The Smith normal form diagonal entries are canonical invariants and control the cokernel module.
Smith normal form invariants: Let R be a PID and let A∈Mm×n(R). Suppose A has Smith normal form diag(d1,…,dr,0,…,0) with d1∣d2∣⋯∣dr. Then the elements di are uniquely determined by A up to multiplication by a unit of R, and they determine the isomorphism class of the cokernel module
Smith normal form theorem: Let R be a PID and let A be an m×n matrix with entries in R. Then there exist invertible matrices U∈GLm(R) and V∈GLn(R) such that
UAV=diag(d1,…,dr,0,…,0),
where di=0 for 1≤i≤r and d1∣d2∣⋯∣dr. The di are determined uniquely up to multiplication by units; they are the Smith invariants (see Smith normal form invariants).
a decomposition as a direct sum of a free part and cyclic torsion factors. The integer r, the number t, and the invariant factors di are unique, with each di determined up to multiplication by a unit.
A principal ideal domain (PID) is an integral domainR such that every idealI⊆R is principal, i.e. I=(a) for some a∈R.
PIDs provide strong control of divisibility and modules, and they sit between Euclidean domains and unique factorization: every Euclidean domain is a PID, and every PID is a UFD (see PID implies UFD).