# An interesting class of ideals in subalgebras of $C\left(X\right)$ containing ${C}^{*}\left(X\right)$

Sudip Kumar Acharyya; Dibyendu De

Commentationes Mathematicae Universitatis Carolinae (2007)

- Volume: 48, Issue: 2, page 273-280
- ISSN: 0010-2628

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topAcharyya, Sudip Kumar, and De, Dibyendu. "An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$." Commentationes Mathematicae Universitatis Carolinae 48.2 (2007): 273-280. <http://eudml.org/doc/250211>.

@article{Acharyya2007,

abstract = {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.},

author = {Acharyya, Sudip Kumar, De, Dibyendu},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {Stone-Čech compactification; rings of continuous functions; maximal ideals; $z^\{\beta \}_A$-ideals; Stone-Čech compactification; rings of continuous functions; maximal ideals; -ideals},

language = {eng},

number = {2},

pages = {273-280},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$},

url = {http://eudml.org/doc/250211},

volume = {48},

year = {2007},

}

TY - JOUR

AU - Acharyya, Sudip Kumar

AU - De, Dibyendu

TI - An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2007

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 48

IS - 2

SP - 273

EP - 280

AB - 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.

LA - eng

KW - Stone-Čech compactification; rings of continuous functions; maximal ideals; $z^{\beta }_A$-ideals; Stone-Čech compactification; rings of continuous functions; maximal ideals; -ideals

UR - http://eudml.org/doc/250211

ER -

## References

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- Byun H.L., Watson S., Prime and maximal ideals of $C\left(X\right)$, Topology Appl. 40 (1991), 45-62. (1991) MR1114090
- De D., Acharyya S.K., Characterization of function rings between ${C}^{*}\left(X\right)$ and $C\left(X\right)$, Kyungpook Math. J. 46 (2006), 503-507. (2006) Zbl1120.54014MR2282652
- Dominguege J.M., Gomez J., Mulero M.A., Intermediate algebras between ${C}^{*}\left(X\right)$ and $C\left(X\right)$ as rings of fractions of ${C}^{*}\left(X\right)$, Topology Appl. 77 (1997), 115-130. (1997) MR1451646
- Gillman L., Jerison M., Rings of Continuous Functions, Springer, New York, 1976. Zbl0327.46040MR0407579
- Henriksen M., Johnson D.G., On the structure of a class of archimedean lattice ordered algebras, Fund. Math. 50 (1961), 73-94. (1961) Zbl0099.10101MR0133698
- Plank D., On a class of subalgebras of $C\left(X\right)$ with application to $\beta X-X$, Fund. Math. 64 (1969), 41-54. (1969) MR0244953
- Redlin L., Watson S., Maximal ideals in subalgebras of $C\left(X\right)$, Proc. Amer. Math. Soc. 100 (1987), 763-766. (1987) Zbl0622.54011MR0894451

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