The torsion subgroup of a group is the subgroup of all those elements , which have finite order, i.e. those for which for some .
A group is torsion-free if there is no such element apart from the neutral element itself, i.e. when the torsion subgroup is trivial.
Given a ring , an element in an -module is torsion element if there is a nonzero element in such that . A torsion module is a module whose elements are all torsion. A torsion-free module is a module whose elements are not torsion, other than .