Let $R$ be a ring, let $M$ be a left $R$-module, and let $(N_i)_{i \in I}$ be a family of submodules of $M$. Define $\sum_{i \in I} N_i$ to be the set of all finite sums of elements drawn from the submodules $N_i$, with the convention that the empty finite sum is $0$. Then $\sum_{i \in I} N_i$ is a submodule of $M$, each $N_i$ is contained in $\sum_{i \in I} N_i$, and if $L \le M$ satisfies $N_i \subset L$ for every $i \in I$, then

\begin{align*} \sum_{i \in I} N_i \subset L. \end{align*}