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Lattice aspects of radical ideals and choice principles. (English) Zbl 0569.16003

The lattices of radical ideals after Levitzki, Jacobson, and Brown-McCoy, of any unitary ring, are shown to be compact frames. In the commutative case, the Levitzki radical ideals are known to form a coherent frame; but this is shown to be false in general. The Brown-McCoy radical ideals are shown to be the best behaved (for noncommutative rings) in that they frame a compact subfit locale. The Axiom of Choice is known to imply that all such locales are spatial; here the authors deduce the result from a weaker maximality principle of P. T. Johnstone, which A. Blass has since shown (to be published) to be equivalent to the Prime Ideal Theorem.
Reviewer: J.R.Isbell

MSC:

16Nxx Radicals and radical properties of associative rings
16Dxx Modules, bimodules and ideals in associative algebras
03E25 Axiom of choice and related propositions
18E40 Torsion theories, radicals
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