Elementary divisors
In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R {\displaystyle R} is a PID and M {\displaystyle M} a finitely generated R {\displaystyle R} -module, then M is isomorphic to a finite direct sum of the form M ≅ R r ⊕ ⨁ i = 1 l R / ( q i ) with r , l ≥ 0 {\displaystyle M\cong R^{r}\oplus \bigoplus _{i=1}^{l}R/(q_{i})\qquad {\text{with }}r,l\geq 0} , where the ( q i ) {\displaystyle (q_{i})} are nonzero primary ideals.