r is the torsion-free rank of M and the di are the invariant factors. This decomposition is unique up to multiplying the invariant factors by units. When R = ℤ, then the classification of finite abelian groups follows. When R = F[x], where F is a field, the rational canonical form follows, and when R = ℂ[x], a similar version of the structure theorem gives the Jordan normal form. See also Smith normal form.