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On the countably generated $ z$-ideals of $ C(X)$


Author: G. De Marco
Journal: Proc. Amer. Math. Soc. 31 (1972), 574-576
DOI: https://doi.org/10.1090/S0002-9939-1972-0288563-3
MathSciNet review: 0288563
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Abstract | References | Additional Information

Abstract: A necessary and sufficient condition for the countable generation of certain $ z$-ideals of $ C(X)$ is given. In particular, for $ X$ compact, the countably generated $ z$-ideals of $ C(X)$ are the sets of all functions which vanish on a neighborhood of some zero-set of $ X$. Any finitely generated semiprime ideal of $ C(X)$ is generated by an idempotent.


References [Enhancements On Off] (What's this?)

  • [D] W. E. Dietrich, Jr., On the ideal structure of $ C(X)$, Trans. Amer. Math. Soc. 152 (1970), 61-77. MR 0265941 (42:850)
  • [G] L. Gillman, Countably generated ideals in rings of continuous functions, Proc. Amer. Math. Soc. 11 (1960), 660-666. MR 27 #6120. MR 0156189 (27:6120)
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [K] C. W. Kohls, A note on countably generated ideals in rings of continuous functions, Proc. Amer. Math. Soc. 12 (1961), 744-749. MR 26 #617. MR 0143051 (26:617)
  • [M] M. Mandelker, Round $ z$-filters and round subsets of $ \beta X$, Israel J. Math. 7 (1969), 1-8. MR 39 #6264. MR 0244951 (39:6264)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0288563-3
Keywords: $ z$-ideal, countably generated $ z$-ideal, semiprime ideal
Article copyright: © Copyright 1972 American Mathematical Society

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