Do a matrix and its transpose have the same invariant factors over a PID?

Question

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,

Answer 1

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.