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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.

user26857
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hopeless
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    The first answer here appears to answer your question in the affirmative (though that answer has a downvote and I have not checked it). – Julian Rosen Sep 30 '15 at 02:45
  • Dear Julian, I have checked the answer you refer to and it is absolutely correct and very well written. The same goes for the alternative answer given by Makoto. – Georges Elencwajg Mar 13 '16 at 09:42

1 Answers1

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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.

user26857
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