Rocky Mountain Journal of Mathematics

On maximal ideals of $C_c(X)$ and the uniformity of its localizations

F. Azarpanah, O.A.S. Karamzadeh, Z. Keshtkar, and A.R. Olfati

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A similar characterization, as the Gelfand-Kolmogoroff theorem for the maximal ideals in $C(X)$, is given for the maximal ideals of $C_c(X)$. It is observed that the $z_c$-ideals in $C_c(X)$ are contractions of the $z$-ideals of $C(X)$. Using this, it turns out that maximal ideals (respectively, prime $z_c$-ideals) of $C_c(X)$ are precisely the contractions of maximal ideals (respectively, prime $z$-ideals) of $C(X)$, as well. Maximal ideals of $C^*_c(X)$ are also characterized, and two representations are given. We reveal some more useful basic properties of $C_c(X)$. In particular, we observe that, for any space $X$, $C_c(X)$ and $C^*_c(X)$ are always clean rings. It is also shown that $\beta _0X$, the Banaschewski compactification of a zero-dimensional space $X$, is homeomorphic with the structure spaces of $C_c(X)$, $C^F(X)$, $C_c(\beta _0X)$, as well as with that of $C(\beta _0 X)$. $F_c$-spaces are characterized, the spaces $X$ for which $C_c(X)_P$, the localization of $C_c(X)$ at prime ideals $P$, are uniform (or equivalently are integral domain). We observe that $X$ is an $F_c$-space if and only if $\beta _0X$ has this property. In the class of strongly zero-dimensional spaces, we show that $F_c$-spaces and $F$-spaces coincide. It is observed that, if either $C_c(X)$ or $C^*_c(X)$ is ax Bezout ring, then $X$ is an $F_c$-space. Finally, $C_c(X)$ and $C^*_c(X)$ are contrasted with regards to being an absolutely Bezout ring. Consequently, it is observed that the ideals in $C_c(X)$ are convex if and only if they are absolutely convex if and only if $C_c(X)$ and $C^*_c(X)$ are both unitarily absolute Bezout rings.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 2 (2018), 345-384.

Dates
First available in Project Euclid: 4 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1528077621

Digital Object Identifier
doi:10.1216/RMJ-2018-48-2-345

Mathematical Reviews number (MathSciNet)
MR3809150

Zentralblatt MATH identifier
06883471

Subjects
Primary: 54C40: Algebraic properties of function spaces [See also 46J10]

Keywords
Bézout ring $z_c$-ideal clean zero-dimensional $F_c$-space $F$-space

Citation

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 Mountain J. Math. 48 (2018), no. 2, 345--384. doi:10.1216/RMJ-2018-48-2-345. https://projecteuclid.org/euclid.rmjm/1528077621


Export citation

References

  • S. Afrooz, F. Azarpanah and O.A.S. Karamzadeh, Goldie dimension of rings of fractions of $C(X)$, Quaest. Math. 38 (2015), 139–154.
  • F. Azarpanah, When is $C(X)$ a clean ring?, Acta Math. Hungar. 94 (2002), 53–58.
  • F. Azarpanah and O.A.S. Karamzadeh, Algebraic characterizations of some disconnected spaces, Italian J. Pure Appl. Math. 12 (2002), 155–168.
  • G. Bezhanishvili, V. Marra, P.J. Morandi and B. Olberding, Idempotent generated algebras and Boolean powers of commutative rings, Alg. Univ. 73 (2015), 183–204.
  • G. Bezhanishvili, P.J. Morandi and B. Olberding, Bounded Archimedian $\ell$-algebras and Gelfand-Neumark-Stone duality, Theory Appl. Categ. 28 (2013), 435–476.
  • P. Bhattacharjee, M.L. Knox and W.W. McGovern, The classical ring of quotient of $C_c(X)$, Appl. Gen. Topol. 15 (2014), 147–154.
  • M.M. Choban, Functionally countable spaces and Baire functions, Serdica. Math. J. 23 (1997), 233–247.
  • G. De Marco and R.G. Wilson, Rings of continuous functions with values in an archimedian ordered field, Rend. Sem. Math. Padova 44 (1970), 263–272.
  • R. Engelking, General topology, Sigma Pure Math. 6, Heldermann Verlag, Berlin, 1989.
  • M. Ghadermazi, O.A.S. Karamzadeh and M. Namdari, On the functionally countable subalgebra of $C(X)$, Rend. Sem. Mat. Univ. Padova 129 (2013), 47–69.
  • ––––, $C(X)$ versus its functionally countable subalgebra, Bull. Iranian Math. Soc., to appear.
  • L. Gillman, Convex and pseudoprime ideals in $C(X)$, Lect. Notes Pure Appl. Math. 123 (1990), 87–95.
  • L. Gillman and M. Jerison, Rings of continuous functions, D. Van Nostrand Publishers, Princeton, NJ, 1960.
  • A.W. Hager, On inverse-closed subalgebras of $C(X)$, Proc. Lond. Math. Soc. 19 (1969), 233–257.
  • P.T. Johnstone, Stone spaces, Cambridge Stud. Adv. Math. 3, Cambridge University Press, Cambridge, 1982.
  • O.A.S. Karamzadeh, MR3451352 (Review), Math. Rev. 90161 (2016).
  • O.A.S. Karamzadeh, M. Namdari and S. Soltanpour, On the locally functionally countable subalgebra of $C(X)$, Appl. Gen. Topol. 16 (2015), 183–207.
  • W.W. McGovern, Clean semiprime f-rings with bounded inversion, Comm. Alg. 31 (2003), 3295–3304.
  • J.R. Porter and R.G. Woods, Extensions and absolutes of Hausdorff, Springer-Verlag, New York, 1988.
  • K. Varadarajan, Study of Hopficity in certain classical rings, Comm. Alg. 28 (2000), 771–783.
  • R.C. Walker, The Stone-Čech compactification, Springer-Verlag, New York, 1974.