Nilpotent elements

Let $R$ be a commutative ring.

Prove that the set $N$ of all nilpotent elements of $R$ is an ideal.
Prove that $R / N$ is a ring with no nonzero nilpotent elements.
Show that $N$ is contained in every prime ideal of $R$ .

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L A T E X

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Let $R$ be a commutative ring.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that the set $N$ of all nilpotent elements of $R$ is an ideal.
	\item Prove that $R/N$ is a ring with no nonzero nilpotent elements.
	\item Show that $N$ is contained in every prime ideal of $R$.
\end{enumerate}