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 $d(xvee y,,xvee z)leq d(y,z)$ or 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. 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. We end with two open questions.
Highlights
The Boolean ring B of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals that are closed under the natural metric, but has no prime ideal closed under that metric; closed radical ideals are not, in general, intersections of closed prime ideals
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 ∨ y, x ∨ z) ≤ d(y, z) or the inequality d(x ∧ y, x ∧ z) ≤ d(y, z), and if in L every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, every Cauchy sequence in L converges; that is, L is complete as a metric space
A standard result of ring theory says that if I is an ideal of a commutative ring R, the nil radical of I
Summary
Most of the desired properties of the example sketched above are straightforward to verify. Recall that for any set X, the subsets of X form a Boolean ring under the operations. The operations of the Boolean ring B of measurable subsets of [0, 1] modulo sets of measure zero are continuous in the metric d of (2.11). (For example, one can take S = U ∩ [0, t] and T = U ∩ Such a t exists by continuity of μ(U ∩ [0, t]) in t.) Since [1 + S][1 + T ] = [1 + U ] ∈ P, one of [1 + S], [1 + T ] belongs to P. Our metrized ring has additive-translation-invariant metric by (2.8), completeness in the uniform structure so arising from the metric topology is equivalent to completeness in the metric.)
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