TITLE: When does intersection commute with tensor product
QUESTION [9 upvotes]: Given two submodules $U,V \subseteq M$ over a (commutative) ring $R$, and a flat $R$-module $A$, I can interpret $U \otimes_R A$ and $V \otimes_R A$ as submodules of $M \otimes_R A$. Is it necessarily true that $$(U \cap V) \otimes_R A \cong (U \otimes_R A) \cap (V \otimes_R A) ?$$
I think it should be true in many cases, with intuition coming from $\mathbb{Z}$-modules and $A = \mathbb{Q}$, but I'm unsure about what happens in general.

REPLY [6 votes]: Finite intersections are pullbacks, so in particular are finite limits, and a module is flat iff tensoring with it commutes with all finite limits (exercise).