= Torsion element of a module
{title2=$\exists r\ne0:\ rm=0$}
= Torsion elements of a module
{synonym}
For a <module> $M$ over an <integral domain> $R$, a torsion element is an $m\in M$ with $rm=0$ for some $0\ne r\in R$. This includes zero. Equivalently, its <annihilator> $\operatorname{Ann}_R(m)=\{r\in R:rm=0\}$ is a nonzero <ideal>. The <torsion elements of a module> form the <torsion submodule>.
For $M=\mathbb Z/6\mathbb Z$, computed <annihilators> are
$$
\operatorname{Ann}_{\mathbb Z}(\overline1)=6\mathbb Z,\qquad \operatorname{Ann}_{\mathbb Z}(\overline2)=3\mathbb Z,\qquad \operatorname{Ann}_{\mathbb Z}(\overline3)=2\mathbb Z.
$$
For these three elements, the additive <order of a group element> is respectively $6$, $3$, and $2$: these are the smallest positive <integers> killing them. Torsion over $\mathbb Z$ agrees exactly with finite additive <order of a group element>. Over a general <integral domain>, torsion concerns the chosen scalars instead; the <polynomial module from a linear operator> gives a torsion example whose nonzero elements have infinite additive <order of a group element>.
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