torsionfree

Torsionfree is a term used in algebra to describe abelian groups and modules that have no nonzero elements of finite order, i.e., no torsion. For an abelian group G, G is torsion-free if nx = 0 with n a nonzero integer implies x = 0 for every x in G. Equivalently, G has no nontrivial elements of finite order. For a module M over an integral domain R, M is torsion-free if rm = 0 with r ≠ 0 implies m = 0 for every m in M. The two notions align when R = Z and M is viewed as an abelian group.

Examples: The additive group Z is torsion-free; the group Q is torsion-free. Direct sums of torsion-free groups are torsion-free. The group Z ⊕ Z/pZ is not torsion-free because the element (0,1) has order p. In module language, Z is a torsion-free Z-module, whereas Z/pZ is torsion as a Z-module.

Key properties: Subgroups of a torsion-free abelian group are torsion-free. Over a principal ideal domain, every finitely generated torsion-free module is free; in particular, every finitely generated torsion-free abelian group is isomorphic to Z^n. Over a general domain R, a finitely generated torsion-free module need not be free.

Localization viewpoint: If R is a domain with field of fractions K, a torsion-free R-module M injects into the vector space K ⊗_R M, and M embeds into a module that is free over K. This provides a framework for understanding the structure and behavior of torsion-free modules.

Torsion-free groups are characterized by their torsion subgroup T(G); a group is torsion-free precisely when T(G) = {0}.