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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 compactification 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 have shown that every nonregular prime ideal of
$C(X)$ is a $z^\circ$-ideal if and only if $X$ is $\partial$-space
(a space in which the boundary of any zeroset is again a zeroset).
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