Powers of maximal ideal in local ring with a single prime ideal.

Question

Let R be a non zero, local, Noetherian ring with $\mathfrak{m}$ the maximal ideal of R. If we assume that $\mathrm{Spec}R=\{\mathfrak{m}\}$, what can we say about powers of $\mathfrak{m}$ ? I have found that if $R$ is an integral domain then the powers of $\mathfrak{m}$ are distinct (Application of Nakayama's Lemma).
On the other hand if R is not integral domain i have found example such that powers of $\mathfrak{m}$ are not distinct. For example, if $R=\mathbb{Z}_{4}$ then $R$ is local with maximal ideal $\mathfrak{m}=2\mathbb{Z}_{4}$ and $\mathrm{Spec}R=\{2\mathbb{Z}_{4}\}$.
Any insight on how i can generalize this for any R be a non zero, local, Noetherian ring ?