Exercise 20, § 2

Let $\mathscr{F}$ be a topologizing and idempotent set of left ideals of a ring $A$. (a) Suppose that every ideal of $\mathscr{F}$ contains a finitely generated ideal belonging to $\mathscr{F}$. Let $M$ be an $A$-module and $(N_i)_{i \in I}$ a right directed family of submodules of $M$, whose union is $M$. Show that $M_{\mathscr{F}}$ is the union of its sub-$A_{\mathscr{F}}$-modules $(N_i)_{\mathscr{F}}$. In particular, for every family $(M_\lambda)_{\lambda \in L}$ of $A$-modules, $\left( \bigoplus_{\lambda \in L} M_\lambda \right)_{\mathscr{F}}$ is canonically isomorphic to $\bigoplus_{\lambda \in L} (M_\lambda)_{\mathscr{F}}$. (b) Suppose that condition (a) holds and also that, for every surjective $A$-module homomorphism $u : M \to N$, $u_{\mathscr{F}} : M_{\mathscr{F}} \to N_{\mathscr{F}}$ is surjective. Then, the canonical mapping $A_{\mathscr{F}} \otimes_A M \to M_{\mathscr{F}}$ defined by the external law on $M_{\mathscr{F}}$ as an $A_{\mathscr{F}}$-module is an $A_{\mathscr{F}}$-module isomorphism (reduce it to the case where $M$ is free); the ring $A_{\mathscr{F}}$ is a flat left $A$-module and the left $A_{\mathscr{F}}$-modules coincide with the left $A$-modules $M$ such that the mapping $j_M : M \to M_{\mathscr{F}}$ is bijective. (c) Suppose that $A$ is an infinite product $\prod_{t \in I} A_t$, of rings; let e, denote the unit element of $A$, and take $\mathscr{F}$ to be the set of left ideals of $A$ containing the two-sided ideal $a = \bigoplus_{t \in I} A_t$, (the $A_t$, being canonically identified with ideals of $A$). Show that, for every left $A$-module $M$, $M_{\mathscr{F}}$ is identified with the product $\prod_{t \in I} (e_t M)$ and in particular $A_{\mathscr{F}} = A$. Deduce that, for every surjective $A$-homomorphism $u : M \to N$, $u_{\mathscr{F}} : M_{\mathscr{F}} \to N_{\mathscr{F}}$ is surjective, but give examples of $A$-modules $M$ such that $j_M$ is not bijective. (d) Suppose that condition (a) holds and also that $A$ is commutative. Show that, if $M$ is 9-negligible, $M_{(\mathscr{F})} = 0$.