An abelian group is said to be torsion-free if no element other than the identity is of finite order.[2][3][4] Explicitly, for any , the only element for which is .
A natural example of a torsion-free group is , as only the integer 0 can be added to itself finitely many times to reach 0. More generally, the free abelian group is torsion-free for any . An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a .
A non-finitely generated countable example is given by the additive group of the polynomial ring (the free abelian group of countable rank).
More complicated examples are the additive group of the rational field , or its subgroups such as (rational numbers whose denominator is a power of ). Yet more involved examples are given by groups of higher rank.