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:

1. 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 $E_p$

1. Lang says he does it by induction, induction over what?

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

• Do you need to understand Lang's version of the proof? There are better and more understandable proofs elsewhere IMO. – Qudit Dec 7 '17 at 21:46
• I would prefer Lang's version, the thing is that in the course I am taking, most of it is a build up to this theorem, and probably other proofs use tools that escape the course. I have seen a really short proof using exact sequences and Jordan matrix, but it seems like ''cheating'' if you will. – Temporary Dec 8 '17 at 0:19