Proposition 8

If a family $(M_i)_{i \in I}$ of submodules of $E$ has intersection reduced to $0$ E is canonically isomorphic to a submodule of $\prod_{i \in I} (E / M_i)$.

Given a subset $X$ of an $A$-module $E$, the intersection $F$ of the submodules of $E$ containing $X$ is called the submodule generated by $X$ and $X$ is called a generating set (or generating system) of $F$ (I, § 4, no. 3); for a family $(a_i)_{i \in I}$ of elements of $E$, the submodule generated by the set of $a_i$ is called the submodule generated by the family $(a_i)$.

An $A$-module is called finitely generated if it has a finite generating set.