Proposition 4

Let $N$ and $P$ be two submodules of an $A$-module $M$ such that their sum $N + P$ is a direct factor in $M$ and their intersection $N \cap P$ is a direct factor in $N$ and in $P$. Then the homomorphisms $\Lambda(N) \to \Lambda(M)$, $\Lambda(P) \to \Lambda(M)$ and $\Lambda(N \cap P) \to \Lambda(M)$, canonical extensions of the canonical injections, are injective; if $\Lambda(N)$, $\Lambda(P)$ and $\Lambda(N \cap P)$ are identified with subalgebras of $\Lambda(M)$ by means of these homomorphisms, then

The proof is derived from that of § 5, no. 2, Proposition 4 replacing $T$ by $\Lambda$ throughout. The hypotheses of Proposition 4 are always satisfied by arbitrary submodules $N, P$ of $M$ when $A$ is a field.