Corollary 2

Let $M$ be an A-module and, for all $m \in \Omega$, let $f_m$ be the canonical mapping $M \to M_m$. The homomorphism $x \mapsto (f_m(x))$ of $M$ to $\prod_{m \in \Omega} M_m$ is injective.

Applying Corollary 1 in the case $N = 0$, it is seen that the relation $x = 0$ is equivalent to $f_m(x) = 0$ for all $m \in \Omega$.