Reference for $(N_1\cap N_2)\otimes_A M = (N_1\otimes_A M)\cap ( N_2\otimes_A M)$

Question

Where can I find a canonical proof of the following statement?

If $M$ is a flat $A$-module and $N$ is an $A$-module with submodules $N_1, N_2$, then $$(N_1\cap N_2)\otimes_A M = (N_1\otimes_A M)\cap (N_2\otimes_A M)$$ inside $N \otimes_A M$.

This is a standard proposition in commutative algebra, so I expect it can be found in a textbook. I would like to know so that I can study this proof in detail until I understand it.

Answer 1

This is Theorem 7.4(i) in Matsumura's Commutative Ring Theory (on page 48).

The proof is simple enough that I may as well include it here.

Take $$\begin{array}{rcrcc} \varphi&:&N&\rightarrow &N/N_1\oplus N/N_2 \\ &&x&\mapsto &(x+N_1,x+N_2)\end{array}$$ and the sequence $$0\rightarrow N_1\cap N_2 \rightarrow N \overset{\varphi}{\rightarrow} \frac{N}{N_1}\oplus \frac{N}{N_2}\rightarrow 0$$ is exact. Since $M$ is flat, we then have that $$0\rightarrow (N_1\cap N_2)\otimes_A M \rightarrow N\otimes_A M \overset{\varphi\otimes 1}{\rightarrow} \frac{N\otimes_A M}{N_1\otimes_A M} \oplus \frac{N\otimes_A M}{N_2\otimes_A M} \rightarrow 0$$ is exact, and that's what we wanted to show.