Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 11G / a / Solution

A Smith normal form of an $m\times n$ matrix $A$ over a principal ideal domain $R$ is a diagonal matrixwhere $U\in G{L}_m(R)$ and $V\in G{L}_n(R)$. It is guaranteed to exist for every matrix over a principal ideal domain. The entries $d_i$ are determined up to multiplication by units; over $\mathbb{Z}$ they are conventionally chosen positive.