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. The result that B is complete in its metric is generalized to show that if L is a lattice given with a metric satisfying identically either the inequality d(x\vee y,\,x\vee z)\leq d(y,z) or the inequality d(x\wedge y,x\wedge z)\leq d(y,z), and if in L every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in L converges; that is, L is complete as a metric space. We show by example that if the above inequalities are replaced by the weaker conditions d(x,\,x\vee y)\leq d(x,y), respectively d(x,\,x\wedge y)\leq d(x,y), the completeness conclusion can fail. We end with two open questions.