Characterizations of Ideals in Intermediate -Rings via the -Compactifications of

Let be a completely regular topological space. An intermediate ring is a ring of continuous functions satisfying . In Redlin and Watson (1987) and in Panman et al. (2012), correspondences and are defined between ideals in and -filters on , and it is shown that these extend the well-known correspondences studied separately for and , respectively, to any intermediate ring. Moreover, the inverse map sets up a one-one correspondence between the maximal ideals of and the -ultrafilters on . In this paper, we define a function that, in the case that is a -ring, describes in terms of extensions of functions to realcompactifications of . For such rings, we show that maps -filters to ideals. We also give a characterization of the maximal ideals in that generalize the Gelfand-Kolmogorov theorem from to . 1. Introduction Let be a completely regular space and an intermediate ring of continuous real-valued functions; that is, . It is well known that there is a natural correspondence between ideals of and -filters on X as described in [1, pages 26-27]. Such a correspondence also exists for [1, Problem 2L]. In [2], a correspondence between the ideals of any and the -filters on was introduced, and its properties were further investigated in [3–5]. In [6], another correspondence between ideals of any and -filters on is introduced. It is shown in [6] that the correspondences and extend the correspondences and from and , respectively, to all intermediate rings, and an explicit formula is stated that relates the two correspondences. In this paper, we give a characterization (Definition 3 and Theorem 6) of the correspondence for intermediate -rings in terms of the -compactifications of introduced in [7]. In this setting, we show (Theorem 14) that the inverse map of the set map maps ideals in to -filters on . We also give a characterization of the maximal ideals in . This characterization generalizes from to the Gelfand-Kolmogorov theorem (Theorem 8). We follow the notation in [1, 6]. 2. Preliminaries For convenience we state some of the definitions and results needed in this paper. Following the notation in [1], we set to be the collection of the zero sets of all functions . In this paper, we generally work with functions on a fixed set , as well as some extensions of to larger domains. As expected, then denotes the zero set of on the larger domain. A -filter ( -ultrafilter, resp.) on is the intersection of with a filter (ultrafilter, resp.) on . The kernel of a set of -ultrafilters is defined by One can verify that the kernel of a set of -ultrafilters is a -filter. The hull