The pointfree version of 𝓒 _c (X) via the ranges of functions

Abstract

Let R α := {r ∈ ℝ : coz(α − r) ≠ → p} for every α ∈ 𝓡(L). The ring 𝓒 c (L) is introduced as a pointfree version of the subring 𝓒 c (X) of C(X) by R α . In this paper, we show that 𝓒 c (X) is a z-good ring and every radical ideal in it is an absolutely convex ideal. Also, we study this result which for any frame L, there exists a zero-dimensional frame M, which is a continuous image of L and 𝓒 c (L) ≅ 𝓒 c (M).