## Rocky Mountain Journal of Mathematics

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

#### 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

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

#### 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

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