Examining a module as both an $F$ and $F[x]$ module I am studying for a comp from a large comprehensive exam bank and I ran across the following problem which I feel I should be able to do yet I have no idea where to even start.  
Suppose we have a field $F$ and an indeterminate $x$.  Also, suppose we have a cyclic $F[x]$-module $M$ which is not free.  Finally assume that we have that the ideal $I=\lbrace{ p(x)\in F[x]| pm=0, \forall m\in M\rbrace}$ is generated by $g(x)=f_0+ \ldots + f_nx^n$. 
There are a couple things I wish to prove related to $M$.  I want to show that M is finite dimensional as an $F$-module and give a basis for $M$ as an $F$-module.  Also, I want to prove the map m \to xm is F-linear.  Obviously our ideal $I$ is important as is its generator but I don't see their significance.  
Thanks for the help.

Let $v$ be the generator of your module and consider the elements $v,xv,\ldots,x^{n-1} v$. We claim that these generate your module as an $F$ - vector space. Take any element $f(x)v \neq 0 \in M$ for some polynomial $f(x) \in F[x]$. 
Then by the division algorithm we can write $f(x) = g(x)q(x) + r(x)$ where $ 0 \leq \deg r(x) < \deg g(x)$. Thus $f(x)v = r(x)v$ where $r$ is a polynomial of degree less than $g$, since $g$ is the generator of $\text{Ann}(M)$. That is, we can write $r(x) = a_0 + a_1x + \ldots + a_kx^k$ where $k < n$. Thus $r(x)v = a_0 + a_1xv + \ldots + a_kx^kv$ as claimed.
