= Reduced locus of a ring
{title2=$\{P\in\operatorname{Spec}R:R_P\text{ is reduced}\}$}
The subset of the <spectrum of a ring> where the <localization at a prime ideal> is <reduced>. Since the <nilradical commutes with localization>, its complement is the <support of a module> $N=\operatorname{Nil}(R)$. When $N$ is a <finitely generated module>, <vanishing of a finitely generated module near a prime> gives
$$
\mathcal R=\operatorname{Spec}R\setminus V(\operatorname{Ann}_R(N)),
$$
so the locus is a <Zariski-open subset>. In particular this holds for every <Noetherian ring>.
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