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I am looking for an example or application of this theorem that isn't completely trivial or too involved. I am giving a seminar talk and I've got a few examples. I'm looking for one more. But I can't find any good examples surfing the web. Examples so far.

1) Structure theorem for finitely generated abelian groups. 2) If $k$ is an algebraically closed field and $T$ is a linear operater on the finite dimensional vector space $\mathcal{V}$ then we can turn $\mathcal{V}$ into a $k[x]$ module and express it using the structure theorem.

Any ideas would be much appreciated. Thanks

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how about the ring of integers of a number field? though that is a little bit more "involved" –  user641 Oct 18 '13 at 1:28
Well I'm more interested in taking a module over a DD and factoring it using the theorem. –  TheNumber23 Oct 18 '13 at 1:42
ok what about the well-definedness of the ideal class? –  user641 Oct 18 '13 at 1:46

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