Proposition 4

Let E be an A-module admitting a basis $(a_\lambda)_{\lambda \in L}$; if $\phi : x \mapsto 1 \otimes x$ is the canonical mapping of E into $\rho^*(E)$, then $(\phi(a_\lambda))_{\lambda \in L}$ is a basis of $\rho^*(E)$. If $\rho$ is injective, so is $\phi$.

The first assertion follows immediately from § 3, no. 7, Corollary 1 to Proposition 7. Also, for every family $(\xi_\lambda)_{\lambda \in L}$ of elements of A of finite support,

and the relation $\phi \left( \sum_{\lambda \in L} \xi_\lambda a_\lambda \right) = 0$ is therefore equivalent to $\rho(\xi_\lambda) = 0$ for all $\lambda \in L$, whence the second assertion.