Intersections of minimal prime ideals in the rings of continuous functions

Swapan Kumar Ghosh

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 4, page 623-632
  • ISSN: 0010-2628

Abstract

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A space X is called μ -compact by M. Mandelker if the intersection of all free maximal ideals of C ( X ) coincides with the ring C K ( X ) of all functions in C ( X ) with compact support. In this paper we introduce φ -compact and φ ' -compact spaces and we show that a space is μ -compact if and only if it is both φ -compact and φ ' -compact. We also establish that every space X admits a φ -compactification and a φ ' -compactification. Examples and counterexamples are given.

How to cite

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Ghosh, Swapan Kumar. "Intersections of minimal prime ideals in the rings of continuous functions." Commentationes Mathematicae Universitatis Carolinae 47.4 (2006): 623-632. <http://eudml.org/doc/249883>.

@article{Ghosh2006,
abstract = {A space $X$ is called $\mu $-compact by M. Mandelker if the intersection of all free maximal ideals of $C(X)$ coincides with the ring $C_K(X)$ of all functions in $C(X)$ with compact support. In this paper we introduce $\phi $-compact and $\phi ^\{\prime \}$-compact spaces and we show that a space is $\mu $-compact if and only if it is both $\phi $-compact and $\phi ^\{\prime \}$-compact. We also establish that every space $X$ admits a $\phi $-compactification and a $\phi ^\{\prime \}$-compactification. Examples and counterexamples are given.},
author = {Ghosh, Swapan Kumar},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {minimal prime ideal; $P$-space; $F$-space; $\mu $-compact space; $\phi $-compact space; $\phi ^\{\prime \}$-compact space; round subset; almost round subset; nearly round subset; minimal prime ideal; -space; -space; -compact space; -compact space},
language = {eng},
number = {4},
pages = {623-632},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Intersections of minimal prime ideals in the rings of continuous functions},
url = {http://eudml.org/doc/249883},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Ghosh, Swapan Kumar
TI - Intersections of minimal prime ideals in the rings of continuous functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 4
SP - 623
EP - 632
AB - A space $X$ is called $\mu $-compact by M. Mandelker if the intersection of all free maximal ideals of $C(X)$ coincides with the ring $C_K(X)$ of all functions in $C(X)$ with compact support. In this paper we introduce $\phi $-compact and $\phi ^{\prime }$-compact spaces and we show that a space is $\mu $-compact if and only if it is both $\phi $-compact and $\phi ^{\prime }$-compact. We also establish that every space $X$ admits a $\phi $-compactification and a $\phi ^{\prime }$-compactification. Examples and counterexamples are given.
LA - eng
KW - minimal prime ideal; $P$-space; $F$-space; $\mu $-compact space; $\phi $-compact space; $\phi ^{\prime }$-compact space; round subset; almost round subset; nearly round subset; minimal prime ideal; -space; -space; -compact space; -compact space
UR - http://eudml.org/doc/249883
ER -

References

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  1. Gillman L., Jerison M., Rings of Continuous Functions, University Series in Higher Math., Van Nostrand, Princeton, New Jersey, 1960. Zbl0327.46040MR0116199
  2. Henriksen M., Jerison M., The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130. (1965) Zbl0147.29105MR0194880
  3. Johnson D.G., Mandelker M., Functions with pseudocompact support, General Topology Appl. 3 (1973), 331-338. (1973) Zbl0277.54009MR0331310
  4. Mandelker M., Round z -filters and round subsets of β X , Israel J. Math. 7 (1969), 1-8. (1969) Zbl0174.25604MR0244951
  5. Mandelker M., Supports of continuous functions, Trans. Amer. Math. Soc. 156 (1971), 73-83. (1971) Zbl0197.48703MR0275367

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