I am trying to prove that if $A$ and $B$ are matrices over the integers of the same dimension $m\times n$ and they have the same row space (that is, the set of all linear combinations with integer coefficients of the rows of the matrix), then there must exist a unimodular matrix $U$ (that is, an integer matrix with determinant equal to $\pm 1$) such that $A=UB$.
I am really stuck at this. I only have an argument that would work if I could assume that $A$ and $B$ have full row rank:
Since the row spaces of the matrices are the same, the rows of each matrix can be expressed as a combination of the rows of the other matrix, which means that there exist integer matrices $V$ and $W$ such that $A=VB$ and $B=WA$. From this, we have that $A=VWA$ and $B=WVB$. If $A$ and $B$ have full row rank, that implies that $VW=I$ and $WV=I$, so $V$ and $W$ are inverse of each other, so they are unimodular and I have the result.
Any hints as to how I could prove the result without having to assume full row rank would be very helpful. Thanks in advance.