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