completeness results for metrized rings and lattices

The boolean ring b of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, {0}) that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. moreover, b is known to be complete in its metric. together, these facts answer a question posed by j. gleason. from this example, rings of arbitrary characteristic with the same properties are obtained.