I've been getting a little lost in algebra today. Let $M$ be a finitely generated $R[x]$-module where $R$ is a PID. There is a short exact sequence $$0\to tM \to M \to F \to 0 $$ where $tM$ is the torsion submodule (consisting of those $m\in M$ with $p\cdot m=0$ for some non-zero $p\in R[x]$) and $F$ is torsion-free.
Question: Does the above sequence split? In other words, can I write $M\cong tM \oplus F$?
I believe the answer is yes if $R$ is a field, because then $R[x]$ is a PID and in that case finitely generated torsion-free modules should be projective. But what happens if $R$ is not a field? I don't expect a positive answer, in general, but I'm too dense to think of a counterexample.
By the way, I'm mostly interested in the case $R=\mathbb Z$, if it makes a difference. Any pointers are appreciated.