torsion (module theory)

Definition

For a ring $R$ and a module $M$, an element $m\in M$ is torsion if $rm=0$ for some $r\ne 0$.

If $R$ is commutative then the subset $M_{\text{tor}}\subset M$ consisting of torsion elements is called the torsion submodule. Note if $R$ is not commutative, then this set is not a submodule.

Examples

• For $k$ a field, then any $k$-module is just a vector space and the only torsion element is $0$.

• The torsion elements of any $\mathbb{Z}$-module are elements of finite order.