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The ring R c L is introduced as a sub-f-ring of RL as a pointfree analogue to the subring C c (X) of C(X) consisting of elements with the countable image. We introduce z c-ideals in R c L and study their properties. We prove that for any frame L, there exists a space X such that βL OX with C c (X) R c (OX) R c βL R * c L, and from this, we conclude that if α, β ∈ R c L, |α| ≤ |β| q for some q > 1, then α is a multiple of β in R c L. Also, we show that IJ = I ∩ J whenever I and J are z c-ideals. In particular, we prove that an ideal of R c L is a z c-ideal if and only if it is a z-ideals. In addition, we study the relation between z c-ideals and prime ideals in R c L. Finally, we prove that R c L is a Gelfand ring.

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.2010 Mathematics Subject Classification: primary 06D22; secondary 13A15, 13C99. Key words and phrases: frame, ring of real-valued continuous functions, weakly spatial frame, fixed and strongly fixed ideal.

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There are many results proved using the Axiom of Choice. Using point-free topology, we can prove some of these results without using this axiom. B. Banaschewski in [Pointfree Topology and the Spectra of f-rings, Ordered algebraic structures (Curacoa, 1995), Kluwer, Dordrecht, 123-148], studying the spectra of f-rings, describes the point-free version of the classical Gelfand duality without using the Axiom of Choice In this paper, referring to [Ebrahimi, M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct. 12 (2004), 397-409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of '-Modules, To appear in J. Pure Appl. Algebra], we describe a point-free version of the classical Kakutani duality. For this, using one of the spectra given in [Ebrahimi, M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct. 12 (2004), 397-409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of l-Modules, To appear in J. Pure Appl. Algebra], we find an adjunction between the category of compact completely regular frames with frame maps and the category of Archimedean bounded Riesz spaces with continuous Riesz maps. Mathematics Subject Classification (2000): 06D22, 46A40, 46B40, 46B42, 03E25, 03E99.