Torsion module - Imported Knowlpedia Content

Let $R$ be an integral domain and $M$ an $R$ -module. The module $M$ is a torsion module if every element of $M$ is a torsion element.

Torsion modules sit opposite to torsion-free modules and often decompose into primary pieces over suitable rings.

Equivalent characterizations

Equivalently, for each $m\in M$ there exists $0\ne r\in R$ with $rm=0$ .

Examples

Any finite abelian group, viewed as a $\mathbb Z$ -module, is torsion.
For $R=k[x]$ and nonzero $f\in R$ , the module $R/(f)$ is torsion as an $R$ -module (every class is killed by $f$ ).
(Nonexample) $\mathbb Z$ is not a torsion $\mathbb Z$ -module.