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The spaces $X$ in which every prime $z^{\circ }$-ideal of $C\left(X\right)$ is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces $X$, such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime $z^{\circ }$-ideal in $C\left(X\right)$ is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in $C\left(X\right)$ a $z^{\circ }$-ideal? When is every nonregular (prime) $z$-ideal in $C\left(X\right)$ a $z^{\circ }$-ideal? For...

An ideal I in a commutative ring R is called a z°-ideal if I consists of zero divisors and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We characterize topological spaces X for which z-ideals and z°-ideals coincide in , or equivalently, the sum of any two ideals consisting entirely of zero divisors consists entirely of zero divisors. Basically disconnected spaces, extremally disconnected and P-spaces are characterized in terms of z°-ideals. Finally,...

Let $C_F\left(X\right)$ be the socle of C(X). It is shown that each prime ideal in $C\left(X\right)/C_F\left(X\right)$ is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that $dim(C\left(X\right)/C_F\left(X\right))\ge dimC\left(X\right)$, where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points. For each essential...

We prove that a Hausdorff space $X$ is locally compact if and only if its topology coincides with the weak topology induced by $C_{\infty }\left(X\right)$. It is shown that for a Hausdorff space $X$, there exists a locally compact Hausdorff space $Y$ such that $C_{\infty }\left(X\right)\cong C_{\infty }\left(Y\right)$. It is also shown that for locally compact spaces $X$ and $Y$, $C_{\infty }\left(X\right)\cong C_{\infty }\left(Y\right)$ if and only if $X\cong Y$. Prime ideals in $C_{\infty }\left(X\right)$ are uniquely represented by a class of prime ideals in $C^{*}\left(X\right)$. $\infty$-compact spaces are introduced and it turns out that a locally compact space $X$ is $\infty$-compact if and only if every...

In this article we define the $m$-topology on some rings of quotients of $C\left(X\right)$. Using this, we equip the classical ring of quotients $q\left(X\right)$ of $C\left(X\right)$ with the $m$-topology and we show that $C\left(X\right)$ with the $r$-topology is in fact a subspace of $q\left(X\right)$ with the $m$-topology. Characterization of the components of rings of quotients of $C\left(X\right)$ is given and using this, it turns out that $q\left(X\right)$ with the $m$-topology is connected if and only if $X$ is a pseudocompact almost $P$-space, if and only if $C\left(X\right)$ with $r$-topology is connected. We also observe that...