Let $R$ be a commutative ring and let $\mathfrak{m} \subseteq R$ be a finitely generated ideal. If $(M_n)_{n \in \mathbb{N}}$ is a family of submodules of some $R$-module with $M_0 \supseteq M_1 \supseteq \dotsc$, do we have $\mathfrak{m} \cdot \cap_{n \in \mathbb{N}} M_n = \cap_{n \in \mathbb{N}} (\mathfrak{m} \cdot M_n)$? Of course only $\supseteq$ needs a proof. It is not even clear to me when $\mathfrak{m}$ is principal.
Background: This seems to be used in Borger's first paper on the geometry of Witt vectors, page 8. In that situation $\mathfrak{m}$ is actually maximal, and invertible as an $R$-module, and some other nice conditions are given. But the argument says explicitly "because $\mathfrak{m}$ is finitely generated ideal, we have ...".
