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Let $R$ be an euclidean domain, and $A$ a $m\times n$ matrix. I want to prove two things:

1) The torsion submodules of $\mathrm{Coker}\;A$ and $\mathrm{Coker}\;A^T$ are isomorphic.

2) $\mathrm{Coker}\;A$ and $\mathrm{Coker}\;A^T$ are isomorphic is and only if $n=m$.

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Up to multiplying on the left and right by invertible square matrices, you can assume that $A$ is diagonal—a keyword to find this is «Smith normal form». With that hypothesis, your two things are easy.

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