Corollary 3

If $(M_\lambda)_{\lambda \in L}$ is a right directed family of submodules of an A-module E, the sum $\sum_{\lambda \in L} M_\lambda$ is identical with the union $\bigcup_{\lambda \in L} M_\lambda$.

$\bigcup_{\lambda \in L} M_\lambda \subset \sum_{\lambda \in L} M_\lambda$ is always true without hypothesis; on the other hand, for every finite subfamily $(M_\lambda)_{\lambda \in J}$ of $(M_\lambda)_{\lambda \in L}$, there exists by hypothesis a $\mu \in L$ such that $M_\lambda \subset M_\mu$ for all $\lambda \in J$, hence $\sum_{\lambda \in L} M_\lambda \subset M_\mu$ and it thus follows from Corollary 2 that $\sum_{\lambda \in L} M_\lambda \subset \bigcup_{\lambda \in L} M_\lambda$.