I'm thinking of an alternative proof for Structure theorem for finitely generated modules over a principal ideal domain (also called Fundamental theorem of finitely generated modules over a PID in some books), but I don't know where I am wrong.
Statement: let $R$ be a PID, $M$ be a finitely generated $R$ module, then $M \cong \prod_{i=1}^n R/Ra_i$, where $n>0$ and each $a_i \in R$
So here's my idea: Since a finitely generated $R$-module $M$ is a quotient of a finite free $R$-module $R^n$, we may write $M$ as $R^n/N$, where $N$ is a submodule of $R^n$;
As $R^n$ is a finite direct sum it also has a product ring structure, hence $N$ can also be viewed as an ideal over the product ring $R^n$, while we know that every ideal over $R^n$ can be written as $\prod_{i=1}^{n} I_i$, where each $I_i$ is an ideal of $R$, $M$ is isomorphic to $R^n/\prod_{i=1}^nI_i$;
With some results from ring theory, we may prove that $M\cong R^n/\prod_{i=1}^nI_i \cong \prod_{i=1}^n R/I_i \cong \prod_{i=1}^n R/(a_i)$. The last equivalence is true since $R$ is a PID. As $(a_i) = Ra_i$, the proof is finished.
I searched for this approach but haven't seen any book or references using it, so I guess there's probably something wrong with the proof. So could anyone tell me where the problem is?
