Lemma 10.38.2. Let $R$ be a ring. Let $I, J \subset R$ be ideals. Let $M$ be a flat $R$-module. Then $IM \cap JM = (I \cap J)M$.
Proof. Consider the exact sequence $0 \to I \cap J \to R \to R/I \oplus R/J$. Tensoring with the flat module $M$ we obtain an exact sequence
Since the kernel of $M \to M/IM \oplus M/JM$ is equal to $IM \cap JM$ we conclude. $\square$
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