Equivalence of Definitions of Nilradical of Ring

Theorem
Let $A$ be a commutative ring.

The following definitions of its nilradical are equivalent:

Proof
By Nilpotent Element is Contained in Prime Ideals, $\operatorname{Nil}(A)$ is contained in the intersection of all prime ideals.

It remains to prove the other inclusion.

Let $f \in A$ be not nilpotent.

Let $S$ be the set of ideals of $A$ that are disjoint from $\{f^n : n \in \N\}$.

By Zorn's Lemma, $S$ has a maximal element $P$.

In particular, $f\notin P$.

We want to show that $P$ is prime.

Let $a,b\in A$ with $a,b\notin P$.

Then the sums of ideals $(a)+P$ and $(b)+P$ contain $P$ strictly.

By the maximality of $P$, there exist $n,m\in\N$ with $f^n\in (a)+P$ and $f^m\in (b)+P$.

Then $f^{m+n}\in (ab)+P$.

Thus $(ab)+P\notin S$; in particular $ab\notin P$.

Thus $P$ is prime, and $f\notin \bigcap_{\mathfrak p \in \operatorname{Spec} \left({A}\right)}\mathfrak p$.