Let $R$ be a commutative ring, let $M$ be an $R$-module, let $n \in \mathbb{N}$, and let $f_1, \ldots, f_n \in R$ be elements such that the ideal $(f_1, \ldots, f_n)$ is equal to $R$. Suppose that, for each $i \in \{1, \ldots, n\}$, the localization $M_{f_i}$ is finitely generated as an $R_{f_i}$-module. Then $M$ is finitely generated as an $R$-module.