# Article

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Keywords:
$z^\circ$-ideal; prime $z$-ideal; nonregular ideal; almost ${P}$-space; $\partial$-space; $m$-space
Summary:
The spaces $X$ in which every prime $z^\circ$-ideal of $C(X)$ 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 $C(X)$ is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in $C(X)$ a $z^\circ$-ideal? When is every nonregular (prime) $z$-ideal in $C(X)$ a $z^\circ$-ideal? For instance, we show that every nonregular prime ideal of $C(X)$ is a $z^\circ$-ideal if and only if $X$ is a $\partial$-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior).
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