Torsion module

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

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

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.