Corollary 1

We keep the assumptions of Theorem 2. Let $(x_i)_{i\in I}$ be a family of elements of M, and let $\overline{x}_i$ be the canonical image of $x_i$ in $M/\mathfrak{a}M$. If the family $(\overline{x}_i)_{i\in I}$ generates the $(A/\mathfrak{a})$-module $M/\mathfrak{a}M$, then the family $(x_i)_{i\in I}$ generates the A-module M.

This follows from Theorem 2 applied to the submodule N of M generated by the family $(x_i)_{i\in I}$.