| Title:
|
Intersections of minimal prime ideals in the rings of continuous functions (English) |
| Author:
|
Ghosh, Swapan Kumar |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
623-632 |
| . |
| Category:
|
math |
| . |
| Summary:
|
A space $X$ is called $\mu $-compact by M. Mandelker if the intersection of all free maximal ideals of $C(X)$ coincides with the ring $C_K(X)$ of all functions in $C(X)$ with compact support. In this paper we introduce $\phi $-compact and $\phi '$-compact spaces and we show that a space is $\mu $-compact if and only if it is both $\phi $-compact and $\phi '$-compact. We also establish that every space $X$ admits a $\phi $-compactification and a $\phi '$-compactification. Examples and counterexamples are given. (English) |
| Keyword:
|
minimal prime ideal |
| Keyword:
|
$P$-space |
| Keyword:
|
$F$-space |
| Keyword:
|
$\mu$-compact space |
| Keyword:
|
$\phi $-compact space |
| Keyword:
|
$\phi '$-compact space |
| Keyword:
|
round subset |
| Keyword:
|
almost round subset |
| Keyword:
|
nearly round subset |
| MSC:
|
46E25 |
| MSC:
|
46J20 |
| MSC:
|
54C40 |
| idZBL:
|
Zbl 1150.54018 |
| idMR:
|
MR2337417 |
| . |
| Date available:
|
2009-05-05T17:00:08Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119623 |
| . |
| Reference:
|
[1] Gillman L., Jerison M.: Rings of Continuous Functions.University Series in Higher Math., Van Nostrand, Princeton, New Jersey, 1960. Zbl 0327.46040, MR 0116199 |
| Reference:
|
[2] Henriksen M., Jerison M.: The space of minimal prime ideals of a commutative ring.Trans. Amer. Math. Soc. 115 (1965), 110-130. Zbl 0147.29105, MR 0194880 |
| Reference:
|
[3] Johnson D.G., Mandelker M.: Functions with pseudocompact support.General Topology Appl. 3 (1973), 331-338. Zbl 0277.54009, MR 0331310 |
| Reference:
|
[4] Mandelker M.: Round $z$-filters and round subsets of $\beta X$.Israel J. Math. 7 (1969), 1-8. Zbl 0174.25604, MR 0244951 |
| Reference:
|
[5] Mandelker M.: Supports of continuous functions.Trans. Amer. Math. Soc. 156 (1971), 73-83. Zbl 0197.48703, MR 0275367 |
| . |