Structure theorem for finitely generated modules over a PID

Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$ -module. Then there exist an integer $r\ge 0$ and nonzero nonunits $d_1,\dots,d_t\in R$ , with $d_1\mid d_2\mid\cdots\mid d_t$ , such that

M \;\cong\; R^{\,r}\;\oplus\;\bigoplus_{i=1}^t R/(d_i),

a decomposition as a direct sum of a free part and cyclic torsion factors. The integer $r$ , the number $t$ , and the invariant factors $d_i$ are unique, with each $d_i$ determined up to multiplication by a unit.

Consequences

The theorem gives the classification of finitely generated abelian groups when $R=\mathbb Z$ . It is equivalent to Smith normal form and also has an elementary-divisor form.