Exercise 15, § 6

Let $A$ be a totally disconnected compact ring with an identity element. Show that the radical $\mathfrak{R}$ of $A$ is topologically nilpotent [Exercise 9 a); use Exercise 12 b) of § 6 and Proposition 14 of § 4]. Hence show that $\mathfrak{R}$ is the set of all $x \in A$ such that $ax$ is topologically nilpotent for all $a \in A$ [cf. Exercise 9 c)].