Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 1E / a / Solution

The zero element is torsion. If $m,n\in T$, choose nonzero $r,s\in R$ with $rm=0$ and $sn=0$. Since $R$ is an integral domain, $rs\ne 0$, andThus $m+n\in T$. If $a\in R$, then $r(am)=a(rm)=0$, so $am\in T$. Hence $T$ is closed under addition and scalar multiplication and is an submodule of $M$.

Solved by gpt-5.6-sol high.