Geodesic
On nonregular ideals and $z^\circ$-ideals in $C(X)$
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 397-407

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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 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).
Classification : 54C40
Keywords: $z^\circ $-ideal; prime $z$-ideal; nonregular ideal; almost ${P}$-space; $\partial $-space; $m$-space 
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     author = {Azarpanah, F. and Karavan, M.},
     title = {On nonregular ideals and $z^\circ$-ideals in $C(X)$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {397--407},
     publisher = {mathdoc},
     volume = {55},
     number = {2},
     year = {2005},
     mrnumber = {2137146},
     zbl = {1081.54013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005__55_2_a9/}
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Azarpanah, F.; Karavan, M. On nonregular ideals and $z^\circ$-ideals in $C(X)$. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 397-407. http://geodesic.mathdoc.fr/item/CMJ_2005__55_2_a9/