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.
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What is $N = \sum_g g$ ? You gave no set of summation over. – Daniel Donnelly Jul 15 '14 at 22:55
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$\sum_{g\in G}g$. – Berci Jul 15 '14 at 22:57
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Precisely! Fixed it. Thanks! – Max Jul 15 '14 at 23:31