 Minimal primes are associated primes

For a Noetherian ring and a proper ideal $I$, every minimal prime $P$ over $I$ is an associated prime of a module $R/I$. The localized quotient has one prime, so its finitely generated maximal ideal is nilpotent and its socle is nonzero. An element with annihilator $P{R}_P$ can be lifted to the quotient. Clearing denominators for a finite generating set of $P$ gives a nonzero element with annihilator exactly $P$.