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, Volume 2, Issue 2, pp 211–213 | Cite as

Prime elements from prime ideals

  • B. Banaschewski
Article

Abstract

The prime ideal theorem for distributive lattices (PIT) is shown to imply that any complete distributive lattice with a compact unit has a prime element, which is then used to deduce from PIT that (1) every nontrivial ring with unit has a prime ideal, and (2) every Wallman locale is spatial.

AMS (MOS) subject classifications (1980)

03E25 06D99 16A66 

Key words

Prime Ideal Theorem Boolean Ultrafilter Theorem distributive lattice prime ideals in rings 

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References

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    B.Banaschewski and R.Harting (1985) Lattice aspects of radical ideals and choice principles. Proc. London Math. Soc. 50, 385–404.Google Scholar
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    A. Blass (1984) Prime ideals yield almost maximal ideals. Preprint.Google Scholar
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    N.Bourbaki (1949) Sur le théorème de Zorn, Arch. Math. 2, 434–437.Google Scholar
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    P. T. Johnstone (to appear) Almost maximal ideals, Fund. Math. Google Scholar
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    P. T.Johnstone (1982) Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge Univ. Press, Cambridge.Google Scholar
  6. 6.
    H.Wallman (1938) Lattices and topological spaces, Ann. Math. 39, 112–126.Google Scholar

Copyright information

© D. Reidel Publishing Company 1985

Authors and Affiliations

  • B. Banaschewski
    • 1
  1. 1.Department of Mathematical SciencesHamiltonCanada

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