Sublocale-based ideals of rings of integer-valued functions

Abstract

Let L be a zero-dimensional frame and ZL be the ring of continuous integer-valued functions on L. We associate with each sublocale of L, the Banaschewski compacti cation 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 continuous real-valued functions on 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 the ideals of the larger ring can be contracted to the smaller one. We show that the contraction of the socle of RL to ZL is the ideal of ZL associated with the join (in the coframe of sublocales of L) of all nowhere dense sublocales of L. This ideal also appears in other guises which give it a socle-like appearance. For a zero-dimensional Hausdor space X, denote, as usual, by C(X; Z) the ring of continuous integer-valued functions on X. If f 2 C(X; Z), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set CK(X; Z) = ff 2 C(X; Z) j clX(X r Z(f)) is compactg is the integer-valued analogue of the ideal of functions with compact support in C(X). By rst working in the category of locales and then interpreting the results in spaces, we characterise this ideal in several ways. Writing X for the Banaschewski compacti cation of X, we also explore some properties of ideals of C(X; Z) associated with subspaces of X analogously to how one associates, for any Tychono space Y , subsets of Y with ideals of C(Y ). If A is a commutative ring with identity and B is a subring of A, the pair B A is said to satisfy the Lying Over property (abbreviated \LO property") if Spec(B) = fB \ P j P 2 Spec(A)g. This concept was introduced by Kaplansky. For such a pair, we study the relationship between the radical ideals of the subring and the contractions of radical ideals of the bigger ring.

This we do in the category AFrm of algebraic frames. We show that the pair B A satis es the LO property if and only if the induced morphism RId(B) ! RId(A) is a monomorphism in this category. A stronger property is one that requires over and above the LO property that Max(B) = fB \ M j M 2 Max(A)g. We call it the Strong Lying Over property (SLO property). As shown by Rudd, it is satis ed by any pair I + R C(X), where I is an ideal of C(X). We show, among other things, that in a class of rings properly containing all the rings C(X), if B A satis es the SLO property, then the z-ideals of the smaller ring are precisely the contractions to it of the z-ideals of the bigger ring.