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pasting topological spaces at one point; rings of continuous (bounded) real functions on $X$; $z$-ideal; $z^\circ $-ideal; $C$-embedded; $P$-space; $F$-space.
Let $\lbrace X_\alpha \rbrace _{\alpha \in \Lambda }$ be a family of topological spaces and $x_{\alpha }\in X_{\alpha }$, for every $\alpha \in \Lambda $. Suppose $X$ is the quotient space of the disjoint union of $X_\alpha $’s by identifying $x_\alpha $’s as one point $\sigma $. We try to characterize ideals of $C(X)$ according to the same ideals of $C(X_\alpha )$’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let $m$ be an infinite cardinal. (1) Is there any ring $R$ and $I$ an ideal in $R$ such that $I$ is an irreducible intersection of $m$ prime ideals? (2) Is there any set of prime ideals of cardinality $m$ in a ring $R$ such that the intersection of these prime ideals can not be obtained as an intersection of fewer than $m$ prime ideals in $R$? Finally, we answer an open question in [11].
[1] A. R. Aliabad: $z^\circ $-ideals in  $C(X)$. PhD. Thesis, 1996.
[2] F. Azarpanah, O. A. S. Karamzadeh, and A.  Rezaei Aliabad: On ideals consisting entirely of zero divisors. Comm. Algebra 28 (2000), 1061–1073. DOI 10.1080/00927870008826878 | MR 1736781
[3] F.  Azarpanah, O, A. S.  Karamzadeh, and A.  Rezaei Aliabad: On $z^o-ideals$ in  $C(X)$. Fundamenta Math. 160 (1999), 15–25. MR 1694400
[4] F. Azarpanah, O. A. S.  Karamzadeh: Algebraic characterizations of some disconnected spaces. Italian  J.  Pure Appl. Math. 10 (2001), 9–20. MR 1962109
[5] R. Engelking: General Topology. PWN—Polish Scientific Publishing, , 1977. MR 0500780 | Zbl 0373.54002
[6] A. A. Estaji, O, A. S.  Karamzadeh: On $C(X)$ modulo its socle. Comm. Algebra 31 (2003), 1561–1571. DOI 10.1081/AGB-120018497 | MR 1972881
[7] L. Gillman, M.  Jerison: Rings of Continuous Functions. Van Nostrand Reinhold, New York, 1960. MR 0116199
[8] M. Henriksen, R. G. Wilson: Almost discrete $SV$-space. Topology and its Application 46 (1992), 89–97. MR 1184107
[9] M. Henriksen, S.  Larson, J. Martinez, and R. G. Woods: Lattice-ordered algebras that are subdirect products of valuation domains. Trans. Amer. Math. Soc. 345 (1994), 195–221. DOI 10.1090/S0002-9947-1994-1239640-0 | MR 1239640
[10] O. A. S.  Karamzadeh, M.  Rostami: On the intrinsic topology and some related ideals of  $C(X)$. Proc. Amer. Math. Soc. 93 (1985), 179–184. MR 0766552
[11] S. Larson: $f$-rings in which every maximal ideal contains finitely many prime ideals. Comm. Algebra 25 (1997), 3859–3888. DOI 10.1080/00927879708826092 | MR 1481572
[12] R. Levy: Almost $P$-spaces. Can. J.  Math. 2 (1977), 284–288. MR 0464203 | Zbl 0342.54032
[13] S. Willard: General Topology. Addison Wesley, Reading, 1970. MR 0264581 | Zbl 0205.26601
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