Nilradical commutes with localization ($\mathrm{Nil}(S^{-1}R)=S^{-1}\mathrm{Nil}(R)$)

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 $t{a}^n=0$. Then $(ta{)}^n=t^{n-1}t{a}^n=0$ and $a/s=ta/(ts)$ has a nilpotent numerator. This proves the formula for the nilradical without any finiteness assumption.