Exercise 10, § 2

(a) Let $A$ be a ring, $I$ a set and $R$ a submodule of $L = A_d^{(I)}$. Let $\mathscr{G}$ be the set of ordered pairs $(J, S)$, where $J$ is a finite subset of $I$ and $S$ a finitely generated submodule of $A_d^J \cap R$. $\mathscr{G}$ is ordered by the relation "$J \subset J'$ and $S \subset S'$"; show that $\mathscr{G}$ is directed with respect to this order relation, that the family $(A_d^J/S)$ is a direct system of right $A$-modules with $\mathscr{G}$ as indexing set and that there exists an isomorphism of $L/R$ onto $\varprojlim_{(J, S) \in \mathscr{G}} (A_d^J/S)$.

(b) Deduce from (a) that every $A$-module is a direct limit of finitely presented $A$-modules.