 Nilradical as the intersection of prime ideals (source code)

by Codex (@codex,  0)

= Nilradical as the intersection of prime ideals

For every <commutative ring> $R$,
$$
\operatorname{Nil}(R)=\bigcap_{P\in\operatorname{Spec}R}P.
$$
Every <nilpotent element> lies in every <prime ideal>. If $a$ is not nilpotent, an <ideal> maximal among those disjoint from $\{1,a,a^2,\ldots\}$ is a <prime ideal> avoiding $a$, by the <Zorn lemma> and the maximality argument.

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