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\(C(X)\) Versus its Functionally Countable Subalgebra

  • Mostafa Ghadermazi
  • Omid Ali S. Karamzadeh
  • Mehrad Namdari
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Abstract

Let \(C_c(X)\) (resp. \(C^F(X)\)) denote the subring of \(C(X)\) consisting of functions with countable (resp. finite) image and \(C_F(X)\) be the socle of \(C(X)\). We characterize spaces X with \(C^*(X)=C_c(X)\), which generalizes a celebrated result due to Rudin, Pelczynnski, and Semadeni. Two zero-dimensional compact spaces X, Y are homeomorphic if and only if \(C_c(X)\cong C_c(Y)\) (resp. \(C^F(X)\cong \ C^F(Y)\)). The spaces X for which \(C_c(X)=C^F(X)\) are characterized. The socles of \(C_c(X)\), \(C^F(X)\), which are observed to be the same, are topologically characterized and spaces X for which this socle coincides with \(C_F(X)\) are determined, too. A certain well-known algebraic property of \(C(X)\), where X is real compact, is extended to \(C_c(X)\). In contrast to the fact that \(C_F(X)\) is never prime in \(C(X)\), we characterize spaces X for which \(C_F(X)\) is a prime ideal in \(C_c(X)\). It is observed for these spaces, \(C_c(X)\) coincides with its own socle (a fact, which is never true for \(C(X)\), unless X is finite). Finally, we show that a space X is the one-point compactification of a discrete space if and only if \(C_F(X)\) is a unique proper essential ideal in \(C^F(X)\).

Keywords

Functionally countable subring Socle \(z_c\)-ideal Regular ring CP-space 

Mathematics Subject Classification

Primary 54C40 Secondary 13C11 
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References

  1. 1.
    Azarpanah, F.: Intersection of essential ideals in \(C(X)\). Proc. Am. Math. Soc. 125, 2149–2154 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Azarpanah, F., Karamzadeh, O.A.S., Keshtkar, Z., Olfati, A.R.: On maximal ideals of \(C_c(X)\) and the uniformity of its localizations. Rocky Mt. J. Math. 48, 345–384 (2018)Google Scholar
  3. 3.
    Azarpanah, F., Karamzadeh, O.A.S., Rahmati, S.: \(C(X)\) vs. \(C(X)\) modulo its socle. Colloq. Math. 3, 315–336 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Choban, M.M.: Functionally countable spaces and Baire functions. Serdica. Math. J. 23, 233–247 (1997)MathSciNetMATHGoogle Scholar
  5. 5.
    De Marco, G., Wilson, R.G.: Rings of continuous functions with values in an archimedian ordered field. Rend. del Semin. Math. della Univ. di Padova 44, 263–272 (1970)Google Scholar
  6. 6.
    Engelking, R.: General topology. Heldermann, Berlin (1989)MATHGoogle Scholar
  7. 7.
    Ercan, Z., Onal, S.: A remark on the homomorphism on \(C(X)\). Proc. Am. Math. Soc. 133(12), 3609–3611 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Estaji, A.A., Karamzadeh, O.A.S.: On \(C(X)\) modulo its socle. Commun. Algebra 31, 1561–1571 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Galvin, F.: Problem 6444. Am. Math. Mon. 90(9), 648 (1983)Google Scholar
  10. 10.
    Galvin, F.: Solution. Am. Math. Mon. 92(6), 434 (1985)CrossRefGoogle Scholar
  11. 11.
    Ghadermazi, M., Karamzadeh, O.A.S., Namdari, M.: On the functionally countable subalgebra of \(C(X)\). Rend. Sem. Mat. Univ. Padova 129, 47–69 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ghasemzadeh, S., Karamzadeh, O.A.S., Namdari, M.: The super socle of the ring of continuous functions. Math. Slovaca 67(4), 1001–1010 (2017) (to appear)Google Scholar
  13. 13.
    Gillman, L., Jerison, M.: Rings of Continuous Functions. Springer, New York (1976)MATHGoogle Scholar
  14. 14.
    Gillman, L.: Convex and pseudoprime ideals in \(C(X)\), general topology and applications, proceedings of northeast conference. Marcel-Dekker Inc., New York, pp. 87–95 (1988)Google Scholar
  15. 15.
    Hardy, K., Woods, R.G.: On \(c\)-realcompact spaces and locally bounded normal functions. Pac. J. Math. 43(3), 647–656 (1972)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Karamzadeh, O.A.S., Keshtkar, Z.: On c-realcompact spacs. Quaest. Math. (2018).  https://doi.org/10.2989/16073606.2018.1441919
  17. 17.
    Karamzadeh, O.A.S., Motamedi, M., Shahrtash, S.M.: On rings with a unique proper essential right ideal. Fund. Math. 183, 229–244 (1985)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Karamzadeh, O.A.S., Motamedi, M., Shahrtash, S.M.: Erratum to On rings with a unique proper essential right ideal. Fund. Math. 205, 289–291 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Karamzadeh, O.A.S., Namdari, M., Soltanpour, S.: On the locally functionally countable subalgebra of C(X). Appl. Gen. Topol. 16(2), 183–207 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Karamzadeh, O.A.S., Rostami, M.: On the intrinsic topology and some related ideals of \(C(X)\). Proc. Am. Math. Soc. 93, 179–184 (1985)MathSciNetMATHGoogle Scholar
  21. 21.
    Levy, R., Rice, M.D.: Normal \(P\)-spaces and the \(G_\delta \)-topology. Colloq. Math. 47, 227–240 (1981)CrossRefMATHGoogle Scholar
  22. 22.
    Mehran, S., Namdari, M.: The \(\lambda \)-super socle of the ring of continuous functions. Categories Gen. Algebraic Struct. Appl. 6, 37–50 (2017) (Special issue on the occasion of Banaschewski’s 90th Birthday)Google Scholar
  23. 23.
    Mulero, M.A.: Algebraic properties of rings of continuous functions. Fund. Math. 149, 55–66 (1996)MathSciNetMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  • Mostafa Ghadermazi
    • 1
  • Omid Ali S. Karamzadeh
    • 2
  • Mehrad Namdari
    • 2
  1. 1.Department of MathematicsUniversity of KurdistanSanandajIran
  2. 2.Department of MathematicsShahid Chamran University of AhvazAhvazIran

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