Geodesic
An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 273-280 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In the present paper we give a duality between a special type of ideals of subalgebras of $C(X)$ containing $C^*(X)$ and $z$-filters of $\beta X$ by generalization of the notion $z$-ideal of $C(X)$. We also use it to establish some intersecting properties of prime ideals lying between $C^*(X)$ and $C(X)$. For instance we may mention that such an ideal becomes prime if and only if it contains a prime ideal. Another interesting one is that for such an ideal the residue class ring is totally ordered if and only if it is prime.
In the present paper we give a duality between a special type of ideals of subalgebras of $C(X)$ containing $C^*(X)$ and $z$-filters of $\beta X$ by generalization of the notion $z$-ideal of $C(X)$. We also use it to establish some intersecting properties of prime ideals lying between $C^*(X)$ and $C(X)$. For instance we may mention that such an ideal becomes prime if and only if it contains a prime ideal. Another interesting one is that for such an ideal the residue class ring is totally ordered if and only if it is prime.
Classification : 54C30, 54C35, 54C40, 54D35
Keywords: Stone-Čech compactification; rings of continuous functions; maximal ideals; $z^{\beta}_A$-ideals 
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     title = {An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {273--280},
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     zbl = {1199.54153},
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Acharyya, Sudip Kumar; De, Dibyendu. An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 2, pp. 273-280. http://geodesic.mathdoc.fr/item/CMUC_2007_48_2_a8/