Exercise 5, § 3

Let $A$ be a commutative Noetherian ring and $m$ its Jacobson radical. (a) For $A$ to be strictly linearly compact with a linear topology $\mathscr{T}$ (§ 2, Exercise 19), it is necessary and sufficient that $A$ be semi-local and complete with the $m$-adic topology; $\mathscr{T}$ is then necessarily identical with this latter topology. (Use § 2, Exercise 21 (d) and 22 (a) and also § 3, no. 3, Proposition 6.) (b) For $A$ to be linearly compact with a linear topology $\mathscr{T}$, it is necessary and sufficient that $A$ be semi-local and complete with the $m$-adic topology and that $\mathscr{T}$ be finer than this latter topology. (To see that it is necessary, reduce it first to the case where $A$ is a local ring, using Exercise 21 (d) of § 2. Consider first the case where $\mathscr{T}$ is the discrete topology and note that $A$ is then linearly compact with the $m$-adic topology. In the general case, reduce it to the case where $\mathscr{T}$ is a minimal topology (§ 2, Exercise 18 (c)); show that $A$ is then strictly linearly compact and use criterion (y) of § 2, Exercise 19 (a) and Exercise 18 (b) of § 2.)