Some connections between frames of radical ideals and frames of z-ideals
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For any semisimple f-ring A with bounded inversion, we show that the frame of radical ideals of A and the frame of z-ideals of A have isomorphic subfit coreflections. If we assume the Axiom of Choice, then the two coreflections are actually identical. If the f-ring has the property that the sum of two z-ideals is a z-ideal (as in the case of rings of continuous functions), then the epicompletion of the frame of z-ideals is shown to be a dense quotient of the epicompletion of the frame of radical ideals. Baer rings, exchange rings, and normal rings that lie in the class of semisimple f-rings with bounded inversion are shown to have characterizations in terms of frames of z-ideal which are similar to characterizations in terms of frames of radical ideals.
KeywordsFrame Radical ideal z-ideal Saturation Epicompletion Exchange ring Baer ring Normal ring
Mathematics Subject Classification06D22 13A15 18A22 54C30
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