My knowledge of algebra is undergraduate linear algebra, so I apologize for my complete ignorance.
Thinking about Jordan normal forms, I unintensionally came to an idea that turned out to be called a cyclic decomposition of a PID module. More precisely, if I have got it correctly, any finitely generated module of a PID can be written as a direct sum of a finite number of cyclic modules and a simple module. I wonder what is wrong with infinitely generated modules: why can't they be written as infinite directs sums of cyclic modules plus a simple module? Intuitively, it seems to me that this has to work in arbitrary dimensions. What I have in mind is in fact a Euclidean domain, to be honest, because in my arguments I use the Euclidean algorithm. I guess that if smart people have chosen to content themselves to the finite case, then there should be something wrong with infinite. I would like to understand in simple terms (undergraduate level) what the matter is there.
To be more precise, here are two questions.
Whether and why does cyclic decomposition fail for infinitely generated PID modules.
The same question for modules of a Euclidean domain.
Thank you.