Proposition 13

Let $M$ be an $A$-module, $N$ a direct factor submodule of $M$ which is free of dimension $p$ and $\{u\}$ a basis of $\Lambda^p(N)$. For an element $x \in M$ to belong to $N$, it is necessary and sufficient that $u \wedge x = 0$.

Let $P$ be a submodule of $M$ complementary to $N$ and let $y \in N,\ z \in P$ be such that $x = y + z$. Then $u \wedge x = u \wedge z$. As $\Lambda^p(N)$ is free of dimension 1, the mapping $\phi: P \to P \otimes \Lambda^p(N)$ such that $\phi(p) = p \otimes u$ is bijective (II, § 3, no. 4, Proposition 4). On the other hand (no. 7, Proposition 10), the composition of the canonical homomorphisms

is injective. The mapping $\psi \circ \phi$ is therefore injective, whence the proposition.