Problem with Structure Theorem of PID modules proof I am using Lang's Algebra to prove the Structure Theorem for finitely generated modules over PIDs, and I am having difficulties understanding the proof of the existence of the decomposition for $E(p)$. $E$ is a torsion module over a PID $R$, $p \in R$ prime element, and 
$E(p)=\{m \in E\, ;\, p^nm=0 \, \textrm{for some positive interger n}\}$.
Lang starts with Lemma 7.6 which I understand just fine. The next step is what I do not grasp.
From what I could get, the idea is to show that there is an independent generator of $E(p)$, lets say $\{y_1,...,y_1\}$ and by having that, since it is independent you could see it as a direct sum of cyclic modules, i.e.
$(y_1,..,y_n)= (y_1)\bigoplus...\bigoplus (y_n) \cong\frac{R}{(p^{r_1})} \bigoplus....\bigoplus \frac{R}{(p^{r_n})}$
where $p^{r_i}$ is $y_i$'s period.
This is how he does it (its a copy paste):  
Lang's Proof (Sorry for the sloppiness of the picture, but it was the best I could do with my knowledge) 
Apart from the fact that I don't get the overall proof, here are some doubts that pop to mind:


*

*Is $\overline{E_p}$ well defined? ($E_p=\{m \in E \, ; pm=0\}$)


Because he defines $\overline{E}$ using an element $x_1$ with a maximal period, but $x_1$ is not necessarily in $E_p$


*Lang says he does it by induction, induction over what?


Sorry if the question is not very well formulated. Thanks in advanced.

*

*It's actually $\overline E_p$ and not $\overline{E_p}$ : you take $F= \overline{E}$ and then $F_p$, not $F=E_p$ and then $\overline F$.

*The induction is over the number of generators : you assume the result for all modules that have less than $r$ generators, and prove it for those that have $r$. 
