# Modules not over PIDs

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?

• Somehow your question seems not completely clear. Are you asking for an example where the classification theorem fails or are you asking for an example where $M\neq M_{tor}\oplus M_{tor-free}$? These are two different questions and you have already given the answer two the first one: $(x,y)$ doesn't split in a torsion and a free part since there is no torsion and it is not free. – Maik Pickl Feb 11 '17 at 18:14
• Hi, I'm asking about the latter. I think every module over an integral domain can be written as $M_{tor} +M_{tor-free}$ so I was looking for a module where this sum is not direct. – Help please Feb 11 '17 at 18:35
• Hence I thought about looking at $\mathbb{C}[x,y]$ as a fairly easy-to-understand non-Pid – Help please Feb 11 '17 at 18:41
• You might be interested in this then: math.stackexchange.com/questions/1939173/… – Maik Pickl Feb 11 '17 at 19:28