= Nilradical commutes with localization
{title2=$\operatorname{Nil}(S^{-1}R)=S^{-1}\operatorname{Nil}(R)$}
For any <multiplicative subset> $S$ of a <commutative ring> $R$, fractions with <nilpotent> numerators are <nilpotent> in $S^{-1}R$. Conversely, if $(a/s)^n=0$, the <vanishing criterion in a module localization> gives $t\in S$ with $ta^n=0$. Then $(ta)^n=t^{n-1}ta^n=0$ and $a/s=ta/(ts)$ has a <nilpotent> numerator. This proves the formula for the <nilradical> without any finiteness assumption.
\Comment[[
Exact target: vanishing-of-a-finitely-generated-module-near-a-prime
Existing parent: Localization of a module
Intended file: /home/ciro/bak/git/codex-wiki/commutative-algebra.bigb
]]
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