Let $R$ be a commutative ring.

(a) Let $N$ be the set of nilpotent elements of $R$, that is,

$$N=\left\{r \in R \mid r^{n}=0 \text { for some } n \in \mathbb{N}\right\}$$

Show that $N$ is an ideal of $R$.

(b) Assume $R$ is Noetherian and assume $S \subset R$ is a non-empty subset such that if $s, t \in S$, then $s t \in S$. Let $I$ be an ideal of $R$ disjoint from $S$. Show that there is a prime ideal $P$ of $R$ containing $I$ and disjoint from $S$.

(c) Again assume $R$ is Noetherian and let $N$ be as in part (a). Let $\mathcal{P}$ be the set of all prime ideals of $R$. Show that

$$N=\bigcap_{P \in \mathcal{P}} P$$