Torsion-free module

Let $R$ be an integral domain and $M$ an $R$-module . The module $M$ is torsion-free if its only torsion element is $0$, i.e. if $rm=0$ with $0\ne r\in R$ forces $m=0$.

Torsion-freeness is weaker than freeness but is a key hypothesis in many classification results over PIDs and in the theory of lattices.

Examples:

• $\mathbb Z^n$ is torsion-free as a $\mathbb Z$-module.
• Any ideal $I$ in an integral domain $R$, viewed as an $R$-module, is torsion-free.
• (Nonexample) $\mathbb Z/n\mathbb Z$ is not torsion-free for $n>1$.