Corollary 2

Let M be a finitely generated free A-module and N a submodule of M which is a finitely generated free A-module. For N to be a direct factor of M, it is necessary and sufficient that, for all $m \in \Omega$, $mN = N \cap (mM)$.

By definition M/N is finitely presented; on the other hand, N, and M, are finitely generated free $A_n$-modules. For $N$ to be a direct factor of $M$, it is necessary and sufficient that the canonical mapping $N_m / mN_m \to M_m / mM_m$ be injective (no. 2, Proposition 6); this is the same as saying that the canonical mapping $N / mN \to M / mM$ must be injective (Proposition 9), and as its kernel is $(N \cap mM) / mN$, this proves the corollary.

Proposition 12 (resp. its Corollary 1) will be applied in particular when $A$ is Noetherian and $E$ (resp. $M/N$) a finitely generated $A$-module (Chapter I, § 2, no. 8, Lemma 8).