Exercise 8, § 6

Let A be a topological ring with an identity element. a) The set $A^*$ of units of A is open in A if and only if there is a neighbourhood V of 1 all of whose elements are units in A. b) If $A^*$ is open in A then every maximal (left or right) ideal of A is closed, and the radical of A is closed. If A has no closed left ideals other than $\{0\}$ and A, then A is a division ring.

c) Let $A$ be the subring of the topological field $\mathbf{Q}$ consisting of all rational numbers $k/2^n \ (k \in \mathbf{Z},\ n \in \mathbf{N})$. Show that $A$ contains no closed ideal other than $\{0\}$ and $A$, but is not a field.