Corollary 5

Let A be a ring. Every system of generators of the A-module $A^n$ with n elements is a basis of $A^n$.

Let $(e_i)_{1 \leq i \leq n}$ be the canonical basis of $A''$, $(x_i)_{1 \leq i \leq n}$ a system of generators of $A^n$ with n elements and $u : A^n \to A''$ the A-linear mapping such that $u(e_i) = x_i$ for $1 \leq i \leq n$. By hypothesis $u$ is surjective and it is necessary to show that $u$ is injective. By virtue of Theorem 1 this can immediately be reduced to the case where A is a local ring; if $m$ is the maximal ideal of A, the elements $1 \otimes x_i$ ($1 \leq i \leq n$) in $(A/m)^n$ then form a system of generators of the free $(A/m)$-module $(A/m)^n$; as $A/m$ is a field, this system is a basis of $(A/m)^n$; as $A^n$ is a free A-module, we deduce from Proposition 5 that $(x_i)$ is a basis of $A^n$.