Show that R is a PID if R is a local noetherian integral domain, whose unique maximal ideal is non-trivial and principal

Question

Let R be a local noetherian integral domain, whose unique maximal ideal is
non-trivial and principal.

I need to show that R is a PID.

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We have the hint that we should use Nakayamas Lemma. However, I have no idea how that should help.

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What I tried:

Let $(m)$ be the unique maximal ideal in $R$ and $I$ be any ideal. So: $I = (a_1,…,a_n) \subset (m)$.

So we get $m \,|\, a_i$ for all $i$. With Nakayama we get: $r \in (m) \iff m\,|\,r \iff (\forall x\in R: \,1-rx \in R^*)$

But I don't get anything useful here…

Answer 1

Hints. Let $J=\bigcap_{r\ge1}(m^r)$. Show that $J=mJ$, and then use Nakayama. Write $I=m^k(b_1,...,b_n)$ with not all $b_i$ in $(m)$.