SAMS Subject Classification Number: 2

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 show 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 continuous real-valued functions on \(L\) 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 the 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.