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Commutative Singular f-Rings

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Ordered Algebraic Structures

Abstract

This paper examines commutative f-rings A with identity 1, for which 1 is a singular element; (i.e., such that 0 ≤ s ≤ 1 implies that s ∧ (1 — s) = 0.) One of the main results is that for such f-rings the following are equivalent: (a) A is semihereditary; (b) A is a Prüfer ring; (c) the weak dimension of A does not exceed 1; (d) every subring B of the maximum ring of quotients QA which contains A is flat over A; (e) the lattice of all ideals of A is distributive. In the final section singular f-rings which are I-rings are discussed. It is shown that C(X, ℤ) is an I-ring precisely when X is an extremally disconnected almost P-space.

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Author information

Authors and Affiliations

  1. University of Florida, Gainesville, FL, 32611, USA

    Robert T. Finn, Jorge Martinez & Warren W. McGovern

Authors
  1. Robert T. Finn
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  2. Jorge Martinez
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  3. Warren W. McGovern
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Bowling Green State University, Bowling Green, Ohio, USA

    W. Charles Holland

  2. Caribbean Mathematics Foundation, Department of Mathematics, University of Florida, Gainesville, Florida, USA

    Jorge Martinez

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© 1997 Springer Science+Business Media Dordrecht

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Finn, R.T., Martinez, J., McGovern, W.W. (1997). Commutative Singular f-Rings. In: Holland, W.C., Martinez, J. (eds) Ordered Algebraic Structures. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5640-0_6

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  • DOI: https://doi.org/10.1007/978-94-011-5640-0_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6378-4

  • Online ISBN: 978-94-011-5640-0

  • eBook Packages: Springer Book Archive

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