Banaschewski, Bernhard; Harting, Roswitha Lattice aspects of radical ideals and choice principles. (English) Zbl 0569.16003 Proc. Lond. Math. Soc., III. Ser. 50, 385-404 (1985). 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 Cited in 1 ReviewCited in 28 Documents 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 Keywords:lattices of radical ideals; compact frames; Levitzki radical ideals; coherent frame; Brown-McCoy radical ideals; Axiom of Choice; maximality principle; Prime Ideal Theorem PDFBibTeX XMLCite \textit{B. Banaschewski} and \textit{R. Harting}, Proc. Lond. Math. Soc. (3) 50, 385--404 (1985; Zbl 0569.16003) Full Text: DOI