TITLE: For $\mathfrak{m}$ maximal and principal, there's no ideal between $\mathfrak{m}^2$ and $\mathfrak{m}$
QUESTION [5 upvotes]: Let $R$ be a commutative ring with unity. If a maximal ideal $\mathfrak{m}$ of $R$ is principal, prove that there is no ideal $I$ with $\mathfrak{m}^2\subsetneq I\subsetneq \mathfrak{m}$.

I have no idea how to start this one - I can't see why we can say anything specific about $I$.

REPLY [3 votes]: Hint: you may already know this, but if not show that $\mathfrak m$ is principal if and only if there exists a surjection of $R$-modules $\pi: R \twoheadrightarrow \mathfrak m$. What is $\pi(\mathfrak m)$?