Exercise 18, § 2

(a) Let $A$ be a ring and $\mathscr{F}$ and $\mathscr{G}$ two topologizing sets (Exercise 16) of left ideals of $A$. Let $\mathscr{G}$. denote the set of left ideals $I$ of $A$ for which there exists a left ideal $n \in \mathscr{G}$ such that $n/I$ is $\mathscr{G}$-negligible. Show that $\mathscr{G}$. is topologizing; for an $A$-module $M$ to be $\mathscr{F}$.$\mathscr{G}$-negligible, it is necessary and sufficient that there exist an $\mathscr{F}$-negligible submodule $M'$ of $M$ such that $M/M'$ is $\mathscr{G}$-negligible. Show that the composition law $(\mathscr{F}, \mathscr{G}) \mapsto \mathscr{G}$. on the set of topologizing sets of left ideals of $A$ is associative.

$\mathscr{F}$ is called idempotent if $\mathscr{G}$. $\mathscr{G}$.

(b) Show that $\mathscr{G} \mathscr{G}$ contains all the ideals of the form $m.n$ where $m \in \mathscr{F}$ and $n \in \mathscr{G}$.

(c) Let $K$ be a commutative field and $B$ the ring of polynomials with coefficients in $K$ in an infinite system of indeterminates $(X_i)$ ($i \geq 0$). Let $b$ be the ideal of $B$ generated by the elements $X_i X_j$ ($i \neq j$); let $A$ be the ring $B/b$, $\xi_i$ the class mod. $b$ of $X_i$ and $m$ the ideal of $A$ generated by the $\xi_i$; take $\mathscr{F}$ to be the set of ideals containing a power of $m$. Show that $\mathscr{F}$ is topologizing and contains the product of any two ideals of $\mathscr{F}$ but is not idempotent.

(d) Suppose that $A$ is commutative and $\mathscr{F}$ is a topologizing set of ideals of $A$ such that every ideal $m \in \mathscr{F}$ contains a finitely generated ideal $n \in \mathscr{G}$. Show that, if the product of two ideals of $\mathscr{F}$ belongs to $\mathscr{F}$, $\mathscr{F}$ is idempotent.