I was working through a problem that was showing that for a field $F$, $F[x]$-modules correspond to pairs $(V,T)$ of vector spaces over $F$ and linear transformations on that vector space.
The last part of the question asks to find a finitely generated $F[x]$-module which is not projective.
The only examples I really know of modules which are not projective are ones like $\mathbb{Z}/2\mathbb{Z}$ as a $\mathbb{Z}$-module, where I understand it not to be projective because it doesn't have enough homomorphisms going out to $\mathbb{Z}$: every module homomorphism leaving $\mathbb{Z}/2\mathbb{Z}$ has to satisfy $0=f(0)=f(2\cdot z)=2\cdot f(z)$ and $2\cdot f(z)$ implies f(z)=0 in $\mathbb{Z}$. This is a problem since $\mathbb{Z}$ does project onto modules which $\mathbb{Z}/2\mathbb{Z}$ has nontrivial homomorphisms into (e.g.: itself).
I've been trying to think along similar lines for this case of finitely generated $F[x]$ modules, using the above correspondence with pairs $(V,T)$. I've been trying to think of a finite dimensional vector space $V$ and a linear transformation $T_V$ on it, and another vector space $W$ with transformation $T_W$ such that there aren't many $F[x]$-module homomorphisms $(V,T_V)\to (W,T_W)$.
I know such an $F[x]$-module homomorphism has to satisfy, among other things, that $f(T_V \cdot v)=T_W \cdot f(v)$. Using the correspondence with vector spaces and transformations, I've been trying to think of matrices $T_V$ and $T_W$ which only satisfy $f T_V = T_W f$ for very restricted choices of $f$, hoping for an analogous situation to the $\mathbb{Z}/2\mathbb{Z}$ example. I'm not so sure that's a good idea though since I don't know really understand how important it is to that example that there's a nontrivial element ($2$) that annihilates all of the elements.
So, does it make sense to be trying to think of examples this way for this case? Can this be done for arbitrary $F$ or should I be looking at a specific field to get an example? Does anyone have any examples? (I'd prefer to be able to generate my own, but since my current repository of non-projective modules is very sparse, I'd still appreciate any contributions.)