Proposition 3

*If $u: M \to N$ is a surjective $A$-linear mapping, the homomorphism $\Lambda(u): \Lambda(M) \to \Lambda(N)$ is surjective and its kernel is the two-sided ideal of $\Lambda(M)$ generated by the kernel $P \subset M \subset \Lambda(M)$ of $u$.

The proof is derived from that of § 6, no. 2, Proposition 4, replacing $S$ by $\Lambda$ and $\mathfrak{g}'$ by $\mathfrak{g}''$.

If $u: M \to N$ is an injective linear mapping, it is not always true that $\Lambda(u)$ is an injective mapping (§ 6, Exercise 3) (see however below no. 9, Corollary to Proposition 12). However this is so when $u$ is an injection such that $u(M)$ is a direct factor of $N$ and then the image of $\Lambda(u)$ (isomorphic to $\Lambda(M)$) is a direct factor of $\Lambda(N)$; the proof is the same as that for the analogous assertions for $T(u)$ (§ 5, no. 2) replacing $T$ by $\Lambda$.