How Can I prove that this module is isomorphic to the generators with relations...

Question

Let $G$ be a finite group of order $n$. Let $R=\mathbb{Z}G$ and $N=\sum_{g\in G} g$, observe that $gN=Ng=N$, $N^2=nN$. Let $r\in \mathbb{Z}$ be prime to $n$ and let $P_r$ be the ideal of $R$ generated by $r$ and $N$. How can I prove that $P_r$ is the universal $R$-module defined by two generators $u$ and $v$ and the relations $gv=v$ $\forall g\in G$ , $Nu=rv$. The idea is to send $u$ to $r$ and $v$ to $N$. But I'm not able to prove that its injective. Any help is appreciated.