Exercise 22, § 2

(a) Let $A$ be a local ring and $m$ its maximal ideal. Show that, if, for a linear topology $\mathscr{T}$ on $A$, $A$ is strictly linearly compact, $\mathscr{T}$ is coarser than the $m$-adic topology.

(b) Let $A$ be a commutative ring and $m$ an ideal of $A$. For $A$ to be strictly linearly compact with the $m$-adic topology, it is necessary and sufficient that it be Hausdorff and complete with this topology, that $A/m$ be an Artinian ring and $m/m^2$ and $(A/m)$-module of finite length (in other words, $A$ is a complete Noetherian semi-local ring) *.

(c) Let $I$ be an infinite indexing set, $K$ a finite field and $A = K[[X_i]]_{i \in I}$ the algebra of formal power series in the family of indeterminates $(X_i)_{i \in I}$ (Algebra, Chapter IV, § 5, Exercise 1); $A$ is a local ring. As a vector space over $K$, $A$ may be identified with the product space $K^{N(I)}$; if $K$ is given the discrete topology and $A$ the product topology $\mathscr{T}$, show that $\mathscr{T}$ is a linear topology on the ring $A$ with which $A$ is strictly linearly compact; if $m$ is the maximal ideal of $A$, show, by arguing as in Exercise 12 (c), that $A$ is not complete with the $m$-adic topology.