Nilpotent elements
Let
- Prove that the set
of all nilpotent elements of is an ideal. - Prove that
is a ring with no nonzero nilpotent elements. - Show that
is contained in every prime ideal of .
View code
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}