Exercise 13, § 6

Let $A$ be a bounded topological ring (Exercise 12) which has an identity element and is such that the set $A^*$ of units of $A$ is open in $A$. Show that the radical of $A$ is open. Hence if $A$ is without radical, it is discrete; in particular, a bounded Hausdorff topological division ring is discrete. A compact ring $A$ without radical, which has an identity element and in which $A^*$ is open, is finite; in particular, a compact topological division ring is finite.