Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 2 / 2E / ii / Solution

The structure theorem for finitely generated modules over a principal ideal domain applies because every Euclidean domain is a principal ideal domain. In invariant-factor form, every finitely generated module $M$ over such a domain $R$ has a decompositionwhere the $d_i$ are nonzero nonunits, $r\ge 0$, and either list of summands may be empty. The free rank $r$ and the invariant factors $d_i$ are unique up to multiplication of each $d_i$ by a unit. Equivalently the torsion part is a direct sum of cyclic modules with prime-power annihilators. The nonunit convention omits zero summands $R/(1)$.