Torsion (algebra)

This is just a special case of the more general situation, because abelian groups are modules over the ring of integers.

(In fact, this is the origin of the terminology, which was introduced for abelian groups before being generalized to modules.)

Contrary to the commutative case, the torsion elements do not form a subgroup, in general.

It is shown in (Lam 2007) that R is a right Ore ring if and only if T(M) is a submodule of M for all right R-modules.

Any abelian group may be viewed as a module over the ring Z of integers, and in this case the two notions of torsion coincide.

Suppose that R is a (commutative) principal ideal domain and M is a finitely generated R-module.

This corollary does not hold for more general commutative domains, even for R = K[x,y], the ring of polynomials in two variables.

For non-finitely generated modules, the above direct decomposition is not true.

Let Q be the field of fractions of the ring R. Then one can consider the Q-module obtained from M by extension of scalars.

The same interpretation continues to hold in the non-commutative setting for rings satisfying the Ore condition, or more generally for any right denominator set S and right R-module M. The concept of torsion plays an important role in homological algebra.

If M and N are two modules over a commutative domain R (for example, two abelian groups, when R = Z), Tor functors yield a family of R-modules Tori (M,N).

is the kernel of the localisation map of M. The symbol Tor denoting the functors reflects this relation with the algebraic torsion.

On elliptic curves they may be computed in terms of division polynomials.