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.