Corollary 3

Let $E$ be an $A$-module and $(x_\lambda)_{\lambda \in L}$ a family of elements of $E$. For $(x_\lambda)$ to be a free family, it is necessary and sufficient that in the vector $K$-space $E_{(K)}$ the family $(1 \otimes x_\lambda)$ be free.

The family $(x_\lambda)$ defines an $A$-linear mapping $f : A^{(L)} \to E$ such that $f(e_\lambda) = x_\lambda$ for all $\lambda \in L$ ($(e_\lambda)$ being the canonical basis of $A^{(L)}$) and to say that $(x_\lambda)$ is free means that $f$ is injective. It suffices to apply Corollary 2 to $f$, observing that $A^{(L)}$ is torsion-free (Proposition 25).