Geodesic
On $a$-Kasch spaces
Archivum mathematicum, Tome 46 (2010) no. 4, pp. 251-262
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

If $X$ is a Tychonoff space, $C(X)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C(X)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\bar{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $\operatorname{gen}\,(I)
If $X$ is a Tychonoff space, $C(X)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C(X)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\bar{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $\operatorname{gen}\,(I)$ then $I$ is a non-essential ideal. We show that $X$ is an $\aleph _0$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an $\aleph _1$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let $C_F(X)$ denote the socle of $C(X)$. For a topological space $X$ with only a finite number of isolated points, we show that $X$ is an $a$-Kasch space if and only if $\frac{C(X)}{C_F(X)}$ is an $a$-Kasch ring.
Classification : 13A30, 16S60, 46J10, 54C40
Keywords: $a$-Kasch space; almost $P$-space; basically disconnected; $C$-embedded; essential ideal; extremally disconnected; fixed ideal; free ideal; Kasch ring; $P$-space; pseudocompact space; Stone-Čech compactification; socle; realcompactification 
@article{ARM_2010_46_4_a2,
     author = {Estaji, Ali Akbar and Henriksen, Melvin},
     title = {On $a${-Kasch} spaces},
     journal = {Archivum mathematicum},
     pages = {251--262},
     year = {2010},
     volume = {46},
     number = {4},
     mrnumber = {2754064},
     zbl = {1240.54064},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2010_46_4_a2/}
}
TY  - JOUR
AU  - Estaji, Ali Akbar
AU  - Henriksen, Melvin
TI  - On $a$-Kasch spaces
JO  - Archivum mathematicum
PY  - 2010
SP  - 251
EP  - 262
VL  - 46
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ARM_2010_46_4_a2/
LA  - en
ID  - ARM_2010_46_4_a2
ER  - 
%0 Journal Article
%A Estaji, Ali Akbar
%A Henriksen, Melvin
%T On $a$-Kasch spaces
%J Archivum mathematicum
%D 2010
%P 251-262
%V 46
%N 4
%U http://geodesic.mathdoc.fr/item/ARM_2010_46_4_a2/
%G en
%F ARM_2010_46_4_a2
Estaji, Ali Akbar; Henriksen, Melvin. On $a$-Kasch spaces. Archivum mathematicum, Tome 46 (2010) no. 4, pp. 251-262. http://geodesic.mathdoc.fr/item/ARM_2010_46_4_a2/

[1] Azarpanah, F.: Essential ideals in $C(X)$. Period. Math. Hungar. 3 (12) (1995), 105–112. | DOI | MR | Zbl

[2] Azarpanah, F.: Intersection of essential ideals in $C(X)$. Proc. Amer. Math. Soc. 125 (1997), 2149–2154. | DOI | MR | Zbl

[3] Azarpanah, F.: On almost $P$-space. Far East J. Math. Sci. Special volume (2000), 121–132. | MR

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

[5] Dietrich, W. E., Jr., : On the ideal structure of $C(X)$. Trans. Amer. Math. Soc. 152 (1970), 61–77. | MR | Zbl

[6] Donne, A. Le: On a question concerning countably generated $z$-ideal of $C(X)$. Proc. Amer. Math. Soc. 80 (1980), 505–510. | MR

[7] Engelking, R.: General topology. mathematical monographs, vol. 60 ed., PWN Polish Scientific publishers, 1977. | MR | Zbl

[8] Gillman, L., Jerison, M.: Rings of continuous functions. Springer-Verlag, 1979. | MR

[9] Goodearl, K. R.: Von Neumann regular rings. Pitman, San Francisco, 1979. | MR | Zbl

[10] Karamzadeh, O. A. S., Rostami, M.: On the intrinsic topology and some related ideals of $C(X)$. Proc. Amer. Math. Soc. 93 (1985), 179–184. | MR | Zbl

[11] Lam, Tsit-Yuen: Lectures on Modules and Rings. Springer, 1999.

[12] Levy, R.: Almost $P$-spaces. Can. J. Math. 29 (1977), 284–288. | MR | Zbl

[13] Marco, G. De: On the countably generated $z$-ideal of $C(X)$. Proc. Amer. Math. Soc. 31 (1972), 574–576. | MR

[14] Nunzetta, P., Plank, D.: Closed ideal in $C(X)$. Proc. Amer. Math. Soc. 35 (2) (1972), 601–606. | MR