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Keywords:
$z^\circ $-ideal; prime $z$-ideal; nonregular ideal; almost ${P}$-space; $\partial $-space; $m$-space
$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).
References:
[1] M. F. Atiyah and I. G. Macdonald: Introduction to Commutative Algebra. Addison-Wesly, Reading, 1969.
[2] F. Azarpanah: On almost ${P}$-spaces. Far East J. Math. Sci, Special volume (2000), 121–132. MR 1761076 | Zbl 0954.54006
[3] F. Azarpanah, O. A. S. Karamzadeh and A. Rezai Aliabad: On $z^\circ $-ideals in $C(X)$. Fund. Math. 160 (1999), 15–25. MR 1694400
[4] F. Azarpanah, O. A. S. Karamzadeh and A. Rezai Aliabad: On ideals consisting entirely of zero divisors. Comm. Algebra 28 (2000), 1061–1073. DOI 10.1080/00927870008826878 | MR 1736781
[5] F. Dashiel, A. Hager and M. Henriksen: Order-Cauchy completions and vector lattices of continuous functions. Canad. J. Math. XXXII (1980), 657–685. DOI 10.4153/CJM-1980-052-0 | MR 0586984
[6] R. Engelking: General Topology. PWN-Polish Scientific Publishers, Warszawa, 1977. MR 0500780 | Zbl 0373.54002
[7] L. Gillman and M. Jerison: Rings of Continuous Functions. Springer-Verlag, New York-Heidelberg-Berlin, 1976. MR 0407579
[8] M. Henriksen and M. Jerison: The space of minimal prime ideals of a commutative ring. Trans. Amer. Math. Soc. 115 (1965), 110–130. DOI 10.1090/S0002-9947-1965-0194880-9 | MR 0194880
[9] M. Henriksen, J. Martinz and R. G. Woods: Spaces $X$ in which all prime $z$-ideals of $C(X)$ are minimal or maximal. International Conference on Applicable General Topology, Ankara, , , 2001. MR 2026163
[10] R. Levy: Almost ${P}$-spaces. Canad. J. Math. 2 (1977), 284–288. DOI 10.4153/CJM-1977-030-7 | MR 0464203 | Zbl 0342.54032
[11] M. A. Mulero: Algebraic properties of rings of continuous functions. Fund. Math. 149 (1996), 55–66. MR 1372357 | Zbl 0840.54020
[12] A. I. Veksler: $P^{\prime }$-points, $P^{\prime }$-sets, ${P}^{\prime }$-spaces. A new class of order-continuous measures and functionals. Sov. Math. Dokl. 14 (1973), 1445–1450. MR 0341447 | Zbl 0291.54046
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