Math 6310 • Algebra

Let \(R\) be a commutative ring with 1.
\begin{enumerate}
\item
If a maximal ideal \(\mathfrak{m}\) of \(R\) is principal, prove that there is no ideal \(I\) with
\(\mathfrak{m}^2\subsetneq I\subsetneq\mathfrak{m}\).

\item
Give an example where \(\mathfrak{m}\) is maximal, but there exists an ideal \(I\) with
\(\mathfrak{m}^2\subsetneq I\subsetneq\mathfrak{m}\).
\end{enumerate}