On ideals of rings of continuous functions associated with sublocales

摘要

Let X be a Tychonoff space. Associated with every subset A subset of beta X is the ideal O-A of the ring C(X) consisting of all functions that vanish in a neighborhood of X boolean AND A. Now, viewing Tychonoff spaces as objects in the category CRLoc of completely regular locales, we have ideals of the form O-A, where A is a sublocale of beta X. In this paper we study properties of such ideals not only for Tychonoff spaces, but for any object in CRLoc. Carrying out the discussion in this category, we have more function rings (they are denoted RL, for any L is an element of CRLoc) than the class of the rings C(X). Pure ideals of C(X) are known to be exactly the ideals O-A, for A a closed subset of beta X. We characterize the spaces for which the ideals O-A, for A a closed subset of X (note that it need not be closed in beta X) are pure ideals. They properly contain the normal ones. We describe the socle of any ring RL as the ideal O-A, with A equal to the join of all nowhere dense sublocales of beta L. We show that the socle is zero precisely when beta L is dense in itself, and essential if and only if the smallest dense sublocale of beta L has a complement in the lattice of sublocales of beta L. If beta L is scattered, then this is so if and only if L has a smallest nowhere dense sublocale. (C) 2020 Elsevier B.V. All rights reserved.