Proposition 9

The submodule generated by a family $(a_i)_{i \in I}$ of elements of an $A$*-module* $E$ is the set of linear combinations of the family $(a_i)$.

Every submodule of $E$ which contains all the $a_i$ also contains the linear combinations of the $a_i$. Conversely, formulae (1) and (2) of no. 1 prove that the set of linear combinations of the $a_i$ is a submodule of $E$ which obviously contains all the $a_i$ and is therefore the smallest submodule containing them.