Proposition 3

Let A be a ring such that there exists a homomorphism $\rho$ of A into a field D; then for every free A-module E, any two bases of E are equipotent.

Consider the vector space $\rho^*(E) = D \otimes_A E$ over D obtained by extending the ring of scalars to D ($§ 5$, no. 1) and let $\phi : x \mapsto 1 \otimes x$ be the canonical mapping of E into $\rho^*(E)$; if $(a_\lambda)$ is a basis of E, $(\phi(a_\lambda))$ is a basis of $\rho^*(E)$ ($§ 5$, no. 1, Proposition 4); the proposition then follows from Theorem 3.