Lemma 1

— Let L be a module over a commutative ring A, generated by n elements, and let M be a free submodule of L; then M has rank $\leq n$. Suppose first that L is free. Let i denote the canonical injection of M into L. By III, p. 520, Cor., the homomorphism $\bigwedge^n i : \bigwedge^n M \to \bigwedge^n L$ is injective; by III, p. 511, Prop. 6, $\bigwedge^n L = \{0\}$, so $\bigwedge^n M = \{0\}$; it follows that M has rank $\leq n$ (III, p. 518, Cor. 1). Now consider the general case; the module L is a quotient of a free module L' of rank n. There exists a submodule M' of L isomorphic to M (II, p. 218, Prop. 21). By the first part of the argument M' has rank $\leq n$, and the result follows.