# Strongly fixed ideals in $ C (L)$ and compact frames

## A. A. Estaji, A. Karimi Feizabadi, and M. Abedi

Address:

A.A. Estaji and M. Abedi, Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran

A. Karimi Feizabadi, Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran

E-mail:

aaestaji@hsu.ac.ir

ms.abedi@hsu.ac.ir

akarimi@gorganiau.ac.ir

Abstract: Let $C(L)$ be the ring of real-valued continuous functions on a frame $L$. In this paper, strongly fixed ideals and characterization of maximal ideals of $C(L)$ which is used with strongly fixed are introduced. In the case of weakly spatial frames this characterization is equivalent to the compactness of frames. Besides, the relation of the two concepts, fixed and strongly fixed ideals of $C(L)$, is studied particularly in the case of weakly spatial frames. The concept of weakly spatiality is actually weaker than spatiality and they are equivalent in the case of conjunctive frames. Assuming Axiom of Choice, compact frames are weakly spatial.

AMSclassification: primary 06D22; secondary 13A15, 13C99.

Keywords: frame, ring of real-valued continuous functions, weakly spatial frame, fixed and strongly fixed ideal.

DOI: 10.5817/AM2015-1-1