If I have a rectangular matrix $A$ (say $4 \times 6$) with integer entries, is there a way to tell whether it has a right inverse that also has integer entries. I know that if $AA^T$ has determinant $\pm 1$ then $A^T(AA^T)^{-1}$ will give such an inverse, but there are infinitely many right inverses, so is there a way to tell if there is an integral one if this one doesn't work?