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

Karimi Feizabadi, Abolghasem   ; Estaji, Ali Akbar   ; Abedi, Mostafa

Archivum mathematicum, Tome 54 (2018) no. 1, pp. 1-13

Abstract (VO)

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 $L$.

Détail

DOI : 10.5817/AM2018-1-1

Classification : 06D22, 13A15, 54C30
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

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