Definition:Torsion

Groups
Let $(G,*)$ be a Definition:Group.

An element of $G$ is a torsion element if it has finite order.

By Torsion Elements Form Subgroup, the subset of torsion elements of $G$ is a subgroup called the torsion subgroup, often denoted by $T(G)$.

If $T(G) = G$ then $G$ is called a torsion group.

Modules
Let $R$ be a commutative ring and let $M$ be an $R$-module.

An element $m \in M$ is called a torsion element if $a \cdot m = 0$ for some non-zerodivisor $a \in R$.

By Torsion Elements Form Submodule, when $R$ is an integral domain the torsion elements form a submodule $T(M)$ of $M$ called the torsion submodule.

If $T(M) = M$ then $M$ is called a torsion module.