Exercise 2, § 6

Let $(E_\alpha, f_{\beta\alpha})$ be a direct system of A-modules. Show that the A-module $\lim \rightarrow E_\alpha$ is canonically isomorphic to the quotient of the direct sum $\bigoplus_\alpha E_\alpha = F$ by the submodule N generated by the elements of the form $j_\beta(f_{\beta\alpha}(x_\alpha)) - j_\alpha(x_\alpha)$, for every ordered pair $(\alpha, \beta)$ such that $\alpha \leq \beta$ and all $x_\alpha \in E_\alpha$, the $j_\alpha : E_\alpha \to F$ being the canonical injections.