Exercise 14, § 6

Let $A$ be a compact ring with an identity element, totally disconnected (*) and without radical. Then $A$ is isomorphic to the product of a family of finite simple rings. [Show first that $A$ has a fundamental system of neighbourhoods of $o$ which are two-sided ideals, using Exercise 12 b) of § 6 and Proposition 14 of § 4. Hence show that there is a maximal set $\Phi$ of two-sided ideals of $A$ such that (i) every two-sided ideal $\mathfrak{N} \in \Phi$ is a maximal open two-sided ideal of $A$; (ii) no ideal $\mathfrak{N} \in \Phi$ contains a finite intersection of ideals of $\Phi$ other than $\mathfrak{N}$. Show then that $A$ is isomorphic to the product of the rings $A/\mathfrak{N}$ as $\mathfrak{N}$ runs through $\Phi$; prove first that the intersection of all the $\mathfrak{N} \in \Phi$ consists only of the zero element of $A$ by using Exercise 9 b)].