Let $R$ be a commutative Noetherian ring and $A\in M_{n\times m}(R)$.
If $R$ is a field, then $rank(A^t)=rank(A)$.
However, in general Noetherian rings, does $rank(A^t)=rank(A)$ hold?
Since the usual proof technique for proving it in the context of a field cannot be applied to a Noetherian ring, I wonder whether this is true or not. (The usual technique assumes $R$ is a division ring)
What is a counterexample? And if this is indeed true, how do I prove it?