Comaximal graph of C ( X )

Mehdi Badie

Commentationes Mathematicae Universitatis Carolinae (2016)

  • Volume: 57, Issue: 3, page 353-364
  • ISSN: 0010-2628

Abstract

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In this article we study the comaximal graph Γ 2 ' C ( X ) of the ring C ( X ) . We have tried to associate the graph properties of Γ 2 ' C ( X ) , the ring properties of C ( X ) and the topological properties of X . Radius, girth, dominating number and clique number of the Γ 2 ' C ( X ) are investigated. We have shown that 2 ≤ Rad Γ 2 ' C ( X ) ≤ 3 and if | X | > 2 then girth Γ 2 ' C ( X ) = 3 . We give some topological properties of X equivalent to graph properties of Γ 2 ' C ( X ) . Finally we have proved that X is an almost P -space which does not have isolated points if and only if C ( X ) is an almost regular ring which does not have any principal maximal ideals if and only if Rad Γ 2 ' C ( X ) = 3 .

How to cite

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Badie, Mehdi. "Comaximal graph of $C(X)$." Commentationes Mathematicae Universitatis Carolinae 57.3 (2016): 353-364. <http://eudml.org/doc/286784>.

@article{Badie2016,
abstract = {In this article we study the comaximal graph $\Gamma ^\{\prime \}_\{_2\}C(X)$ of the ring $C(X)$. We have tried to associate the graph properties of $\Gamma ^\{\prime \}_\{_2\}C(X)$, the ring properties of $C(X)$ and the topological properties of $X$. Radius, girth, dominating number and clique number of the $\Gamma ^\{\prime \}_\{_2\}C(X)$ are investigated. We have shown that $2\le \operatorname\{Rad\}\Gamma ^\{\prime \}_\{_2\}C(X) \le 3$ and if $|X|> 2$ then $\mathrm \{girth \} \Gamma ^\{\prime \}_\{_2\}C(X)= 3$. We give some topological properties of $X$ equivalent to graph properties of $\Gamma ^\{\prime \}_\{_2\}C(X)$. Finally we have proved that $X$ is an almost $P$-space which does not have isolated points if and only if $C(X)$ is an almost regular ring which does not have any principal maximal ideals if and only if $\operatorname\{Rad\}\Gamma ^\{\prime \}_\{_2\}C(X)= 3$.},
author = {Badie, Mehdi},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {rings of continuous functions; comaximal graph; radius; girth; dominating number; clique number; zero cellularity; $P$-space; almost $P$-space; connected space; regular ring},
language = {eng},
number = {3},
pages = {353-364},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Comaximal graph of $C(X)$},
url = {http://eudml.org/doc/286784},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Badie, Mehdi
TI - Comaximal graph of $C(X)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 3
SP - 353
EP - 364
AB - In this article we study the comaximal graph $\Gamma ^{\prime }_{_2}C(X)$ of the ring $C(X)$. We have tried to associate the graph properties of $\Gamma ^{\prime }_{_2}C(X)$, the ring properties of $C(X)$ and the topological properties of $X$. Radius, girth, dominating number and clique number of the $\Gamma ^{\prime }_{_2}C(X)$ are investigated. We have shown that $2\le \operatorname{Rad}\Gamma ^{\prime }_{_2}C(X) \le 3$ and if $|X|> 2$ then $\mathrm {girth } \Gamma ^{\prime }_{_2}C(X)= 3$. We give some topological properties of $X$ equivalent to graph properties of $\Gamma ^{\prime }_{_2}C(X)$. Finally we have proved that $X$ is an almost $P$-space which does not have isolated points if and only if $C(X)$ is an almost regular ring which does not have any principal maximal ideals if and only if $\operatorname{Rad}\Gamma ^{\prime }_{_2}C(X)= 3$.
LA - eng
KW - rings of continuous functions; comaximal graph; radius; girth; dominating number; clique number; zero cellularity; $P$-space; almost $P$-space; connected space; regular ring
UR - http://eudml.org/doc/286784
ER -

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