Modules not over PIDs

Question

I've seen that there is a classification of finitely generated modules over a PID in terms of torsion and torsion-free modules. I'm trying to think of examples of finitely generated modules not over a PID where this does not hold, i.e. that cannot be written as $M_{tor} \oplus M_{tor-free}$. My first thought is some form of polynomial ring in two variables e.g. $\mathbb{C}[x,y]$ and somehow define an action of $\mathbb{C}[x,y]$ on itself where the torsion-free component is $(x,y)$ as this is not a free module. Is there a good text on this with examples?