Proposition 5

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 $S(N) \to S(M)$, $S(P) \to S(M)$ and

canonical extensions of the canonical injections, are injective; if $S(N)$, $S(P)$ and $S(N \cap P)$ are identified with subalgebras of $S(M)$ by means of these homomorphisms, then (4)

The proof reduces to that of § 5, no. 2, Proposition 4 replacing $T$ by $S$ throughout. The hypotheses of Proposition 5 always hold for arbitrary submodules $N, P$ of $M$ when $A$ is a field.