Remark

For a family $(a_i)_{i \in I}$ of elements of a vector space $E$ over a field $K$ to be free, it is necessary and sufficient that, for all $\kappa \in I$, $a_\kappa$ belong to no subspace of $E$ generated by the $a_i$ of index $i \neq \kappa$. We know that this condition is necessary in any module (§ 1, no. 11, Remark 1). It is sufficient by virtue of Lemma 1, as is seen immediately arguing by reductio ad absurdum and considering a minimal related subfamily of $(a_i)$.