 Structure theorem for finitely generated modules over a principal ideal domain

Every finitely generated module over a PID is a direct sum of a free module and cyclic prime-power torsion modules.

The Structure Theorem for finitely generated modules over a principal ideal domain (PID) is a fundamental result in abstract algebra, specifically in the study of modules over rings. It describes the classification of finitely generated modules over a PID in terms of simpler components. Here’s a concise statement of the theorem: Let \( R \) be a principal ideal domain, and let \( M \) be a finitely generated \( R \)-module.