Torsion submodule

Summary

Let $M$ be a module over a commutative ring $R$. Its torsion submodule is the set of elements of $M$ that are killed by some non-zero-divisor of $R$. It is a submodule of $M$, and it measures how far $M$ is from being torsion-free. For $R = \Z$ it is the torsion subgroup of an abelian group. Over a principal ideal domain, a finitely generated module splits as its torsion submodule plus a free module.