Annihilaor of a prime is non-zero

Question

If we have a commutative noetherian ring how do we know that the annihilator of a prime ideal is always non-zero?

Answer 1

The annihilator of any nonzero prime ideal of $\Bbb Z$ is zero. (If you clarify that you mean something else by annihilator, i can delete this. I'm just answering based on the standard definition.)