Corollary 1

Let $M$ be an A-module, N a submodule of $M$ and $x$ an element of $M$. For $x \in N$, it is necessary and sufficient that, for all $m \in R$, the canonical image of $x$ in $M$, belong to $N$.

Let $\bar{x}$ be the class of $x$ in $M/N$; to say that $x \in N$ means that the A-linear mapping $u : a \mapsto a\bar{x}$ from $A$ to $M/N$ is zero. Now, $(M/N)_m$ is identified with $M_m/N_m$ (§ 2, no. 4, Theorem 1) and $u_m : A_m \to M_m/N_m$ with the mapping $A \mapsto \lambda \bar{x}_m$, where $\bar{x}_m$ is the class mod. $N$, of the canonical image of $x$ in $M$. As the relation $u = 0$ is equivalent to $u_m = 0$ for all $m$ by Theorem 1, this proves the corollary.