Do a matrix and its transpose have the same invariant factors over a PID? I suspect this is true since it holds in the case over a field. But suppose $A\in M_{m\times n}(R)$ where $R$ is a PID. Does it still hold that $A$ and $A^{T}$ have the same invariant factors?
Regards,
 A: The answer is yes. Indeed, if $d_1,\dots,d_r$ are the invariant factors, and if $d_i$ divides $d_{i+1}$ for $1\le i < r$, then $d_1\cdots d_i$ is the gcd of the $i$-minors of any presentation matrix.
A: If you know that the invariant factors are the entries on the diagonal after diagonalizing the matrix by (perhaps generalized) row and column operations, then, given row and column operations that diagonalize A, ask yourself what the corresponding column and row operations do to the transpose.
The moral I am trying to convey is, that to answer this question, one should begin with a definition or computation of the invariant factors, and then ask what happens when that same characterization is applied to the transposed matrix.  I.e. one should ask whether the given construction is in fact unchanged by taking transpose.  This describes also M. Gaillard's answer.
Another equivalent point of view prceeds from the knowledge that the invariant factors of A occur on the diagonal of a diagonal matrix D obtained as a product of form XAY = D.  Then ask if one can find a similar equation involving the transpose of A.
If you have in fact some other characterization of the invariant factors, I urge you to use that one to give another answer.
