Let $G$ be the group $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}=\langle y,z\rangle\rtimes_{\sigma}\langle x\rangle$, where $\sigma(x)=\begin{pmatrix}a, b\\c,d\end{pmatrix}\in SL_2(\mathbb{Z})$, which means that we have relations $xyx^{-1}=y^az^c, xzx^{-1}=y^bz^d$. Then we can form the group ring $R=\mathbb{Z}G$, note that since $G$ is a polycyclic-by-finite group, $R$ is a left Noetherian ring.

I am interested in what a general prime ideal $\wp$ in $R$ looks like, so I want to ask the following question:

**What does Spec(R) generally look like? Especially when $\sigma(x)=\begin{pmatrix}1,0\\1,1\end{pmatrix}$ or $ ~ \sigma(x)=\begin{pmatrix}2,1\\1,1\end{pmatrix}$**