Flat Modules and Intersection

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Let $M$ be a flat module over a commutative ring $A$. Suppose $X_1$ and $X_2$ are submodules of an $A$-module $X$. Prove that $(X_1 \cap X_2) \otimes_A M = (X_1 \otimes_A M) \cap (X_2 \otimes_A M)$ as submodules of $X\otimes_A M$.

 

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Spoiler

There is a short exact sequence $0 \to X_1 \cap X_2 \to X \to X/X_1 \oplus X/X_2 \to 0$. Tensoring with $M$ gives a short exact sequence $$0 \to (X_1 \cap X_2) \otimes_A M \to X \otimes_A M \to \frac{X\otimes_A M}{X_1 \otimes_A M} \oplus \frac{X \otimes_A M}{X_2 \otimes_A M}\to 0$$ The kernel of the third map is $(X_1 \otimes_A M) \cap (X_2 \otimes_A M)$ so indeed $$(X_1 \cap X_2) \otimes_A M = (X_1\otimes_A M) \cap (X_2 \otimes_A M)$$