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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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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| author | Joshua Sack Saleem Watson |
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| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-cbed58ed201046efb8f837e4ae0acee1 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
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| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-cbed58ed201046efb8f837e4ae0acee12025-08-20T02:19:58ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252013-01-01201310.1155/2013/635361635361Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of XJoshua Sack0Saleem Watson1Institute of Logic, Language, and Computation, Universiteit van Amsterdam, P.O. Box 94242, 1090 GE Amsterdam, The NetherlandsDepartment of Mathematics, California State University, Long Beach, CA 90840, USALet 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).http://dx.doi.org/10.1155/2013/635361 |
| spellingShingle | Joshua Sack Saleem Watson Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X International Journal of Mathematics and Mathematical Sciences |
| title | Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X |
| title_full | Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X |
| title_fullStr | Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X |
| title_full_unstemmed | Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X |
| title_short | Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X |
| title_sort | characterizations of ideals in intermediate c rings a x via the a compactifications of x |
| url | http://dx.doi.org/10.1155/2013/635361 |
| work_keys_str_mv | AT joshuasack characterizationsofidealsinintermediatecringsaxviatheacompactificationsofx AT saleemwatson characterizationsofidealsinintermediatecringsaxviatheacompactificationsofx |