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More on the Functor Induced by z-Ideals

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Abstract

An ideal I of a commutative ring A with identity is called a z-ideal if whenever two elements of A belong to the same maximal ideals and one of the elements is in I, then so is the other. For a completely regular frame L we denote by \({{\mathrm{ZId}}}(\mathcal {R}L)\) the lattice of z-ideals of the ring \(\mathcal {R}L\) of continuous real-valued functions on L. It is a coherent frame, and it is known that \(L\mapsto {{\mathrm{ZId}}}(\mathcal {R}L)\) is the object part of a functor \(\mathsf {Z}:\mathbf {CRFrm}\rightarrow \mathbf {CohFrm}\), where \(\mathbf {CRFrm}\) is the category of completely regular frames and frame homomorphisms, and \(\mathbf {CohFrm}\) is the category of coherent frames and coherent maps. We explore when this functor preserves and reflects the property of being a Heyting homomorphism, and also when it preserves and reflects the variants of openness of Banaschewski and Pultr (Appl Categ Struct 2:331–350, 1994). We also record some other properties of this functor that have hitherto not been stated anywhere.

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References

  1. Ball, R.N., Banaschewski, B., Jakl, T., Pultr, A., Walters-Wayland, J.: Tightness relative to some (co)reflections in topology. Quaest. Math. 39, 421–436 (2016)

    Article  MathSciNet  Google Scholar 

  2. Ball, R.N., Walters-Wayland, J.: \(C\)- and \(C^{*}\)-quotients in pointfree topology. Dissert. Math. Rozprawy Mat. 412, 62 (2002)

    MathSciNet  MATH  Google Scholar 

  3. Banaschewski, B.: The real numbers in pointfree topology, Textos de Matemática Série B, No. 12, Departamento de Matemática da Universidade de Coimbra (1997a)

  4. Banaschewski, B.: Pointfree topology and the spectra of f-rings. In: Ordered algebraic structures, pp. 123–148. Kluwer Academic Publishers, Dordrecht (1997b)

  5. Banaschewski, B.: Functorial maximal spectra. J. Pure Appl. Algebra 168, 327–346 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Banaschewski, B., Dube, T., Gilmour, C., Walters-Wayland, J.: Oz in pointfree topology. Quaest. Math. 32, 215–227 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Banaschewski, B., Gilmour, C.: Pseudocompactness and the cozero part of a frame. Comment. Math. Univ. Carolin. 37, 579–589 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Banaschewski, B., Hager, A.W.: Essential completeness in categories of completely regular frames. Appl. Categ. Struct. 21, 167–180 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Banaschewski, B., Pultr, A.: Booleanization. Cah. Topol. Géom. Différ. Catég. 37, 41–60 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Banaschewski, B., Pultr, A.: Variants of openness. Appl. Categ. Struct. 2, 331–350 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Banaschewski, B.: Radical ideals and coherent frames. Comment. Math. Univ. Carolin. 37, 349–370 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Dube, T.: Contracting the socle in rings of continuous functions. Rend. Semin. Mat. Univ. Padova 123, 38–53 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dube, T., Ighedo, O.: On \(z\)-ideals of pointfree function rings. Bull. Iran. Math. Soc. 40, 655–673 (2014a)

    MathSciNet  MATH  Google Scholar 

  14. Dube, T., Ighedo, O.: Two functors induced by certain ideals of function rings. Appl. Categ. Struct. 22, 663–681 (2014b)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dube, T., Naidoo, I.: On openness and surjectivity of lifted frame homomorphisms. Topol. Appl. 157, 2159–2171 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dube, T., Naidoo, I.: Erratum to “On openness and surjectivity of lifted frame homomorphisms” [Topology Appl. 157 (2010) 2159–2171]. Topol. Appl. 158, 2257–2259 (2011)

    Article  MATH  Google Scholar 

  17. Gillman, L., Jerison, M.: Quotient fields of residue class rings of function rings. Ill. J. Math. 4, 425–436 (1960a)

    MathSciNet  MATH  Google Scholar 

  18. Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960b)

    Book  MATH  Google Scholar 

  19. Gutiérrez García, J., Kubiak, T., Picado, J.: Perfectness in locales. Quaest. Math. (2017). doi:10.2989/16073606.2017.1299810

    MathSciNet  Google Scholar 

  20. Ighedo, O.: Concerning ideals of pointfree function rings, Ph.D. thesis, University of South Africa (2014)

  21. Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  22. Madden, J.: \(\kappa \)-frames. J. Pure Appl. Algebra 1–2, 107–127 (1989)

    Google Scholar 

  23. Madden, J., Vermeer, J.: Lindelöf locales and realcompactness. Math. Proc. Camb. Phil. Soc. 99, 473–480 (1986)

    Article  MATH  Google Scholar 

  24. Martínez, J., Zenk, E.R.: Dimension in algebraic frames, II: applications to frames of ideals in \(C(X)\). Comment. Math. Univ. Carolin. 46, 607–636 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Mason, G.: \(z\)-Ideals and prime ideals. J. Algebra 26, 280–297 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mason, G.: Prime \(z\)-ideals of \(C(X)\) and related rings. Can. Math. Bull. 23, 437–443 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  27. Montgomery, R.G.: Structures determined by prime ideals of rings of functions. Trans. Amer. Math. Soc. 147, 367–380 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  28. Niefield, S.B., Rosenthal, K.I.: Componental nuclei. In: Categorical algebra and its applications (Louvain-La-Neuve, 1987), vol. 1348, pp. 299–306. Lecture Notes in Mathematics, Springer, Berlin (1988)

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

    Book  MATH  Google Scholar 

  30. Pultr, A.: Frames. In: Handbook of Algebra, vol. 3, pp. 791–857. North Holland, Amsterdam (2003)

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I thank the referee for the comments and suggestions that have helped improve the paper.

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  1. Department of Mathematical Sciences, University of South Africa, P. O. Box 392, Pretoria, 0003, South Africa

    Oghenetega Ighedo

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  1. Oghenetega Ighedo
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Ighedo, O. More on the Functor Induced by z-Ideals. Appl Categor Struct 26, 459–476 (2018). https://doi.org/10.1007/s10485-017-9498-7

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