Smith normal form invariants

Smith normal form invariants: Let $R$ be a PID and let $A\in M_{m\times n}(R)$. Suppose $A$ has Smith normal form $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0)$ with $d_1\mid d_2\mid\cdots\mid d_r$. Then the elements $d_i$ are uniquely determined by $A$ up to multiplication by a unit of $R$, and they determine the isomorphism class of the cokernel module

The uniqueness part comes from the Smith normal form theorem, and the decomposition agrees with the invariant factors in the structure theorem over a PID, computing cokernels over a PID.