- Characterizations of Ideals in Intermediate C-rings A(X) via the A-compactifications of X
- International journal of mathematics and mathematical sciences
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- Interfacultary Research Institutes
Faculty of Science (FNWI)
- Institute for Logic, Language and Computation (ILLC)
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 ZA and ZA 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 KA that, in the case that A(X) is a C-ring, describes ZA 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).
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