Exercise 21, § 2

One recalls that on a ring A, a topology is said to be linear (on the left) if it is a linear topology on the A-module $A_s$ and if it is compatible with the ring structure of A (chap. II, § 2, exerc. 16). One says that a topological ring A is linearly compact (on the left) (resp. strictly linearly compact (on the left)) if $A_s$ is a linearly compact A-module (resp. strictly linearly compact).

a) Now prove that if A is linearly compact on the left for a topology $\mathscr{C}$, the topology $\mathscr{C}^*$ defined in exerc. 18 b) is again compatible with the ring structure of A (use exerc. 18 d)).

b) Suppose that A is commutative; let $u \neq 0$ be an idempotent of $A/\mathscr{R}$, where $\mathscr{R}$ is the radical of A. Now prove that if A is linearly compact for the discrete topology, there exists one and only one idempotent $e \in A$ whose image in $A/\mathscr{R}$ is equal to $u$. (Now prove that among the affine linear varieties $x + b$, where $b \subset \mathscr{R}$ and $x^2 - x \in b$, which are contained in the class $u$, there is a minimal element $e + a$. Deduce that $e^2 = e$ and $a = 0$, by considering the element $e^2 - e = r$ and proving that $r \in Ar^2$, with the aid of Alg., chap. VIII, § 6, exerc. 10 a). Prove the uniqueness of $e$ by proving that if $e_1, e_2$ are two idempotents of A such that $e_1e_2 \in \mathscr{R}$, one has $e_1e_2 = 0$.)

c) Now prove that every commutative ring A which is linearly compact for the discrete topology is a direct composite of a finite number of local rings (linearly compact for the discrete topology). (Consider first the case where $\mathscr{R} = 0$, using exerc. 15 b) and chap. II, § 1, no 2, prop. 5, then apply b) to the idempotents of $A/\mathscr{R}$.)

d) Now prove that every commutative linearly compact (resp. strictly linearly compact) ring A is a product of a family of local rings which are linearly compact (resp. strictly linearly compact). (Let $(\mathfrak{m}_\lambda)$ be the family of open maximal ideals of A; for every open ideal $a$ not containing $\mathfrak{m}_\lambda$, let $A_\lambda' \times A_\lambda''$ be the decomposition of $A/a$ into a direct composite of two rings, such that $A_\lambda'$ is the local ring with maximal ideal $(\mathfrak{m}_\lambda + a)/a$, defined in c) ; let $e'_\lambda(a)$ be the unit element of $A_\lambda'$, considered as a class mod. $a$ in A ; the $e'_\lambda(a)$ form a filter basis which converges in A to an idempotent $e_\lambda$, and $Ae_\lambda$ is a closed ideal of A of which $e_\lambda$ is the unit element. Now prove that the ring $Ae_\lambda$ is a local ring and that A is isomorphic to the product of the $Ae_\lambda$.