Exercise 2, § 6

a) Let $a$ be a discrete left ideal in a topological ring $A$. Show that, for every $x \in a$, the left annihilator of $x$ in $A$ is open. Hence show that if $A$ is a non-discrete ring with no zero-divisors, it contains no discrete (left or right) ideal other than $\{0\}$. b) Let $A$ be a locally connected topological ring, and let $a$ be a totally disconnected left ideal of $A$. Show that, for each $x \in a$, the left annihilator of $x$ in $A$ is open. Hence show that if $A$ has no zero-divisors it contains no totally disconnected (left or right) ideal other than $\{0\}$.