| Title:
|
Fixed-place ideals in commutative rings (English) |
| Author:
|
Aliabad, Ali Rezaei |
| Author:
|
Badie, Mehdi |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
53-68 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $\alpha \in A$. (English) |
| Keyword:
|
ring of continuous functions |
| Keyword:
|
fixed-place |
| Keyword:
|
anti fixed-place |
| Keyword:
|
irredundant |
| Keyword:
|
semi-prime |
| Keyword:
|
annihilator |
| Keyword:
|
affiliated prime |
| Keyword:
|
fixed-place rank |
| Keyword:
|
Zariski topology |
| MSC:
|
13Axx |
| MSC:
|
54C40 |
| idMR:
|
MR3038071 |
| . |
| Date available:
|
2013-02-21T14:03:38Z |
| Last updated:
|
2015-04-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143152 |
| . |
| Reference:
|
[1] Aliabad A.R.: $z^\circ$-ideals in $C(X)$.Ph.D. Thesis, Chamran University of Ahvaz, Iran. Zbl 1221.54023 |
| Reference:
|
[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. |
| Reference:
|
[3] Aliabad A.R.: Pasting topological spaces at one point.Czechoslovak Math. J. 56 (131) (2006), 1193–1206. Zbl 1164.54338, MR 2280803, 10.1007/s10587-006-0088-3 |
| Reference:
|
[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 |
| Reference:
|
[5] Engelking R.: General Topology.PWN-Polish Scientific Publishing, Warsaw, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[6] Gillman L., Jerison M.: Rings of Continuous Functions.Van Nostrand Reinhold, New York, 1960. Zbl 0327.46040, MR 0116199 |
| Reference:
|
[7] Goodearl K.R., Warfield R.B., Jr.: Introduction to Noncommutative Noetherian Rings.Cambridge University Press, Cambridge, 1989. Zbl 1101.16001, MR 1020298 |
| Reference:
|
[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. MR 1239640, 10.1090/S0002-9947-1994-1239640-0 |
| Reference:
|
[9] Henriksen M., Wilson R.G.: Almost discrete $SV$-space.Topology Appl. 46 (1992), 89–97. Zbl 0791.54049, MR 1184107, 10.1016/0166-8641(92)90123-H |
| Reference:
|
[10] Larson S.: $f$-Rings in which every maximal ideal contains finitely many minimal prime ideals.Comm. Algebra 25 (1997), no. 12, 3859–3888. Zbl 0952.06026, MR 1481572, 10.1080/00927879708826092 |
| Reference:
|
[11] Larson S.: Constructing rings of continuous functions in which there are many maximal ideals with nontrivial rank.Comm. Algebra 31 (2003), 2183–2206. Zbl 1024.54015, MR 1976272, 10.1081/AGB-120018991 |
| Reference:
|
[12] Underwood D.H.: On some Uniqueness questions in primary representations of ideals.Kyoto Math. J. 35 (1969), 69–94. Zbl 0181.05001, MR 0246865 |
| Reference:
|
[13] Willard S.: General Topology.Addison Wesley, Reading, Mass., 1970. Zbl 1052.54001, MR 0264581 |
| . |
