# Article

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Keywords:
ring of continuous functions; fixed-place; anti fixed-place; irredundant; semi-prime; annihilator; affiliated prime; fixed-place rank; Zariski topology
Summary:
Let $I$ be a semi-prime ideal. Then $P_\circ \in \operatorname{Min}(I)$ is called irredundant with respect to $I$ if $I\neq \bigcap_{P_\circ \neq P\in \operatorname{Min}(I)}P$. If $I$ is the intersection of all irredundant ideals with respect to $I$, it is called a fixed-place ideal. If there are no irredundant ideals with respect to $I$, it is called an anti fixed-place ideal. We show that each semi-prime ideal has a unique representation as an intersection of a fixed-place ideal and an anti fixed-place ideal. We say the point $p\in \beta X$ is a fixed-place point if $O^p(X)$ is a fixed-place ideal. In this situation the fixed-place rank of $p$, denoted by FP-$\operatorname{rank}_X(p)$, is defined as the cardinal of the set of all irredundant prime ideals with respect to $O^p(X)$. Let $p$ be a fixed-place point, it is shown that FP-$\operatorname{rank}_X (p)= \eta$ if and only if there is a family $\{Y_\alpha\}_{ \alpha \in A}$ of cozero sets of $X$ such that: 1- $|A|= \eta$, 2- $p\in \operatorname{cl}_{\beta X} Y_\alpha$ for each $\alpha \in A$, 3- $p\notin \operatorname{cl}_{\beta X} (Y_\alpha \cap Y_\beta )$ if $\alpha \neq \beta$ and 4- $\eta$ is the greatest cardinal with the above properties. In this case $p$ is an $F$-point with respect to $Y_\alpha$ for any $\alpha \in A$.
References:
[1] Aliabad A.R.: $z^\circ$-ideals in $C(X)$. Ph.D. Thesis, Chamran University of Ahvaz, Iran. Zbl 1221.54023
[2] Aliabad A.R.: Connections between $C(X)$ and $C(Y)$, where $Y$ is a subspace of $X$. Abstracts of International Conference on Applicable General Topology, August 12–18, 2001, Hacettepe University, Ankara, Turkey.
[3] Aliabad A.R.: Pasting topological spaces at one point. Czechoslovak Math. J. 56 (131) (2006), 1193–1206. DOI 10.1007/s10587-006-0088-3 | MR 2280803 | Zbl 1164.54338
[4] Aliabad A.R., Badie M.: Connection between $C(X)$ and $C(Y)$, where $Y$ is subspace of $X$. Bull. Iranian Math. Soc. 37 (2011), no. 4, 109–126. MR 2915454
[5] Engelking R.: General Topology. PWN-Polish Scientific Publishing, Warsaw, 1977. MR 0500780 | Zbl 0684.54001
[6] Gillman L., Jerison M.: Rings of Continuous Functions. Van Nostrand Reinhold, New York, 1960. MR 0116199 | Zbl 0327.46040
[7] Goodearl K.R., Warfield R.B., Jr.: Introduction to Noncommutative Noetherian Rings. Cambridge University Press, Cambridge, 1989. MR 1020298 | Zbl 1101.16001
[8] Henriksen M., Larson S., Martinez J., Woods R.G.: Lattice-ordered algebras that are subdirect product of valuation domains. Trans. Amer. Math. Soc. 345 (1994), 195–221. DOI 10.1090/S0002-9947-1994-1239640-0 | MR 1239640
[9] Henriksen M., Wilson R.G.: Almost discrete $SV$-space. Topology Appl. 46 (1992), 89–97. DOI 10.1016/0166-8641(92)90123-H | MR 1184107 | Zbl 0791.54049
[10] Larson S.: $f$-Rings in which every maximal ideal contains finitely many minimal prime ideals. Comm. Algebra 25 (1997), no. 12, 3859–3888. DOI 10.1080/00927879708826092 | MR 1481572 | Zbl 0952.06026
[11] Larson S.: Constructing rings of continuous functions in which there are many maximal ideals with nontrivial rank. Comm. Algebra 31 (2003), 2183–2206. DOI 10.1081/AGB-120018991 | MR 1976272 | Zbl 1024.54015
[12] Underwood D.H.: On some Uniqueness questions in primary representations of ideals. Kyoto Math. J. 35 (1969), 69–94. MR 0246865 | Zbl 0181.05001
[13] Willard S.: General Topology. Addison Wesley, Reading, Mass., 1970. MR 0264581 | Zbl 1052.54001

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