Torsion is exactly having a nonzero annihilator.Suggest a slogan
Let \(R\) be a commutative ring and let \(M\) be an \(R\)-module. An element \(m\in M\) is a torsion element if and only if its annihilator \(\Ann_R(m)\coloneqq\theset{r\in R\suchthat rm=0}\) is a nonzero ideal of \(R\).
By definition, \(m\) is a torsion element if and only if there is a nonzero \(r\in R\) with \(rm=0\), that is, a nonzero element of \(\Ann_R(m)\).
When is $m \in M$ a torsion element?