Past exam of the commutative algebra course of the University of Oxford / 2023 / 3 / b / Solution

Write $N=\sqrt{(0)}$. Every element of the extension of an ideal $N{R}_{\mathfrak{p}}$ is a fraction $a/s$ with $a\in N$ and $s\notin \mathfrak{p}$: a finite sum of such fractions has a common denominator and numerator still in the ideal $N$. If $a^n=0$, then $(a/s{)}^n=0$, soConversely, let $a/s$ be a nilpotent element of the localization at a prime ideal $R_{\mathfrak{p}}$. For some $n\ge 1$, $a^n/s^n=0$. The vanishing criterion in a module localization supplies $t\notin \mathfrak{p}$ with $t{a}^n=0$. It follows thatso $ta\in N$. Since $ts\notin \mathfrak{p}$,This proves that the nilradical commutes with localization:The multiplication by $t$ is necessary: the numerator $a$ of a nilpotent element in a localization need not itself be nilpotent in $R$.