Exercise 23, § 2

Let $A$ be a commutative ring with a linear topology $\mathscr{T}$.

(a) Show that the ideals $a$ of $A$, which are closed under $\mathscr{T}$ and such that $A/a$ is Artinian, form a fundamental system of neighbourhoods of 0 under a linear topology $\mathscr{T}^{(c)}$ on $A$, which is coarser than $\mathscr{T}$ (use Algebra, Chapter VIII, § 2, Exercise 12); then $(\mathscr{T}^{(c)})^{(c)} = \mathscr{T}^{(c)}$. Let $B$ be the Hausdorff completion ring of $A$ with respect to the topology $\mathscr{T}^{(c)}$; show that $B$ is strictly linearly compact; $B$ is said to be the strictly linearly compact ring associated with $A$.

(b) Let $\mathscr{T}_\omega(A)$ be the discrete topology on $A$, $\mathscr{T}_c(A)$ the topology $(\mathscr{T}_\omega(A))^{(c)}$. For the topology $\mathscr{T}^{(c)}$ on $A$ to be Hausdorff, it is necessary and sufficient that $\mathscr{T}$ be Hausdorff and that, for every ideal $b$ of $A$ open under $\mathscr{T}$, the topology $\mathscr{T}_c(A/b)$ be Hausdorff. \* If $A$ is the ring of a non-discrete valuation of height 1 (Chapter VI), $\mathscr{T}_c(A)$ is not Hausdorff. \*

(c) Suppose that $A$ is linearly compact under $\mathscr{T}$; then $A$ is complete (but in general not Hausdorff) with $\mathscr{T}^{(c)}$. Then for $\mathscr{T}^{(c)}$ to be Hausdorff, it is necessary and sufficient that there exist on $A$ a linear topology which is strictly linearly compact and coarser than $\mathscr{T}$; $\mathscr{T}^{(c)}$ is then the unique topology with these properties.

(d) Let $E$ be a linearly topologized and strictly linearly compact $A$-module. Show that, if $A$ is given the topology $\mathscr{T}_c(A)$, $E$ is a topological ring (note that if $F$ is an Artinian $A$-module, then, for all $x \in F$, the annihilator $a$ of $x$ is such that $A/a$ is Artinian). Deduce that, if $B$ is the Hausdorff completion of $A$ with respect to the topology $\mathscr{T}_c(A)$, $E$ can be considered as a topological $B$-module; the ring $B$ being isomorphic to a product of strictly linearly compact local rings

B_\lambda (Exercise 21 (d)), show that E is isomorphic to a product of strictly linearly compact submodules E_\lambda, where E_\lambda is annihilated by the B_\mu of index \mu \neq \lambda and can therefore be considered as a topological B_\lambda-module (if e_\lambda is the unit element of B, take E = e_\lambda \cdot E).