Completeness results for metrized rings and lattices

Abstract

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.

Keywords

• complete topological ring without closed prime ideals
• measurable sets modulo sets of measure zero
• lattice complete under a metric