Corollary 2

The submodule generated by the union of a family $(M_i)_{i \in I}$ of submodules of a module $E$ is identical with the set of sums $\sum_{i \in I} x_i$, where $(x_i)_{i \in I}$ runs through the set of families of elements of $E$ of finite support such that $x_i \in M_i$ for all $i \in I$.

Clearly every linear combination of elements of $\bigcup_{i \in I} M_i$ is of the above form; the converse is obvious.

The submodule of E generated by the union of a family $(M_i)_{i \in I}$ of submodules of E is called the sum of the family $(M_i)$ and is denoted by $\sum_{i \in I} M_i$.

If for all $i \in I$, $h_i$ is the canonical injection $M_i \to E$ and $h : (x_i) \mapsto \sum_i h_i(x_i)$ the linear mapping of $\bigoplus_{i \in I} M_i$ into E corresponding to the family $(h_i)$ (no. 6, Proposition 6), $\sum_{i \in I} M_i$ is the image of $h$; in other words, there is an exact sequence