Exercise 22, § 2

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

(b) Let $A$ be a commutative ring, $\mathfrak{m}$ an ideal of $A$. For $A$ to be strictly linearly compact for the $\mathfrak{m}$-adic topology, it is necessary and sufficient that it be separated and complete for this topology, that $A/\mathfrak{m}$ be an artinian ring and $\mathfrak{m}/\mathfrak{m}^2$ a $(A/\mathfrak{m})$-module of finite length (in other words, A is a complete noetherian semi-local ring).

(c) Let I be an infinite set of indices, K a finite field, $A = K[[X_i]]_{i \in I}$ the algebra of formal power series with respect to the family of indeterminates $(X_i)_{i \in I}$ (Alg., chap. IV, § 5, exerc. 1); A is a local ring. As a vector space over K, one can identify A with the product space $K^{N(I)}$; if K is provided with the discrete topology and A with the product topology $\mathscr{C}$, show that $\mathscr{C}$ is a linear topology on the ring $A$, for which $A$ is strictly linearly compact; if $m$ is the maximal ideal of $A$, now prove, by reasoning as in exerc. 12 c), that $A$ is not complete for the $m$-adic topology.