Suppose that $F$ is a field and consider the $F[x]$-module $(V,f)$, with $V$ an $F$-vector space and $f\in\operatorname{End}_F(V)$ a linear operator.
Q1. The most common way to define $f$ is by setting $f(v):=x\cdot v$. Are there other possible definitions that will yield a different module structure?
Now, let's suppose that $V$ is a finite dimensional vector space over $F$.
Q2. Under this condition, can we guarantee that this $F[x]$-module is finitely generated?
Thanks.