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 .