Elementary divisor theorem

Let $R$ be a principal ideal domain and $M$ a finitely generated $R$ -module. Then there are an integer $r\ge0$ , prime elements $p_1,\ldots,p_t\in R$ , and positive integers $e_1,\ldots,e_t$ such that

M \;\cong\; R^{\,r}\;\oplus\;\bigoplus_i R/(p_i^{e_i}),

The integer $r$ and the multiset of ideals $(p_i^{e_i})$ are uniquely determined by $M$ , up to reordering.

Relation to invariant factors

This is equivalent to the invariant-factor form of the structure theorem over a PID: factor the invariant factors into prime powers and regroup their primary summands. For $R=\mathbb Z$ , it gives the primary decomposition in the classification of finitely generated abelian groups.