torsion element

Let $R$ be a commutative ring, and $M$ an $R$-module. We call an element $m\in M$ a torsion element if there exists a non-zero-divisor $\alpha \in R$ such that $\alpha \cdot m=0$. The set is denoted by $tor(M)$.

$tor(M)$ is not empty since $0\in tor(M)$. Let $m,n\in tor(M)$, so there exist $\alpha ,\beta \ne 0\in R$ such that $0=\alpha \cdot m=\beta \cdot n$. Since $\alpha \beta \cdot (m-n)=\beta \cdot \alpha \cdot m-\alpha \cdot \beta \cdot n=0,\alpha \beta \ne 0$, this implies that $m-n\in tor(M)$. So $tor(M)$ is a subgroup of $M$. Clearly $\tau \cdot m\in tor(M)$ for any non-zero $\tau \in R$. This shows that $tor(M)$ is a submodule of $M$, the torsion submodule of $M$. In particular, a module that equals its own torsion submodule is said to be a torsion module.