Is the Ring of Continuous Functions Injective

• Published: 30 September 2021

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

Algebra universalis volume 82, Article number:60 (2021) Cite this article

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.

Access options

Buy single article

Instant access to the full article PDF.

39,95 €

Price includes VAT (Indonesia)

References

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

   Book  Google Scholar

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

   Article  MathSciNet  Google Scholar

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

   Article  MathSciNet  Google Scholar

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

   Article  MathSciNet  Google Scholar

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

   MATH  Google Scholar

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)

    Article  MathSciNet  Google Scholar

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)

    Article  MathSciNet  Google Scholar

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)

    Article  MathSciNet  Google Scholar

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

    MATH  Google Scholar

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

    Article  MathSciNet  Google Scholar

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

    Article  MathSciNet  Google Scholar

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

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

    Article  MathSciNet  Google Scholar

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

Download references

Acknowledgements

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

Author information

Authors and Affiliations

1. Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran

   Ali Akbar Estaji

2. Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

   Mostafa Abedi

Corresponding author

Correspondence to Mostafa Abedi.

Additional information

Communicated by Presented by W. Wm. McGovern.

Publisher's Note

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

Rights and permissions

About this article

Cite this article

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

Download citation

• Received: 11 August 2020

• Accepted: 30 August 2021

• Published: 30 September 2021

• DOI : https://doi.org/10.1007/s00012-021-00753-2

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

parkerofflus.blogspot.com

Source: https://link.springer.com/article/10.1007/s00012-021-00753-2

Share this post