torsion module
A torsion module is one made entirely of elements that can be 'killed' by scaling: for each element there is some nonzero ring scalar that sends it to zero. These are the elements with no analogue in a vector space — over a field the only way to scale something to zero is to multiply by zero — so torsion measures exactly how far a module is from being a vector space.
Let R be an integral domain and M an R-module. An element m is a torsion element if rm = 0 for some nonzero r in R. The torsion elements form a submodule T(M), the torsion submodule; M is a torsion module if T(M) = M, and torsion-free if T(M) = 0. (For this to behave well one needs R to have no zero divisors, which is why one usually restricts to domains.) Every module M fits into 0 -> T(M) -> M -> M/T(M) -> 0 with torsion-free quotient.
Over a PID the torsion submodule is precisely the finite-torsion part of the structure theorem: the direct sum of the cyclic blocks R/(d_i), with the free part R^r being torsion-free. Each torsion element m has an annihilator ideal, generated by its 'order'. Examples: every finite abelian group is a torsion Z-module; Q is torsion-free but not free over Z; and torsion-free is equivalent to flat for modules over a PID.
Z/12Z is a torsion Z-module: every element x is annihilated by 12 (and the element 3 already by 4). Its torsion submodule is all of it. By contrast Z is torsion-free, since rn = 0 with r, n nonzero is impossible in an integral domain.
Finite abelian groups are torsion modules; the free part of the structure theorem is torsion-free.