Torsion (algebra)

In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion free if its torsion submodule is reduced to the zero element.
This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements.
This terminology applies to abelian groups (with "module" and "submodule" replaced by "group" and "subgroup"). This is allowed by the fact that the abelian groups are the modules over the ring of integers (in fact, this is the origin of the terminology, that has been introduced for abelian groups before being generalized to modules).
In the case of groups that are noncommutative, a "torsion element" is an element of finite order. Contrarily to the commutative case, the torsion elements do not form a subgroup, in general.
Definition.
An element "m" of a module "M" over a ring "R" is called a "torsion element" of the module if there exists a regular element "r" of the ring (an element that is neither a left nor a right zero divisor) that annihilates "m", i.e., 
In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element, but this definition does not work well over more general rings.
A module "M" over a ring "R" is called a "torsion module" if all its elements are torsion elements, and "torsion-free" if zero is the only torsion element. If the ring "R" is an integral domain then the set of all torsion elements forms a submodule of "M", called the "torsion submodule" of "M", sometimes denoted T("M"). If "R" is not commutative, T("M") may or may not be a submodule. It is shown in that "R" is a right Ore ring if and only if T("M") is a submodule of "M" for all right "R" modules. Since right Noetherian domains are Ore, this covers the case when "R" is a right Noetherian domain (which might not be commutative).
More generally, let "M" be a module over a ring "R" and "S" be a multiplicatively closed subset of "R". An element "m" of "M" is called an "S"-torsion element if there exists an element "s" in "S" such that "s" annihilates "m", i.e., In particular, one can take for "S" the set of regular elements of the ring "R" and recover the definition above.
An element "g" of a group "G" is called a "torsion element" of the group if it has finite order, i.e., if there is a positive integer "m" such that "g""m" = "e", where "e" denotes the identity element of the group, and "g""m" denotes the product of "m" copies of "g". A group is called a "torsion (or periodic) group" if all its elements are torsion elements, and a if the only torsion element is the identity element. 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.
Case of a principal ideal domain.
Suppose that "R" is a (commutative) principal ideal domain and "M" is a finitely-generated "R"-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module "M" up to isomorphism. In particular, it claims that
where "F" is a free "R"-module of finite rank (depending only on "M") and T("M") is the torsion submodule of "M". As a corollary, any finitely-generated torsion-free module over "R" is free. 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. The torsion subgroup of an abelian group may not be a direct summand of it.
Torsion and localization.
Assume that "R" is a commutative domain and "M" is an "R"-module. Let "Q" be the quotient field of the ring "R". Then one can consider the "Q"-module
obtained from "M" by extension of scalars. Since "Q" is a field, a module over "Q" is a vector space, possibly, infinite-dimensional. There is a canonical homomorphism of abelian groups from "M" to "M""Q", and the kernel of this homomorphism is precisely the torsion submodule T("M"). More generally, if "S" is a multiplicatively closed subset of the ring "R", then we may consider localization of the "R"-module "M",
which is a module over the localization "R""S". There is a canonical map from "M" to "M""S", whose kernel is precisely the "S"-torsion submodule of "M".
Thus the torsion submodule of "M" can be interpreted as the set of the elements that 'vanish in the localization'. 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".
Torsion in homological algebra.
The concept of torsion plays an important role in homological algebra. If "M" and "N" are two modules over a commutative ring "R" (for example, two abelian groups, when "R" = Z), Tor functors yield a family of "R"-modules Tor"i"("M","N"). The "S"-torsion of an "R"-module "M" is canonically isomorphic to TorR1("M", "R""S"/"R") by the long exact sequence of TorR*: The short exact sequence formula_4 of "R"-modules yields an exact sequence formula_5, hence formula_6 is the kernel of the localisation map of "M". The symbol Tor denoting the functors reflects this relation with the algebraic torsion. This same result holds for non-commutative rings as well as long as the set "S" is a right denominator set.
Abelian varieties.
The torsion elements of an abelian variety are "torsion points" or, in an older terminology, "division points". On elliptic curves they may be computed in terms of division polynomials.