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

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