| Title:
|
Pasting topological spaces at one point (English) |
| Author:
|
Aliabad, Ali Rezaei |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
1193-1206 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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]. (English) |
| Keyword:
|
pasting topological spaces at one point |
| Keyword:
|
rings of continuous (bounded) real functions on $X$ |
| Keyword:
|
$z$-ideal |
| Keyword:
|
$z^\circ $-ideal |
| Keyword:
|
$C$-embedded |
| Keyword:
|
$P$-space |
| Keyword:
|
$F$-space. |
| MSC:
|
54B15 |
| MSC:
|
54C40 |
| MSC:
|
54C45 |
| MSC:
|
54G05 |
| MSC:
|
54G10 |
| idZBL:
|
Zbl 1164.54338 |
| idMR:
|
MR2280803 |
| . |
| Date available:
|
2009-09-24T11:42:09Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128139 |
| . |
| Reference:
|
[1] A. R. Aliabad: $z^\circ $-ideals in $C(X)$.PhD. Thesis, 1996. |
| Reference:
|
[2] F. Azarpanah, O. A. S. Karamzadeh, and A. Rezaei Aliabad: On ideals consisting entirely of zero divisors.Comm. Algebra 28 (2000), 1061–1073. MR 1736781, 10.1080/00927870008826878 |
| Reference:
|
[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 |
| Reference:
|
[4] F. Azarpanah, O. A. S. Karamzadeh: Algebraic characterizations of some disconnected spaces.Italian J. Pure Appl. Math. 10 (2001), 9–20. MR 1962109 |
| Reference:
|
[5] R. Engelking: General Topology.PWN—Polish Scientific Publishing, , 1977. Zbl 0373.54002, MR 0500780 |
| Reference:
|
[6] A. A. Estaji, O, A. S. Karamzadeh: On $C(X)$ modulo its socle.Comm. Algebra 31 (2003), 1561–1571. MR 1972881, 10.1081/AGB-120018497 |
| Reference:
|
[7] L. Gillman, M. Jerison: Rings of Continuous Functions.Van Nostrand Reinhold, New York, 1960. MR 0116199 |
| Reference:
|
[8] M. Henriksen, R. G. Wilson: Almost discrete $SV$-space.Topology and its Application 46 (1992), 89–97. MR 1184107 |
| Reference:
|
[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. MR 1239640, 10.1090/S0002-9947-1994-1239640-0 |
| Reference:
|
[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 |
| Reference:
|
[11] S. Larson: $f$-rings in which every maximal ideal contains finitely many prime ideals.Comm. Algebra 25 (1997), 3859–3888. MR 1481572, 10.1080/00927879708826092 |
| Reference:
|
[12] R. Levy: Almost $P$-spaces.Can. J. Math. 2 (1977), 284–288. Zbl 0342.54032, MR 0464203 |
| Reference:
|
[13] S. Willard: General Topology.Addison Wesley, Reading, 1970. Zbl 0205.26601, MR 0264581 |
| . |
