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

Change the generators to $a=m_1$, $b=m_3-m_2$ and $c=m_2$. This is an invertible integer change of generators, since $m_1=a$, $m_2=c$, $m_3=b+c$. The relations become $2b=0$ and $4c=0$, with no relation involving $a$. ConsequentlyThis is already the invariant-factor order required by the structure theorem for finitely generated modules over a principal ideal domain, because $2\mid 4$. The inverse change of generators shows that there are no hidden extra relations.