# 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 $\alpha,\beta\neq 0\in R$ such that $0=\alpha\cdot m=\beta\cdot n$. Since $\alpha\beta\cdot(m-n)=\beta\cdot\alpha\cdot m-\alpha\cdot\beta\cdot n=0,\alpha% \beta\neq 0$, 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.

Title torsion element TorsionElement 2013-03-22 13:54:41 2013-03-22 13:54:41 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 13C12 torsion submodule torsion module