Let $(R,\mathfrak{m})$ be a commutative local ring, let $M$ be a finitely generated $R$-module, and let $x_1,\ldots,x_k \in M$ for some integer $k \ge 0$. If the residue classes $x_1+\mathfrak{m}M,\ldots,x_k+\mathfrak{m}M$ generate the $R/\mathfrak{m}$-[vector space](/page/Vector%20Space) $M/\mathfrak{m}M$, then $x_1,\ldots,x_k$ generate $M$ as an $R$-module.