Quotient of Dedekind rings by a power of a prime ideal

Question

Let $ A $ be a Dedekind ring, let $ P$ be a nonzero prime ideal of $ A $. What can we say about $ P^n $ or $ A/P^n $ in general? For example, is it true that $ A/P^n $ is a vector space over $ A/P $ of dimension $ n $?

Answer 1

No, it's not true since this ring, as an $A$-module, is not killed by $P$ as an $A$-module (as a counter-example, consider $\mathbf Z/p^n\mathbf Z$: is it a $\mathbf Z/p\mathbf Z$-vectorspace.

The nearest assertion which is true is that, for instance $P^{n-1}/P^n$ is an $A/P$-vector space.

The only other thing I can see that can be said about $A/P^n$ is that it's a local (noetherian) ring of Krull dimension $0$.