Geodesic
On nonregular ideals and $z^\circ$-ideals in $C(X)$
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 397-407
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The spaces $X$ in which every prime $z^\circ $-ideal of $C(X)$ is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces $X$, such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime $z^\circ $-ideal in $C(X)$ is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in $C(X)$ a $z^\circ $-ideal? When is every nonregular (prime) $z$-ideal in $C(X)$ a $z^\circ $-ideal? For instance, we show that every nonregular prime ideal of $C(X)$ is a $z^\circ $-ideal if and only if $X$ is a $\partial $-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior).
The spaces $X$ in which every prime $z^\circ $-ideal of $C(X)$ is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces $X$, such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime $z^\circ $-ideal in $C(X)$ is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in $C(X)$ a $z^\circ $-ideal? When is every nonregular (prime) $z$-ideal in $C(X)$ a $z^\circ $-ideal? For instance, we show that every nonregular prime ideal of $C(X)$ is a $z^\circ $-ideal if and only if $X$ is a $\partial $-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior).
Classification : 54C40
Keywords: $z^\circ $-ideal; prime $z$-ideal; nonregular ideal; almost ${P}$-space; $\partial $-space; $m$-space 
@article{CMJ_2005_55_2_a9,
     author = {Azarpanah, F. and Karavan, M.},
     title = {On nonregular ideals and $z^\circ$-ideals in $C(X)$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {397--407},
     year = {2005},
     volume = {55},
     number = {2},
     mrnumber = {2137146},
     zbl = {1081.54013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005_55_2_a9/}
}
TY  - JOUR
AU  - Azarpanah, F.
AU  - Karavan, M.
TI  - On nonregular ideals and $z^\circ$-ideals in $C(X)$
JO  - Czechoslovak Mathematical Journal
PY  - 2005
SP  - 397
EP  - 407
VL  - 55
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2005_55_2_a9/
LA  - en
ID  - CMJ_2005_55_2_a9
ER  - 
%0 Journal Article
%A Azarpanah, F.
%A Karavan, M.
%T On nonregular ideals and $z^\circ$-ideals in $C(X)$
%J Czechoslovak Mathematical Journal
%D 2005
%P 397-407
%V 55
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2005_55_2_a9/
%G en
%F CMJ_2005_55_2_a9
Azarpanah, F.; Karavan, M. On nonregular ideals and $z^\circ$-ideals in $C(X)$. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 397-407. http://geodesic.mathdoc.fr/item/CMJ_2005_55_2_a9/

[1] M. F.  Atiyah and I. G.  Macdonald: Introduction to Commutative Algebra. Addison-Wesly, Reading, 1969.

[2] F. Azarpanah: On almost ${P}$-spaces. Far East J.  Math. Sci, Special volume (2000), 121–132. | MR | Zbl

[3] F. Azarpanah, O. A. S.  Karamzadeh and A.  Rezai Aliabad: On $z^\circ $-ideals in  $C(X)$. Fund. Math. 160 (1999), 15–25. | MR

[4] F. Azarpanah, O. A. S.  Karamzadeh and A.  Rezai Aliabad: On ideals consisting entirely of zero divisors. Comm. Algebra 28 (2000), 1061–1073. | DOI | MR

[5] F.  Dashiel, A.  Hager and M.  Henriksen: Order-Cauchy completions and vector lattices of continuous functions. Canad. J.  Math. XXXII (1980), 657–685. | DOI | MR

[6] R.  Engelking: General Topology. PWN-Polish Scientific Publishers, Warszawa, 1977. | MR | Zbl

[7] L.  Gillman and M.  Jerison: Rings of Continuous Functions. Springer-Verlag, New York-Heidelberg-Berlin, 1976. | MR

[8] M.  Henriksen and M.  Jerison: The space of minimal prime ideals of a commutative ring. Trans. Amer. Math. Soc. 115 (1965), 110–130. | DOI | MR

[9] M.  Henriksen, J.  Martinz and R. G. Woods: Spaces  $X$ in which all prime $z$-ideals of  $C(X)$ are minimal or maximal. International Conference on Applicable General Topology, Ankara, , , 2001. | MR

[10] R.  Levy: Almost ${P}$-spaces. Canad. J.  Math. 2 (1977), 284–288. | DOI | MR | Zbl

[11] M. A.  Mulero: Algebraic properties of rings of continuous functions. Fund. Math. 149 (1996), 55–66. | MR | Zbl

[12] A. I. Veksler: $P^{\prime }$-points, $P^{\prime }$-sets, ${P}^{\prime }$-spaces. A new class of order-continuous measures and functionals. Sov. Math. Dokl. 14 (1973), 1445–1450. | MR | Zbl