Algebra universalis

, 79:7 | Cite as

Some connections between frames of radical ideals and frames of z-ideals

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Abstract

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.

Keywords

Frame Radical ideal z-ideal Saturation Epicompletion Exchange ring Baer ring Normal ring 

Mathematics Subject Classification

06D22 13A15 18A22 54C30 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of South AfricaPretoriaSouth Africa

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