On minimal ideals in the ring of real-valued [4pt] continuous functions on a frame

Abolghasem Karimi Feizabadi, Ali Akbar Estaji, and Mostafa Abedi *

Address:
A. Karimi Feizabadi, Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran
A. A. Estaji, Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran
M. Abedi, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

E-mail:
akarimi@gorganiau.ac.ir
aaestaji@hsu.ac.ir
abedi@esfarayen.ac.ir
ms_abedi@yahoo.com

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 $L$.

AMSclassification: primary 06D22; secondary 54C30, 13A15.

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.

DOI: 10.5817/AM2018-1-1