Pasting topological spaces at one point. (English) Zbl 1164.54338

Aliabad, Ali Rezaei

Czech. Math. J. 56, No. 4, 1193-1206 (2006).

Summary: Let \(\{X_\alpha \}_{\alpha \in \Lambda }\) 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 the \(X_\alpha \)’s by identifying the \(x_\alpha \)’s as one point \(\sigma \). We try to characterize ideals of \(C(X)\) according to the same ideals of the \(C(X_\alpha )\)’s. In addition we generalize the concept of rank of a point 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 \(R\)?

MSC:

54C40	Algebraic properties of function spaces in general topology
54C45	\(C\)- and \(C^*\)-embedding
54G05	Extremally disconnected spaces, \(F\)-spaces, etc.
54G10	\(P\)-spaces

Keywords:

rings of continuous (bounded) real functions on  \(X\); \(z\)-ideal

References:

[1]	A. R. Aliabad: z{\(\deg\)}-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. · Zbl 0970.13002 · doi:10.1080/00927870008826878
[3]	F. Azarpanah, O, A. S. Karamzadeh, and A. Rezaei Aliabad: On z{\(\deg\)}-ideals in C(X). Fundamenta Math. 160 (1999), 15–25. · Zbl 0991.54014
[4]	F. Azarpanah, O. A. S. Karamzadeh: Algebraic characterizations of some disconnected spaces. Italian J. Pure Appl. Math. 10 (2001), 9–20. · Zbl 1117.54030
[5]	R. Engelking: General Topology. PWN–Polish Scientific Publishing, 1977. · Zbl 0373.54002
[6]	A. A. Estaji, O, A. S. Karamzadeh: On C(X) modulo its socle. Comm. Algebra 31 (2003), 1561–1571. · Zbl 1025.54012 · doi:10.1081/AGB-120018497
[7]	L. Gillman, M. Jerison: Rings of Continuous Functions. Van Nostrand Reinhold, New York, 1960. · Zbl 0093.30001
[8]	M. Henriksen, R. G. Wilson: Almost discrete SV-space. Topology and its Application 46 (1992), 89–97. · Zbl 0791.54049 · doi:10.1016/0166-8641(92)90123-H
[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. · Zbl 0817.06014 · doi:10.2307/2154601
[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. · Zbl 0524.54013
[11]	S. Larson: f-rings in which every maximal ideal contains finitely many prime ideals. Comm. Algebra 25 (1997), 3859–3888. · Zbl 0952.06026 · doi:10.1080/00927879708826092
[12]	R. Levy: Almost P-spaces. Can. J. Math. 2 (1977), 284–288. · Zbl 0342.54032 · doi:10.4153/CJM-1977-030-7
[13]	S. Willard: General Topology. Addison Wesley, Reading, 1970. · Zbl 0205.26601