The nilradical of a ring
Let
- Prove that
is an ideal of . - Prove that
is contained in the intersection of all prime ideals of .
View code
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}