Exercise 24, § 2

On a commutative ring $A$, denote by $\mathscr{C}_m(A)$ the linear topology for which a fundamental system of neighbourhoods of $0$ is formed by the products (equal to the intersections) of a finite number of powers of maximal ideals (cf. No. 13, Prop. 17); denote by $\mathscr{C}_u(A)$ the linear topology for which a fundamental system of neighbourhoods of $0$ is formed by all the ideals $\neq \{0\}$ of $A$.

a) Now prove that $\mathscr{C}_c(A)$ (Exercise 23) is less fine than $\mathscr{C}_m(A)$ (observe that the radical of an artinian ring is nilpotent); give an example where $\mathscr{C}_c(A) \neq \mathscr{C}_m(A)$ (cf. Exercise 22 c)). If $A$ is a non-discrete valuation ring of rank 1 (Chapter VI), one has $\mathscr{C}_c(A) = \mathscr{C}_m(A)$ and $\mathscr{C}_u(A) \neq \mathscr{C}_m(A)$.

b) The topology $\mathscr{C}_m(A)$ is the least fine of the linear topologies on $A$ for which the maximal ideals of $A$ are closed and every power of an open ideal of $A$ is an open ideal (note that for a linear topology on $A$ for which a maximal ideal $m$ is closed, $m$ is necessarily open).

c) If $A$ is a noetherian ring, one has $\mathscr{C}_m(A) = \mathscr{C}_e(A)$ (observe that if $m$ is a maximal ideal of $A$, $A/m^k$ is an artinian ring for every integer $k \geq 1$, by noting that $m^n/m^{n+1}$ is necessarily a module of finite length over $(A/m)$). Give an example of a noetherian local ring $A$ for which $\mathscr{C}_m(A) \neq \mathscr{C}_u(A)$ (consider a local ring of a polynomial ring over a field).