# Article

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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
Summary:
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 $\operatorname{gen}\,(I)<a$ 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.
References:
[1] Azarpanah, F.: Essential ideals in $C(X)$. Period. Math. Hungar. 3 (12) (1995), 105–112. DOI 10.1007/BF01876485 | MR 1609417 | Zbl 0869.54021
[2] Azarpanah, F.: Intersection of essential ideals in $C(X)$. Proc. Amer. Math. Soc. 125 (1997), 2149–2154. DOI 10.1090/S0002-9939-97-04086-0 | MR 1422843 | Zbl 0867.54023
[3] Azarpanah, F.: On almost $P$-space. Far East J. Math. Sci. Special volume (2000), 121–132. MR 1761076
[4] Azarpanah, F., Karamzadeh, O. A. S., Aliabad, A. R.: On $z^{\circ }$-ideals in $C(X)$. Fund. Math. 160 (1999), 15–25. MR 1694400 | Zbl 0991.54014
[5] Dietrich, W. E., Jr., : On the ideal structure of $C(X)$. Trans. Amer. Math. Soc. 152 (1970), 61–77. MR 0265941 | Zbl 0205.42402
[6] Donne, A. Le: On a question concerning countably generated $z$-ideal of $C(X)$. Proc. Amer. Math. Soc. 80 (1980), 505–510. MR 0581015
[7] Engelking, R.: General topology. mathematical monographs, vol. 60 ed., PWN Polish Scientific publishers, 1977. MR 0500780 | Zbl 0373.54002
[8] Gillman, L., Jerison, M.: Rings of continuous functions. Springer-Verlag, 1979. MR 0407579
[9] Goodearl, K. R.: Von Neumann regular rings. Pitman, San Francisco, 1979. MR 0533669 | Zbl 0411.16007
[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 0766552 | Zbl 0524.54013
[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 0464203 | Zbl 0342.54032
[13] Marco, G. De: On the countably generated $z$-ideal of $C(X)$. Proc. Amer. Math. Soc. 31 (1972), 574–576. MR 0288563
[14] Nunzetta, P., Plank, D.: Closed ideal in $C(X)$. Proc. Amer. Math. Soc. 35 (2) (1972), 601–606. MR 0303496

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