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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...