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Torsion

Definitions

A member of a module $m \in M$ is called a torsion element if $\exists r \in R\setminus\{0\}: r.m=0$.

If all elements of $M$ are torsion elements, we say $M$ is a torsion module.

If the only torsion element of $M$ is zero, we say $M$ is a torsion-free module.

Properties

An element $m \in M$ is a torsion element if and only if the annihilator $Ann_R(m) = \{r \in R: r.m=0\}$ is not $\{0\}$.

The torsion elements $M^{tor}$ of a module (over an integral domain) form a submodule. The quotient module $M/M^{tor}$ is torsion-free.

all these pages adapted with probably insufficient credit from my university's lecture notes