I have a problem on Atiyah and MacDonald's commutative algebra book, the exercise 3.1:
Let $S$ be a multiplicatively closed subset of a ring $A$ and $M$ a finitely generated $A$ - module. Prove that $S^{-1}M = 0$ iff there exists $s \in S$ such that $sM = 0$.
I know the correct proof, but I have confusion about the following proof of the sufficiency:
Let $S^{-1}M = 0$, then $\frac{m}{t}=0=\frac{0}{1} , \forall m \in M, \forall t \in S$. So $\exists s \in S$ such that $s(m \ast 1-0 \ast t)=0$, that is $sm=0, \forall m \in M$ $\Rightarrow $ $sM=0$.
I am not sure it is right. Any help will be appreciated.
What you have shows that for a fixed $m \in M$, there is $s \in S$ so that $sm = 0$. It does not show that $sM = 0$. To show this, you need to use the fact that $M$ is finitely generated. Apply what you have to a generating set of $M$.