On minimal ideals in the ring of real-valued continuous functions on a frame

Abolghasem Karimi Feizabadi; Ali Akbar Estaji; Mostafa Abedi

Archivum Mathematicum (2018)

  • Volume: 054, Issue: 1, page 1-13
  • ISSN: 0044-8753

Abstract

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Let ℛ L be the ring of real-valued continuous functions on a frame L . The aim of this paper is to study the relation between minimality of ideals I of ℛ L and the set of all zero sets in L determined by elements of I . To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame L , it is proved that the f -ring ℛ L is isomorphic to the f -ring C ( Σ L ) of all real continuous functions on the topological space Σ L . Finally, a one-one correspondence is presented between the set of isolated points of Σ L and the set of atoms of L .

How to cite

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Karimi Feizabadi, Abolghasem, Estaji, Ali Akbar, and Abedi, Mostafa. "On minimal ideals in the ring of real-valued continuous functions on a frame." Archivum Mathematicum 054.1 (2018): 1-13. <http://eudml.org/doc/294640>.

@article{KarimiFeizabadi2018,
abstract = {Let $\mathcal \{R\}L$ be the ring of real-valued continuous functions on a frame $L$. The aim of this paper is to study the relation between minimality of ideals $I$ of $\mathcal \{R\}L$ and the set of all zero sets in $L$ determined by elements of $I$. To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame $L$, it is proved that the $f$-ring $\mathcal \{R\}L$ is isomorphic to the $f$-ring $ C(\Sigma L)$ of all real continuous functions on the topological space $\Sigma L$. Finally, a one-one correspondence is presented between the set of isolated points of $\Sigma L$ and the set of atoms of $L$.},
author = {Karimi Feizabadi, Abolghasem, Estaji, Ali Akbar, Abedi, Mostafa},
journal = {Archivum Mathematicum},
keywords = {ring of real-valued continuous functions on a frame; coz-disjoint; coz-dense and coz-spatial frames; zero sets in pointfree topology; $z$-ideal; strongly $z$-ideal},
language = {eng},
number = {1},
pages = {1-13},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On minimal ideals in the ring of real-valued continuous functions on a frame},
url = {http://eudml.org/doc/294640},
volume = {054},
year = {2018},
}

TY - JOUR
AU - Karimi Feizabadi, Abolghasem
AU - Estaji, Ali Akbar
AU - Abedi, Mostafa
TI - On minimal ideals in the ring of real-valued continuous functions on a frame
JO - Archivum Mathematicum
PY - 2018
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 054
IS - 1
SP - 1
EP - 13
AB - Let $\mathcal {R}L$ be the ring of real-valued continuous functions on a frame $L$. The aim of this paper is to study the relation between minimality of ideals $I$ of $\mathcal {R}L$ and the set of all zero sets in $L$ determined by elements of $I$. To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame $L$, it is proved that the $f$-ring $\mathcal {R}L$ is isomorphic to the $f$-ring $ C(\Sigma L)$ of all real continuous functions on the topological space $\Sigma L$. Finally, a one-one correspondence is presented between the set of isolated points of $\Sigma L$ and the set of atoms of $L$.
LA - eng
KW - ring of real-valued continuous functions on a frame; coz-disjoint; coz-dense and coz-spatial frames; zero sets in pointfree topology; $z$-ideal; strongly $z$-ideal
UR - http://eudml.org/doc/294640
ER -

References

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