Let $\mathbb{F}$ be a field (of characteristic 0, if needed) and $\mathbf{V}$ an $\mathbb{F}$-vector space. Let $T\in\mbox{End}_\mathbb{F}(\mathbf{V})$ be an endomorphism, and let (following Bourbaki) $\mathbf{V}_T$ be the $\mathbb{F}[X]$-module given by $p(X)\cdot v=p(T)v$.
Question: Is there a non-trivial but tractable sufficient condition on $\mathbf{V}$ and/or $T$ for the torsion submodule of $\mathbf{V}_T$ to be a direct factor?
To be honest, I don't even know a counterexample where this fails to hold. For a general PID I found a counterexample here.
This is trivially true when $\dim\mathbf{V}<\infty$ because then $\mathbf{V}_T$ is torsion (Jordan normal form for matrices). By non-triviality I mean that $\mathbf{V}_T$ is not torsion.
Comments are welcome. I apologize for my illiteracy in algebra in advance.
Thank you.
Edit: Thanks for all comments. Now I have understood that one such condition is that $\mathbf{V}_T$ is finitely generated over $\mathbb{F}[X]$. What about the second, apparently more general case mentioned by Denis Nardin below? Is it true that if $\mathbf{V}_T/\mathbf{V}_T^{tors}$ is free then it is a direct factor?
Edit 2: Thanks again for the comments. It seems to me that every Noetherian torsion free module of a PID is necessarily free. Is there an example of a torsion free $\mathbb{F}[X]$-module that is not Noetherian?