Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X. (17th July 2013)

• Record Type: Journal Article
• Authors: Sack, Joshua; Watson, Saleem
• Other Names: Dobbs David Academic Editor.
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 2013(2013)
• Publication Date: 2013-07-17
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1155/2013/635361
• Languages: English
• ISSNs: 0161-1712

Abstract

Abstract : Let X be a completely regular topological space. An intermediate ring is a ring A ( X ) of continuous functions satisfying C * ( X ) ⊆ A ( X ) ⊆ C ( X ) . In Redlin and Watson (1987) and in Panman et al. (2012), correspondences 𝒵 A and ℨ A are defined between ideals in A ( X ) and z -filters on X, and it is shown that these extend the well-known correspondences studied separately for C ∗ ( X ) and C ( X ), respectively, to any intermediate ring. Moreover, the inverse map 𝒵 A ← sets up a one-one correspondence between the maximal ideals of A ( X ) and the z -ultrafilters on X . In this paper, we define a function 𝔎 A that, in the case that A ( X ) is a C -ring, describes ℨ A in terms of extensions of functions to realcompactifications of X . For such rings, we show that ℨ A ← maps z -filters to ideals. We also give a characterization of the maximal ideals in A ( X ) that generalize the Gelfand-Kolmogorov theorem from C ( X ) to A ( X ) .