I think the following statement is not true in the general situations, but consider it:

$R$ is a ring, $\mathfrak{p}$ is a prime ideal, then the unit group of $\dfrac{R}{\mathfrak{p}^nR}$ is isomorphic to $(\dfrac{R}{\mathfrak{p}R})^{*}\times\dfrac{R}{\mathfrak{p}^{n-1}R}$.

This statement holds if $R=\mathbb{Z}$, and $\mathfrak{p}=p\mathbb{Z}$ for some odd prime number. What can we say if we consider $R=\mathcal{O}_K$, where $\mathcal{O}_K$ is the ring of integers of a number field? Can we say anything similar, if we consider the localization of $\mathcal{O}_K$ at some prime ideal? I mean are there some sufficient conditions under which the above statement holds?

Is there any relation with the roots of unity? I mean can we do something similar to the following:

Let $K$ be a number field, and assume that the order of torsion elements in the multiplicative group $K^*$ divides $n$. Then for any prime ideal $\mathfrak{p}$ with $\gcd(\mathfrak{p}, n)=1$ we have: $$(\dfrac{\mathcal{O}_K}{\mathfrak{p}^n})^* \cong (\dfrac{\mathcal{O}_K}{\mathfrak{p}})^{*}\times\dfrac{\mathcal{O}_K}{\mathfrak{p}^{n-1}}.$$