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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?

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For the future readers: $\mathbb Z_p$ does NOT mean the $p$-adic integers here... –  darij grinberg Jul 25 '10 at 13:48
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1) More generally, if $F$ is a field, and $A$ is a finitely generated $F\left[t,t^{-1}\right]$-module and every element of $A$ is divisible by $t-1$, then $A$ is a finite-dimensional $F$-vector space. Proof: Nakayama. –  darij grinberg Jul 25 '10 at 14:22
    
Thank you. Now I am convinced –  Topologieee Jul 25 '10 at 14:40
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For (2). $A=\mathbf{Z}[\mathbf{Z}_m]$ is the ring of integers of $K$; $K$ is the field of fractions of $A$. And $K(t)$ is evidently the field of fractions of $A[t,1/t]$, hence flat over $A[t,1/t]$. –  George McNinch Jul 25 '10 at 15:28

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