Torsion (algebra)

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In abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be free.

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[edit] Definition

Let G be a group. An element g of G is called a torsion element if g has finite order. If all elements of G are torsion, then G is called a torsion group. If the only torsion element is the identity element, then the group G is called torsion-free.

Let M be a module over a ring R without zero divisors. An element m of M is called a torsion element if the cyclic submodule of M generated by m is not free. Equivalently, m is torsion if and only if it has a non-zero annihilator in R. If the ring R is commutative, then the set of all torsion elements forms a submodule of M, called the torsion submodule of M, sometimes denoted T(M). The module M is called a torsion module if T(M) = M, and is called torsion-free if T(M) = 0. If the ring R is non-commutative then the situation is more complicated, and the set of torsion elements need not be a submodule. Nevertheless, it is a submodule given the assumption that the ring R satisfies the Ore condition. This covers the case when R is a Noetherian domain.

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.

More generally, let R be an arbitrary ring and S ⊂ R be a multiplicatively closed subset. Then one defines the notion of S-torsion as follows. An element m of an R-module M is called an S-torsion element if there exists s in S such that s annihilates m, i.e., sm = 0. In particular, one can take for S to be the set of all non-zero divisors of the ring R. In this case, S-torsion is frequently called simply torsion, extending the definition above from the case of domains to general rings.

[edit] Examples

1. Let M be a free module over any ring R. Then it follows immediately from the definitions that M is torsion-free (if the ring R is not a domain then torsion is considered with respect to the set S of non-zero divisors of R). In particular, any free abelian group is torsion-free and any vector space over a field K is torsion-free when viewed as the module over K.

2. By contrast with Example 1, any finite group (abelian or not) is periodic and finitely generated. Burnside problem asks whether, conversely, any finitely generated periodic group must be finite. (The answer is "no" in general, even if the period is fixed.)

3. In the modular group, Γ obtained from the group SL(2,Z) of two by two integer matrices with unit determinant by factoring out its center, any nontrivial torsion element either has order two and is conjugate to the element S or has order three and is conjugate to the element ST. In this case, torsion elements do not form a subgroup, for example, S · ST = T, which has infinite order.

4. The abelian group Q/Z, consisting of the rational numbers (mod 1), is periodic, i.e. every element has finite order. Analogously, the module K(t)/K[t] over the ring R = K[t] of polynomials in one variable is pure torsion. Both these examples can be generalized as follows: if R is a commutative domain and Q is its field of fractions, then Q/R is a torsion R-module.

5. The torsion subgroup of (R/Z,+) is (Q/Z,+) while the groups (R,+),(Z,+) are torsion-free. The quotient of a torsion-free abelian group by a subgroup is torsion-free exactly when the subgroup is a pure subgroup.

[edit] 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

M \simeq F\oplus T(M),

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 polynomial in two variables.

[edit] 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

M_Q = M \otimes_R Q,

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 MQ, 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,

M_S = M \otimes_R R_S,

which is a module over the localization RS. There is a canonical map from M to MS, 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.

[edit] 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 Tori(M,N). The S-torsion of an R-module M is canonically isomorphic to Tor1(M, RS/R). 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.

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