Proofs of the structure theorem for finitely generated modules over a PID
I'm looking for different proofs (references or sketch of main ideas) of the structure theorem for finitely generated modules over a PID.
If possible, a comparison in terms of clarity, elegance or usefulness would be appreciated.
Solution 1:
Fred Goodman has a freely downloadable algebra text that contains this result in section 8.5. Unfortunately, I am not knowledgeable enough to offer a useful comparison to other references. But hey, it's free!
(I studied draft versions of this material as it was being written. I was taking a class from the author, and I found it useful. It became "Edition 2.5" with substantial additions because the 2nd Edition didn't have enough to cover the full first-year graduate level algebra sequence at Iowa.)
Solution 2:
I started from Goodman (2.6/e), then Jacobson, then Weintraub. I completely agree rschwieb's opinion.
These three books not only proof the structure theorem, but also apply the structure theorem on F[x]-module and get the Rational Canonical Form and Jordan canonical form and give some computational examples. Most books just only give you the abstract proof, you still don't know how to compute the rational form and Jordan form after you read the proof. That's why I choose these books.
If you don't understand the proof of a theorem in Jacobson, you can refer Goodman's proof. For example, the proof of the existence of the smith normal form (Jacobson's Theorem 3.8 or Goodman's Corollary 8.4.6).
The proof of Goodman is more readable than Jacobson's. Because Goodman uses a homomorphism $f$ from a free module of rank $n$ to the module (over a PID) which generated by $n$ elements (Section 8.5). Then we can use the theorem about the submodule $\ker{f}$ of a free module (Goodman's Theorem 8.4.12 or Jacobson's Theorem 3.7). But this approach cause confusion when I compute the Rational Canonical Form (Exercise 8.6.3).
Jacobson's proof is a little hard to read (for me). But it will clear when you read it second time. A disadvantage of Jacobson is that he doesn't specify the index of the summation in the proof of the structure theorem.
In my experience, you can skip the proof of the uniqueness of the structure theorem when you learn the theorem first time. Because I always don't know where am I after I proved the uniqueness. If you want to know that which lemma is necessarily for proving the existence, this chart maybe helpful. (Goodman's proof.)
