Exercise 21, § 2

Recall that on a ring A a topology is called (left) linear if it is a linear topology on the A-module A, and it is compatible with the ring structure on A (Chapter II, § 2, Exercise 16). A topological ring A is called (left) linearly compact (resp. (left) strictly linearly compact) if A, is a linearly compact (resp. strictly linearly compact) A-module.

(a) Show that, if A is left linearly compact with the topology $\mathscr{T}$, the topology $\mathscr{T}^*$ defined in Exercise 18(b) is also compatible with the ring structure on A (use Exercise 18(d)).

(b) Suppose that A is commutative; let $u \neq 0$ be an idempotent of $A/\mathfrak{R}$, where $\mathfrak{R}$ is the Jacobson radical of A. Show that if A is linearly compact with the discrete topology, there exists a unique idempotent $e \in A$ whose image in $A/\mathfrak{R}$ is equal to $u$. (Show that among the linear affine varieties $x + b$, where $b \subset \mathfrak{R}$ 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 showing that $r \in Ar^2$, using Algebra, Chapter VIII, § 6, Exercise 10 (a). Prove the uniqueness of $e$ by showing that if $e_1, e_2$ are two idempotents of A such that $e_1e_2 \in \mathfrak{R}$, then $e_1e_2 = 0$.)

(c) Show that every commutative ring A which is linearly compact with the discrete topology is a direct composition of a finite number of local rings (which are linearly compact with the discrete topology). (Consider first the case, where $31 = 0$, using Exercise 15(b) and Chapter II, § 1, no. 2, Proposition 5, then apply (b) to the idempotents of $A/\%$.)

(d) Show that every linearly compact (resp. strictly linearly compact) commutative ring A is the product of a family of linearly compact (resp. strictly linearly compact) local rings. (Let $(m_\lambda)$ be the family of open maximal ideals of A; for every open ideal $a$ not containing $m_\lambda$, let $A_l, x \ A_\lambda''$ be the decomposition of $A/a$ as a direct composition of two rings such that $A_l$, is the local ring with maximal ideal $(m_\lambda + a)/a$ 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 base which converges in $A$ to an idempotent $e_\lambda$ and $Ae$, is a closed ideal of $A$ of which $e_\lambda$ is the unit element. Show that the ring $Ae_\lambda$ is a local ring and that $A$ is isomorphic to the product of the $Ae_\lambda$.)