Let $F \subseteq K$ be fields, and suppose $f_1, ... , f_t \in F[X_1, ... , X_n]$. Let $R = F[X_1, ... , X_n]$, and let $S = K[X_1, ... , X_n]$. Is it always true that $(f_1S + \cdots + f_t S) \cap R = f_1R + \cdots + f_t R$?
This seems like a natural thing to conclude, and I think I can prove this in the case $t = 1$, but I also know that intersections don't always distribute over sums.
Yes, it's true.
The ring extension $R = F[X_1, ... , X_n]\subset S = K[X_1, ... , X_n]$ has the property that $S$ is a free $R$-module (since $K$ is a free $F$-module). This shows that it is faithfully flat, and we are done.