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1 results

  • REAL IDEALS IN POINTFREE RINGS OF CONTINUOUS FUNCTIONS

  • Part of
  • THEMBA DUBE
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 83 / Issue 2 / April 2011
    Published online by Cambridge University Press:
    06 December 2010, pp. 338-352
    Print publication:
    April 2011
    • Article
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  • Real ideals of the ring ℜL of real-valued continuous functions on a completely regular frame L are characterized in terms of cozero elements, in the manner of the classical case of the rings C(X). As an application, we show that L is realcompact if and only if every free maximal ideal of ℜL is hyper-real—which is the precise translation of how Hewitt defined realcompact spaces, albeit under a different appellation. We also obtain a frame version of Mrówka’s theorem that characterizes realcompact spaces.