Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X
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 thes...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2013-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2013/635361 |
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| Summary: | 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 𝒵A←
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 ℨA← 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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| ISSN: | 0161-1712 1687-0425 |