torsion module

An element of a module is torsion if some nonzero ring scalar annihilates it — there is a nonzero r with r . m = 0. In a vector space this never happens (you cannot kill a nonzero vector with a nonzero scalar, since scalars are invertible), so torsion is a genuinely new phenomenon that appears only once you drop to a ring. A torsion module is one in which every element is torsion.

Over a PID the torsion elements always form a submodule, the torsion submodule, and the structure theorem separates a module cleanly into its free part plus its torsion part. The torsion part is the sum of the cyclic pieces R/(d_i); the free part R^r is torsion-free. A module is a torsion module exactly when its free rank r is zero.

For the operator picture this is the key observation: a finite-dimensional space V, viewed as an F[x]-module via T, is ALWAYS a torsion module. Reason: by Cayley-Hamilton (or just finite-dimensionality) the characteristic polynomial p satisfies p(T) = 0, so the nonzero scalar p(x) annihilates every vector. There is no free part, which is why V decomposes purely into companion/Jordan blocks with no leftover.

A clarifying caveat: torsion is about being killed by some scalar, not about every scalar. And the contrast term torsion-free is weaker than free — over Z the rationals Q are torsion-free but not free. Over a PID, finitely generated and torsion-free does imply free, which is what makes the clean split possible.