Let $A$ be a Noetherian ring, commutative and with unity. In the online notes of Ravi Vakil, he first states the properties of associated points, one of which is
(A) associated points of $A$ are precisely the generic points of irreducible components of the support of some $m \in A$ on Spec $A$.
I know that Supp $m$ is a closed subset of Spec $A$. I also know that in a scheme, each irreducible set has a unique generic point. What is not clear to me is: how do we know that there is always a generic point to the irreducible components of Supp $m$ which is implicit in the statement (A)? Thank you!
The generic points of irreducible components of the support of $m$ is nothing else but the set of minimal prime ideals of the support of $m$.
So let us prove the statement. One part of the equivalence is trivial:
Let $\mathfrak p$ be an associated prime of $A$. We have $\mathfrak p = Ann(m)$ for some $m \in A$, in particular $\mathfrak p$ is the unique minimal prime of the support of $m$, since $Supp(m) = V(Ann(m)) = V(\mathfrak p) =$ primes containing $\mathfrak p$.
The other direction is less trivial. In general if $M$ is a finite $R$-module, the minimal primes of $Supp M$ and $Ass M$ coincide (You can find this is any textbook about commutative algebra). Apply this on $(m) \subset A$. If $\mathfrak p \in Supp(m)$ is minimal, we get $\mathfrak p \in Ass(m) \subset Ass A$.