I'm reading section 5.5 from Vakil's FOAG, and I got a bit confused about the associated points of a module $M$. The way he develops this theory, is by first stating some axioms and then proving that they hold. The first axiom states the following:
(A) The associated primes/points of $M$ are precisely the generic points of irreducible components of the support of some element of $M$ (on $\operatorname{Spec} A$).
Now, the very next exercise states the following:
Suppose $A$ is an integral domain. Show that the generic point is the only associated point of $\operatorname{Spec} A$.
This is supposed to be done assuming property A. Now, the exercise itself doesn't state what module $M$ we are working with, as in property (A). Does this mean $M = A$, or do we have to show this for all $A$ modules $M$? If $M=A$, I think that the exercise is trivial. However, I am unable to show this holds for all $A$ modules $M$. In fact I think that this latter statement is false. Consider $A = \mathbb{Z}$ and $M = \mathbb{Z}/4\mathbb{Z}$. As far as I can see, the ideal $(2)$ is in the support of $\overline{1}\in M$, whereas $(3)$ isn't, because $(3)$ doesn't contain $4$, and $4\cdot \overline{1} = 0\in M$ means $\overline{1}$ is $0$ in $M_{(3)}$. This shows that the support of $\overline{1}$ is nonempty, while the it isn't the entirety of $\operatorname{Spec} A$, and this means that $(0)$ can't be the generic point of the support.
I would be very grateful if someone could point out where I am going wrong here.
