# Correspondences between ideals and $z$-filters for rings of continuous functions between ${C}^{\ast}$ and $C$

Phyllis Panman; Joshua Sack; Saleem Watson

Commentationes Mathematicae (2012)

- Volume: 52, Issue: 1
- ISSN: 2080-1211

## Access Full Article

top## Abstract

top## How to cite

topPhyllis Panman, Joshua Sack, and Saleem Watson. "Correspondences between ideals and $z$-filters for rings of continuous functions between $C^∗$ and $C$." Commentationes Mathematicae 52.1 (2012): null. <http://eudml.org/doc/291864>.

@article{PhyllisPanman2012,

abstract = {Let $X$ be a completely regular topological space. Let $A(X)$ be a ring of continuous functions between $C^∗(X)$ and $C(X)$, that is, $C^∗(X) \subset A(X) \subset C(X)$. In [9], a correspondence $\mathcal \{Z\}_A$ between ideals of $A(X)$ and $z$-filters on $X$ is defined. Here we show that $\mathcal \{Z\}_A$ extends the well-known correspondence for $C^∗(X)$ to all rings $A(X)$. We define a new correspondence $\mathcal \{Z\}_A$ and show that it extends the well-known correspondence for $C(X)$ to all rings $A(X)$. We give a formula that relates the two correspondences. We use properties of $\mathcal \{Z\}_A$ and $\mathcal \{Z\}_A$ to characterize $C^∗(X)$ and $C(X)$ among all rings $A(X)$. We show that $\mathcal \{Z\}_A$ defines a one-one correspondence between maximal ideals in $A(X)$ and the $z$-ultrafilters in $X$.},

author = {Phyllis Panman, Joshua Sack, Saleem Watson},

journal = {Commentationes Mathematicae},

keywords = {Rings of continuous functions; Ideals; $z$-filters; Kernel; Hull},

language = {eng},

number = {1},

pages = {null},

title = {Correspondences between ideals and $z$-filters for rings of continuous functions between $C^∗$ and $C$},

url = {http://eudml.org/doc/291864},

volume = {52},

year = {2012},

}

TY - JOUR

AU - Phyllis Panman

AU - Joshua Sack

AU - Saleem Watson

TI - Correspondences between ideals and $z$-filters for rings of continuous functions between $C^∗$ and $C$

JO - Commentationes Mathematicae

PY - 2012

VL - 52

IS - 1

SP - null

AB - Let $X$ be a completely regular topological space. Let $A(X)$ be a ring of continuous functions between $C^∗(X)$ and $C(X)$, that is, $C^∗(X) \subset A(X) \subset C(X)$. In [9], a correspondence $\mathcal {Z}_A$ between ideals of $A(X)$ and $z$-filters on $X$ is defined. Here we show that $\mathcal {Z}_A$ extends the well-known correspondence for $C^∗(X)$ to all rings $A(X)$. We define a new correspondence $\mathcal {Z}_A$ and show that it extends the well-known correspondence for $C(X)$ to all rings $A(X)$. We give a formula that relates the two correspondences. We use properties of $\mathcal {Z}_A$ and $\mathcal {Z}_A$ to characterize $C^∗(X)$ and $C(X)$ among all rings $A(X)$. We show that $\mathcal {Z}_A$ defines a one-one correspondence between maximal ideals in $A(X)$ and the $z$-ultrafilters in $X$.

LA - eng

KW - Rings of continuous functions; Ideals; $z$-filters; Kernel; Hull

UR - http://eudml.org/doc/291864

ER -

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.