Proposition 13

The nilradical of a ring $A$ is the intersection of all the prime ideals of $A$ and it is also the intersection of the minimal prime ideals of $A$.

Clearly, if $x \in A$ is nilpotent, it is contained in every prime ideal of $A$ (§ 1, no. 1, Definition 1). Conversely, let $x$ be a non-nilpotent element of $A$; the set $S$ of $x^k$ ($k$ an integer $\geqslant 0$) is then a multiplicative subset of $A$ not containing 0 and hence there exists a prime ideal $p$ of $A$ not meeting $S$ (no. 5, Corollary 2 to Proposition 11) and a fortiori $x \notin p$; this establishes the first assertion. To prove second, it is sufficient to prove