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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.

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There was an MO question about this, but I can't find it. – Qiaochu Yuan Sep 24 '10 at 0:39
Perhaps you meant this question about finitely generated abelian groups.… – Weaam Sep 24 '10 at 0:44
I think Jacobson's proofs in Basic Algebra I are a good baseline. I don't know that anything written there is particularly elegant, but I think it's relatively elementary. – rschwieb Aug 1 '14 at 14:35
@Weaam I wonder if the proof Milne refers to directly generalizes :) – rschwieb Aug 1 '14 at 14:39

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.)

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