The context is the uniqueness of finitely generated PID module decomposition. The text I am using is Aluffi's Algebra Chapter 0.
For the uniqueness, it reduces to prove (where $R$ is PID, $q$ are prime elements in $R$, and $r_i,s_i$ are increasing) \begin{align*} \frac{R}{q^{r_1}}\oplus\cdots \oplus \frac{R}{q^{r_n}}\cong \frac{R}{q^{s_1}}\oplus\cdots \oplus \frac{R}{q^{s_m}} \end{align*} implies $n=m$ and $r_i=s_i$ for all $i$. The hint given is consider the homomophism $m\rightarrow qm$. By induction, if $m=n$ we can show $r_i$ and $s_i$ are the same. But how can I show $m=n$?
It seems we have to eliminate the possibility that \begin{align*} \frac{R}{q^{r}}\cong \frac{R}{q^{r}}\oplus \frac{R}{q^{s}} \end{align*} where $s\leq r$. This is simple if $R$ is finite (i.e. finite abelian group case). However, when $R$ is infinite, I stuck at this question. I tried to show $R/q^{r}$ fail to satisfy the universal property of the product, but it seems does not work.
Does anyone have any comments or ideas?
Thanks in advance