Torsion element

Let $R$ be an integral domain and $M$ a (left) $R$-module . An element $m\in M$ is a torsion element if there exists a nonzero $r\in R$ such that $rm=0$. Equivalently, the annihilator $\operatorname{ann}(m)$ is a nonzero ideal of $R$.

Torsion detects “failure of cancellation” under scalar multiplication and is a central invariant in structure theorems over PIDs and Dedekind domains.

Examples:

• In the $\mathbb Z$-module $\mathbb Z/n\mathbb Z$, every element is torsion (killed by $n$).
• In the $\mathbb Z$-module $\mathbb Z$, the only torsion element is $0$.
• (Edge case) Over a field, every nonzero scalar is invertible, so the only torsion element in a vector space is $0$.