Setup: We know that an $R$-module $M$ is finally presented if it is isomorphic to $R^m/AR^n$, where $A$ is an $m$-by-$n$ matrix known as the presentation matrix. Determining $A$ amounts to picking a set of $m$ generators of $M$, which determines a homomorphism $\phi: R^m \rightarrow M$, and then picking a set of $n$ generators for $\ker(\phi)$.
Next, we may perform various procedures to simplify $A$. We can change basis to put $A$ into Smith normal form. We may then remove any columns that are all $0$, and we can also remove the row and column associated with any diagonal entry that is a unit.
Question: after conducting this procedure, do we always end up with the same canonical presentation matrix (up to units) regardless of how we originally chose the generators? If $M$ is an abelian group, I think this amounts to the uniqueness of the fundamental theorem of finite abelian groups, so I am hopeful that this works in general.
