Corollary 1

Let M be an A-module and N a submodule of M such that M/N is finitely presented. For N to be a direct factor of M, it is necessary and sufficient that, for all $m \in S_2$, N, be a direct factor of M.

To say that N is a direct factor of M means that the identity homomorphism on M/N factors into $M/N \xrightarrow{w} M \xrightarrow{\phi} M/N$ where $\phi$ is the canonical homomorphism and $w$ a homomorphism (Algebra, Chapter II, § 1, no. 9, Proposition 14); as $(M/N)_m = M_m/N_m$ and $\phi_m$ is the canonical homomorphism $M_m \to M_m/N_m$, the corollary follows easily from Proposition 12.