On the ideal of functions with compact support in pointfree function rings

Abstract

As usual, let $\mathcal{R}L$ denote the ring of real-valued continuous functions on a completely regular frame L. We consider the ideals $\mathcal{R}_{s}(L)$ and $\mathcal{R}_{K}(L)$ consisting, respectively, of functions with small cozero elements and functions with compact support. We show that, as in the classical case of C(X), the first ideal is the intersection of all free maximal ideals, and the second is the intersection of pure parts of all free maximal ideals. A corollary of this latter result is that, in fact, $\mathcal{R}_{K}(L)$ is the intersection of all free ideals. Each of these ideals is pure, free, essential or zero iff the other has the same feature. We observe that these ideals are free iff L is a continuous frame, and essential iff L is almost continuous (meaning that below every nonzero element there is a nonzero element the pseudocomplement of which induces a compact closed quotient). We also show that these ideals are zero iff L is nowhere compact (meaning that non-dense elements induce non-compact closed quotients).