On ideals of rings of continuous integer-valued functions on a frame

Abstract

Let $L$ be a zero-dimensional frame and $\mathfrak ZL$ be the ring of integer-valued continuous functions on $L$. We associate with each sublocale of $\zeta L$, the Banaschewski compactification of $L$, an ideal of $\mathfrak ZL$, and study the behaviour of these types of ideals. The socle of $\mathfrak ZL$ is shown to be always the zero ideal, in contrast with the fact that the socle of the ring $\mathcal RL$ of real-valued continuous functions of $L$ is not necessarily the zero ideal. The ring $\mathfrak ZL$ has been shown by B. Banaschewski to be (isomorphic to) a subring of $\mathcal RL$, so that ideals of the larger ring can be contracted to the smaller one. We show that the contraction of the socle of $\mathcal RL$ to $\mathfrak ZL$ is the ideal of $\mathfrak ZL$ associated with the join (in the coframe of sublocales of $\zeta L)$ of all nowhere dense sublocales of $\zeta L$. It also appears in other guises.