Exercise 10, § 6

a) Let $B$ be a ring with an identity element, let $E$ be a free $B$-module $B_d^{(I)}$, and let $A$ be the ring $\mathscr{L}(E)$ of endomorphisms of $E$. For every finite subset $F$ of $E$, let $V_F$ be the set of all $u \in A$ such that $u(x) = 0$ for all $x \in F$. Show that the $V_F$ from a fundamental system of neighbourhoods of $0$ for a Hausdorff topology on $A$, which is compatible with the ring structure of $A$, and with respect to which $A$ is totally disconnected.

b) Take $I = \mathbf{N}$ and take $B$ to be a commutative ring which has no zero-divisors but whose radical $\mathfrak{R}$ is not $\{0\}$. Show that the radical of $A$ is not closed. [Let $(e_n)$ be the canonical basis of $E$. Remark on the one hand that each $u \in A$, such that $u(e_n) = 0$ for all but a finite number of indices and such that $u(E) \subset \mathfrak{R} \cdot E$, belongs to the radical of $A$; consider on the other hand the element $u_0$ of $A$ such that $u_0(e_n) = e_n + te_{n+1}$ with $t \neq 0$ in $\mathfrak{R}$, for each $n$].