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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of South Africa, P.O. Box 392, Pretoria, 0003, South Africa

    Themba Dube

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  1. Themba Dube
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Correspondence to Themba Dube.

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Presented by W. Wm. McGovern.

This article is dedicated to Professor Jorge Martínez.

The research was supported by the National Research Foundation of South Africa under Grant no. 93514.

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Dube, T. Some connections between frames of radical ideals and frames of z-ideals. Algebra Univers. 79, 7 (2018). https://doi.org/10.1007/s00012-018-0494-z

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  • DOI: https://doi.org/10.1007/s00012-018-0494-z

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