Asking about a cyclic $D$-module isomorphic to $\frac{D}{(p^k)}$.

Question

I am sorry if the post is duplicated, I have not been able to find the following result:

Let $D$ be a principal ideal domain, let $p\in D$ be an irreducible element. Let $M$ be a cyclic $D$-module isomorphic to $\frac{D}{(p^k)}$ for some integer $k\geq 1$.

Prove that $ann_{M}(p)$ is a vector space over $\frac{D}{(p)}$ with dimension 1.

Could you give me a proof or a source where to find it? Thank you very much.

Answer 1

Let $\mu:M\to pM$ denote the multiplication by $p$; we have to prove $\operatorname{Ker}\mu\cong D/pD$. By third isomorphism theorem we have an exact sequence of $D$-modules: $${0}\to D/pD\to D/p^kD\xrightarrow\mu pD/p^kD\to\{0\}$$ which proves the assertion.