Elementary divisor theorem

Over a PID, a finitely generated module decomposes into primary cyclic summands.

Elementary divisor theorem

Elementary divisor theorem: Let $R$ be a PID and let $M$ be a finitely generated $R$ -module. Then there exist $r\ge 0$ and a finite multiset of prime powers $\{p_i^{e_i}\}\subset R$ such that

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

where each $p_i$ is a prime element of $R$ . The multiset of prime power factors is unique up to associates and reordering.

This is equivalent to the invariant factor decomposition in the structure theorem over a PID by factoring each invariant factor $d_j$ into prime powers and regrouping into primary components. For $R=\mathbb Z$ , it recovers the primary decomposition in the classification of finitely generated abelian groups

Proof sketch (optional): Start from the invariant factor decomposition and use unique factorization in a PID to write each $d_j$ as a product of prime powers. Show that $R/(ab)\cong R/(a)\oplus R/(b)$ when $(a,b)=1$ , then iterate and regroup by primes.