Remark

Let $E$ be an $A$*-module and* $(M_\alpha)_{\alpha \in I}$ an increasing family of submodules of $E$ such that $E$ is the union of the $M_\alpha$; if $j_{\beta \alpha}: M_\alpha \to M_\beta$ (for $\alpha \leq \beta$) and $j_\alpha: M_\alpha \to E$ are the canonical injections, it is immediate that $j = \lim \longrightarrow j_\alpha$ is an isomorphism of $\lim \longrightarrow M_\alpha$ onto $E$ (Set Theory, III, § 7, no. 6, Remark 1). In particular, every $A$*-module is the direct limit of the right directed family of its finitely generated submodules.