Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701Completeness results for metrized rings and lattices1491688263810.29252/cgasa.11.1.149ENGeorge M.BergmanUniversity of California, Berkeley0000-0003-4027-7293Journal Article20180810The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper <em>radical</em> ideals (for example, ${0})$ that are closed under the natural metric, but has no <em>prime</em> 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. <br />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 <em>either</em> the inequality $d(xvee y,,xvee z)leq d(y,z)$ <em>or</em> the inequality $d(xwedge y,xwedge z)leq d(y,z),$ 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. <br />We show by example that if the above inequalities are replaced by the weaker conditions $d(x,,xvee y)leq d(x,y),$ respectively $d(x,,xwedge y)leq d(x,y),$ the completeness conclusion can fail. <br />We end with two open questions.https://cgasa.sbu.ac.ir/article_82638_41c589d665953b3ab2260903c95697c4.pdf