 (2010)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Estaji, A.A., Karamzadeh, O.A.S.: On \(C (X)\) modulo its socle. Commun. Algebra 31, 1561–1571 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Estaji, A.A., Karimi Feizabadi, A., Zarghani, M.: Zero elements and \(z\)-ideals in modified pointfree topology. Bull. Iranian Math. Soc. 43, 2205–2226 (2017)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Estaji, A.A., Karimi Feizabadi, A., Zarghani, M.: The ring of real-continuous functions on a topoframe. Categr. Gen. Algebra Struct. Appl. 4, 75–94 (2016)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Estaji, A.A., Hashemi, E., Estaji, A.A.: On Property (A) and the socle of the \(f\)-ring \(Frm({\cal{P}}({\mathbb{R}}), L)\). Categr. Gen. Algebra Struct. Appl. 8, 61–80 (2018)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Gillman, L., Jerison, M.: Rings of Continuous Functions. Springer, Berlin (1976)

    MATH  Google Scholar 

  10. 10.

    Estaji, A.A., Abedi, M.: On injectivity of the ring of real-valued continuous functions on a frame. Bull. Belg. Math. Soc. Simon Stevin 25, 467–480 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Estaji, A.A., Abedi, M., Draghadam, A.M.: On self-injectivity of the \(f\)-ring \({{ Frm}}({\cal{P}}({\mathbb{R}}), L)\). Math. Slovaca 69, 999–1008 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Estaji, A.A., Taha, M.: \(z\)-ideals in the real continuous function ring \({\cal{R}}L_{\tau }\). Bull. Iranian Math. Soc. 46, 37–51 (2020)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Goodearl, K.R.: Von Neumann Regular Rings. Pitman, London (1979)

    MATH  Google Scholar 

  14. 14.

    Karamzadeh, O.A.S.: On a question of matlis. Commun. Algebra 25, 2717–2726 (1997)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Karamzadeh, O.A.S., Koochakpour, A.A.: On \(\aleph _0\)-selfinjectivity of strongly regular rings. Commun. Algebra 27, 1501–1513 (1999)

    Article  Google Scholar 

  16. 16.

    Karimi Feizabadi, A., Estaji, A.A., Zarghani, M.: The ring of real-valued functions on a frame. Categr. Gen. Algebra Struct. Appl. 5, 85–102 (2016)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Matlis, E.: The minimal prime spectrum of a reduced ring. Illinois J. Math. 27, 353–391 (1983)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Picado, J., Pultr, A.: Frames and Locales: Topology Without Points. Frontiers in Mathematics. Springer, Basel (2012)

  19. 19.

    Smith, P.F., Tercan, A.: Generalizations of \(CS\)-modules. Commun. Algebra 21, 1809–1847 (1993)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Zarghani, M.: The ring of real-continuous functions on a topoframe. PhD Thesis, Hakim Sabzevari University (2017)

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Acknowledgements

Thanks are due to the referee for helpful comments that have improved the readability of this paper and for providing Corollary 4.10.

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Correspondence to Mostafa Abedi.

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Estaji, A.A., Abedi, M. On regularity and injectivity of the ring of real-continuous functions on a topoframe. Algebra Univers. 82, 60 (2021). https://doi.org/10.1007/s00012-021-00753-2

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Keywords

  • Topoframe
  • Frame
  • Ring of all real-continuous functions on a topoframe
  • P-topoframe
  • Extremally disconnected topoframe
  • Injective
  • \(\aleph _0\)-self-injective
  • Regular ring

Mathematics Subject Classification

  • 06D22
  • 06F25
  • 54C30
  • 16E50
  • 16D50
  • 54G05