An interesting class of ideals in subalgebras of C ( X ) containing C * ( X )

Sudip Kumar Acharyya; Dibyendu De

Commentationes Mathematicae Universitatis Carolinae (2007)

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

Abstract

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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 β 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.

How to cite

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Acharyya, 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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  1. Acharyya S.K., Chattopadhyay K.C., Ghosh D.P., A class of subalgebras of C ( X ) and the associated compactness, Kyungpook Math. J. 41 2 (2001), 323-324. (2001) Zbl1012.54024MR1876202
  2. Byun H.L., Watson S., Prime and maximal ideals of C ( X ) , Topology Appl. 40 (1991), 45-62. (1991) MR1114090
  3. De D., Acharyya S.K., Characterization of function rings between C * ( X ) and C ( X ) , Kyungpook Math. J. 46 (2006), 503-507. (2006) Zbl1120.54014MR2282652
  4. Dominguege J.M., Gomez J., Mulero M.A., Intermediate algebras between C * ( X ) and C ( X ) as rings of fractions of C * ( X ) , Topology Appl. 77 (1997), 115-130. (1997) MR1451646
  5. Gillman L., Jerison M., Rings of Continuous Functions, Springer, New York, 1976. Zbl0327.46040MR0407579
  6. 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
  7. Plank D., On a class of subalgebras of C ( X ) with application to β X - X , Fund. Math. 64 (1969), 41-54. (1969) MR0244953
  8. Redlin L., Watson S., Maximal ideals in subalgebras of C ( X ) , Proc. Amer. Math. Soc. 100 (1987), 763-766. (1987) Zbl0622.54011MR0894451

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