On the maximal regular ideal of pointfree function rings, and more

Abstract

Let L be a completely regular frame and, as customary, let RL denote the ring of continuous real-valued functions on L. In the first part of the paper we characterize the maximal regular ideal of RL. We show that it consists precisely of the functions α such that the open sublocale of L associated with coz α is clopen and is a P-frame. We also give a characterization of this ideal in terms of the notion of the localic remainder. In contrast, the maximal regular ideal of the ring ZL of continuous integer-valued functions on L is shown to be always the zero ideal. Purity and regularity of ideals are germane in the description and characterization of the maximal regular ideal of RL. It is for this reason that in the last part of the paper we consider ideals of ideals of RL because, as we prove, it is precisely the pure ideals of RL all of whose ideals are ideals in the whole ring.