Let $K=\mathbb{Q}(\alpha)$ be a number field of degree $4$. Fixed an integer $z$ and a prime $p$, we define the rings $R=\mathbb{Z}[\alpha, z/\alpha]$ and $R_p=\mathbb{Z}[\alpha, z/\alpha, S(\alpha)/p]$ for some polynomial $S\in \mathbb{Q}[X]$ of degree $\leq 3$. I'd like to know the ratio:
$r_{S,p}=\Large \frac{\# \{R_p\text{-modules in }K\}}{\# \{R\text{-modules in }K\}}$
Obs.: Since $R \hookrightarrow R_p$, every $R_p$-module is an $R$-module, and for every $R$-module $M$, the module $pM$ is an $R_p$-module.