Can Direct Sum Isomorphism Imply Module Equality in PID?

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Hi. I'm trying to prove this "little fact": let M, N be finitely generated modules over a PID. Then if M+M=N+N (where = means isomorphism and + means direct sum) then M=N.

I'm sure it can be done with the structure theorem (it is obvious from the hypotheses); it looks like it should be trivially proven, but alas, I don't think it can be.

What do you think?

Cheers,
 
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EDIT: this isn't true the way I originally thought it was...thinking harder!

Well it's certainly not true without the finitely generated over a PID hypothesis (just think about free abelian groups of infinite rank), so I'm guessing it's some particular property of finitely generated modules over PIDs. The structure theorem immediately comes to mind.
 
M + m = n + n.
2m = 2n
m = n
 

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