Proposition 20

Let $\rho$ be a homomorphism of a field $K$ into a ring $A$. For every left vector space $E$ over $K$, the canonical right $A$-module homomorphism

(§ 5, no. 4) is injective; it is bijective when $E$ is finite-dimensional.

The second assertion follows from § 5, no. 4, Proposition 8. To prove the first, we note that every element of $(E^*)_{(A)}$ may be written as $\sum_i x_i^* \otimes \alpha_i$, where $\alpha_i \in A$ and $(x_i^*)_{1 \leq i \leq n}$ is a free family in $E^*$; there corresponds to it in $(E_{(A)})^*$ the linear form $y^*$ such that $y^*(1 \otimes x) = \sum_i \rho(\langle x, x_i^* \rangle) \alpha_i$ for all $x \in E$. But, there exists in $E$ a family $(x_i)_{1 \leq i \leq n}$ such that $\langle x_i, x_j^* \rangle = \delta_{ij}$ (no. 5, Corollary 2 to Theorem 7), whence $y^*(1 \otimes x_i) = \alpha_i$; the relation $y^* = 0$ therefore implies $\alpha_i = 0$ for all $i$, which proves our assertion.