Let $A$ be a noetherian ring, and let $Ass(A/0)=\{{\mathfrak{p}}_1,\dots ,{\mathfrak{p}}_n\}$. Then:

1. ${\mathfrak{p}}_1\bigcap \dots \bigcap {\mathfrak{p}}_n=$1;;the nilradical of $A$
2. ${\mathfrak{p}}_1\bigcup \dots \bigcup {\mathfrak{p}}_{\mathfrak{n}}=$1;; the zero divisors of $A$

Proof:

1. Use the previous proposition
2. Use that the 1;;zero divisors of $A$ are equal to 1;;$\bigcup _{x\ne 0}r(0:x)$