It is a well-known theorem that, for a Noetherian ring $A$, the minimal primes of $A$ are among the associated primes of $A$; i.e., for every minimal prime $\mathfrak{p}$ of $A$, there is an element $f \in A$ such that $Ann(f) = \mathfrak{p}$. Morally, one might think of $f$ as a "characteristic function" of the irreducible component Spec $A/\mathfrak{p}$ of Spec $A$.

The usual proof of this fact is to look at the set of associated primes of $A$ (or more generally of an $A$-module), together with one or more other sets of primes of $A$, and then go on proving various things that finally conclude with "these collections of primes have the same minimal elements." I find this proof not particularly satisfactory. What I would like is a proof that starts off "Let $\mathfrak{p}$ be a minimal prime of $A$" and then proceeds to "construct" an element $f \in A$ such that $Ann(f) = \mathfrak{p}$. Unfortunately, I have not had much success with proceeding along these lines. Is there anyone who knows (or is able to invent) such a proof?