Corollary 2

We keep the assumptions of Theorem 2. Furthermore, let $M'$ be an A-module and $u: M'\rightarrow M$ be a homomorphism. If the homomorphism $\overline{u}$ from $M'/\mathfrak{a}M'$ to $M/\mathfrak{a}M$ deduced from $u$ by passing to the quotients is surjective, then $u$ is surjective.

It suffices to apply Theorem 2 to the image N of $u:$ indeed, the image of $\overline{u}$ is $(N +\mathfrak{a}M)/\mathfrak{a}M$, so $\overline{u}$ is surjective if and only if we have $N +\mathfrak{a}M = M$.