The nilradical of a ring

#ring_theory

Let $R$ be a commutative ring. The nilradical of $R$ is defined to be $N = {r \in R | r^{n} = 0 for some n \in N}$ .
(a) Prove that $N$ is an ideal of $R$ .
(b) Prove that $N$ is contained in the intersection of all prime ideals of $R$ .

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Let $R$ be a commutative ring. The {\bfseries nilradical} of $R$ is defined to be $N=\{r\in R\,|\, r^n=0\text{ for some }n\in {\bf N}\}$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $N$ is an ideal of $R$.
	\item Prove that $N$ is contained in the intersection of all prime ideals of $R$.
\end{enumerate}