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International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 635361, 6 pages
Research Article

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

1Institute of Logic, Language, and Computation, Universiteit van Amsterdam, P.O. Box 94242, 1090 GE Amsterdam, The Netherlands
2Department of Mathematics, California State University, Long Beach, CA 90840, USA

Received 8 February 2013; Accepted 14 June 2013

Academic Editor: David Dobbs

Copyright © 2013 Joshua Sack and Saleem Watson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 .