{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T15:35:44Z","timestamp":1759073744126,"version":"3.40.5"},"reference-count":9,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"The Netherlands Organisation for Scientific Research","award":["639.072.904"],"award-info":[{"award-number":["639.072.904"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>be a completely regular topological space. An intermediate ring is a ring<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>A<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>of continuous functions satisfying<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:msup><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>*<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2286<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2286<\/mml:mo><mml:mi>C<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. In Redlin and Watson (1987) and in Panman et al. (2012), correspondences<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:msub><mml:mrow><mml:mi>\ud835\udcb5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:msub><mml:mrow><mml:mi>\u2128<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>are defined between ideals in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:mi>A<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:mi>z<\/mml:mi><\/mml:mrow><\/mml:math>-filters on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>, and it is shown that these extend the well-known correspondences studied separately for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:msup><mml:mi>C<\/mml:mi><mml:mo>\u2217<\/mml:mo><\/mml:msup><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mrow><mml:mi>C<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>, respectively, to any intermediate ring. Moreover, the inverse map<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>\ud835\udcb5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u2190<\/mml:mo><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>sets up a one-one correspondence between the maximal ideals of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:mi>A<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>and the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mrow><mml:mi>z<\/mml:mi><\/mml:mrow><\/mml:math>-ultrafilters on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>. In this paper, we define a function<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\"><mml:mrow><mml:msub><mml:mrow><mml:mi>\ud835\udd0e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>that, in the case that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M20\"><mml:mrow><mml:mi>A<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>is a<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M21\"><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><\/mml:math>-ring, describes<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M22\"><mml:mrow><mml:msub><mml:mrow><mml:mi>\u2128<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>in terms of extensions of functions to realcompactifications of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M23\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>. For such rings, we show that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M24\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>\u2128<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u2190<\/mml:mo><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>maps<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M25\"><mml:mrow><mml:mi>z<\/mml:mi><\/mml:mrow><\/mml:math>-filters to ideals. We also give a characterization of the maximal ideals in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M26\"><mml:mrow><mml:mi>A<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>that generalize the Gelfand-Kolmogorov theorem from<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M27\"><mml:mrow><mml:mi>C<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>to<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M28\"><mml:mrow><mml:mi>A<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2013\/635361","type":"journal-article","created":{"date-parts":[[2013,7,17]],"date-time":"2013-07-17T17:01:29Z","timestamp":1374080489000},"page":"1-6","source":"Crossref","is-referenced-by-count":3,"title":["Characterizations of Ideals in Intermediate<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><\/mml:math>-Rings<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>A<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:math>via the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:math>-Compactifications of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>"],"prefix":"10.1155","volume":"2013","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6031-0359","authenticated-orcid":true,"given":"Joshua","family":"Sack","sequence":"first","affiliation":[{"name":"Institute of Logic, Language, and Computation, Universiteit van Amsterdam, P.O. 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