Set of zero divisors on a module equals union of associated primes

In terms of sets
Let $$R$$ be a Noetherian ring and $$M$$ be a $$R$$-module. Then the set:

$$ZD(R) := \{ r \in R \mid \exists 0 \ne m \in M, rm = 0 \}$$

equals the union of all the associated primes to the module $$M$$.

In terms of elements
Let $$R$$ be a Noetherian ring and $$M$$ be a $$R$$-module. Then, if an element $$r \in R$$ annihilates some element $$m \in M$$, we can find an element $$m' \in M$$ such that $$r$$ annihilates $$m'$$, and further, such that the annihilator of $$m'$$ is a prime ideal (this will turn out to be the associated prime containing $$r$$).

Related facts

 * Noetherian implies every element in minimal prime is zero divisor
 * Reduced Noetherian implies zero divisor in minimal prime