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

Abstract

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