Over a Noetherian integrally closed domain, any finitely-generated torsion-free module has a free submodule such that the quotient by it is isomorphic to an ideal of the ring.
Over a Dedekind domain, a finitely-generated module is torsion-free if and only if it is projective, but is in general not free.
Over a principal ideal domain, finitely-generated modules are torsion-free if and only if they are free.
-modules such that for any open affine subscheme U = Spec(R) the restriction F|U is associated to some module M over R. The sheaf F is said to be torsion-free if all those modules M are torsion-free over their respective rings.
Alternatively, F is torsion-free if and only if it has no local torsion sections.