Consider $R=\mathbb{Z}/p\mathbb{Z}$ for $p$ prime and $n>1$ as ring and $M_k=\mathbb{Z}/p^k\mathbb{Z}$ as $R$- module ($k \leq n$). I want to find a projective resolution of $M_k$. $R$ is projective as $R$- module and my idea is to use a projective resolution of $M_k$ as $\mathbb{Z}$-module and generalise that.
$$0 \to \mathbb{Z} \to \mathbb{Z} \to 0$$ where the map $\mathbb{Z} \to \mathbb{Z}$ is the product $\cdot n$.
I think that if I consider the product $\cdot p^{n-k}$ I can do something:
$$P^{\bullet}:\cdots \to \mathbb{Z}/p^n\mathbb{Z}\to \mathbb{Z}/p^n\mathbb{Z} \to 0 $$ I obtain $H^0(P^\bullet)= \mathbb{Z}/p^k\mathbb{Z}$, but In this case, $H^{-n}(P^\bullet)$ is not $0$ for $n>0$.
Is there a way to fix it? If not, how can I found a projective resolution of $M_k$ on $R$?