n. a module that possesses as a basis a subset {a₁, a₂,..., a_n} in terms of which every non-zero element can be written uniquely in the form Σ_i u_i a_i, where the u_i are elements of the ring over which the module is a left module. All vector spaces are free, and an Abelian group is free if and only if it has no element of finite period. Submodules of a free module are not in general free unless the underlying ring is a principal ideal domain. See also torsion-free module.