Atiyah MacDonald Exercise 2.2

Question

In this exercise, we are asked to prove that if $\mathfrak{a}$ an ideal of $A$ and $M$ an $R$-module, then $A/\mathfrak{a} \otimes_{A} M \cong M/\mathfrak{a}M$. There is a hint given that asks to tensor the exact sequence $$0 \xrightarrow{} \mathfrak{a} \xrightarrow{} A \xrightarrow{} A/\mathfrak{a} \xrightarrow{} 0$$ with $M$. But this approach doesn't necessarily work when $M$ is not a flat module, so is there a workaround for this certain approach? (I have already done the exercise by showing that the universal property holds for $M/\mathfrak{a}M$, so it has to be isomorphic to the tensor product, but I just want to know if there's a way to fix the above argument).

Answer 1

The idea is not to use the flatness.

Note that $- \otimes_A M : A-Mod \rightarrow A-Mod$, being a left adjoint is always right exact.

Now, take the exact sequence $0 \rightarrow \alpha \rightarrow A \rightarrow A/\alpha \rightarrow 0$ in $A-Mod$.

Applying $- \otimes_A M$ to this gives you a right exact sequence $\alpha \otimes_A M \rightarrow M \rightarrow A/\alpha \otimes_A M \rightarrow 0$.

Now check that the image of the first map is precisely $\alpha M$. Now apply the first isomorphism theorem.