I'm trying some easy questions on module theory it seems true but I cannot fill the detail of proof. Please help me.

Question1. Let $A$ be a finitely generated $\mathbb{Z}_p [t,t^{-1}]$ module and every element of $A$ is divided by $t-1$. Is $|A|<\infty ?$

Question2. Let $K$ be a cyclotomic field $\mathbb{Q}(\zeta_m)$, where $\zeta_m$ is $m$-th root of unity and $K(t)$ be a field of rational functions over $K$. Then, $K(t)$ is module over $\mathbb{Z}[\mathbb{Z}_m\times \mathbb{Z}]$ if we regard $\mathbb{Z}[\mathbb{Z}_m\times \mathbb{Z}]=\mathbb{Z}[\mathbb{Z}_m][t,t^{-1}]$ and $\mathbb{Z}_m=<\zeta_m>$. Is $K(t)$ is flat?