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.