Torsion (Algebra)

Torsion is the phenomenon of commutative algebra, which is the theory of modules over commutative rings, they fundamentally different from the (simpler ) theory of vector spaces. Torsion is related to the concept of zero divisor.

• 2.1 Definitions
• 2.2 Features

Global torsion

Definitions

In the simplest form of a torsion element is nothing but one element of finite order in a group or a monoid, ie a member g, for which there exists a natural number n, such that (or in an additive notation).

For the Torsionsbegriff commutative algebra R is a ( commutative ) ring ( with unit element ) and M is an R- module.

• The twist or Torsionsuntermodul of M is the submodule of those elements m such that, not only contains the core of the map R → M, r r m a zero divisor. m is then called torsion.
• Equivalently, one the Torsionsuntermodul as the kernel of the homomorphism

• M is called torsion-free, if the Torsionsuntermodul is zero.
• M is a torsion module if the Torsionsuntermodul is the same. We then say sometimes briefly, " is torsion ".

If M is an abelian group (ie module), so agree, the two definitions of torsion. One speaks then analog of torsion (sub) groups.

Simple properties

• If T is the Torsionsuntermodul of M, then M / T is torsion. So there is a canonical Torsionsuntermodul and a canonical torsion-free quotient, but not vice versa.
• Making the Torsionsuntermoduls is a functor, ie f: M → N is a Modulhomomorphismus, so f maps the Torsionsuntermodul of M from the Torsionsuntermodul of N. Even in the case of groups is a homomorphism torsion always depend on torsion.
• From the alternative description of the Torsionsuntermoduls as a core localization follows immediately that the formation of the Torsionsuntermoduls is a left exact functor.

Examples

• Torsion of the group SL2 (Z) ( and others) and, on the other hand their product has infinite order. In non- Abelian groups, the torsion form that is not necessarily a subgroup.
• Another example of this fact is the infinite dihedral group

• R itself, or more generally, a free R - module is torsion-free. In particular, if R is a body, then all R-modules are torsion-free.
• Z / n Z is a torsion ( over Z) for every natural number n general, for a ring R and an ideal J of R, which consists not only of zero divisors of R / J module is a torsion module.
• If K is a field, then the Torsionsuntermodul of K ×, regarded as abelian group and Z - module is equal to the group of roots of unity in K.

Abelian Torsionsgruppen

• An abelian torsion group if and only finitely generated if it is finite.
• An abelian torsion group is the direct sum of its p- primary subgroups for each prime p, ie the sub-groups of elements, the order of which is a power of p. The p- group is a primary sub - group p.
• As the example of the factor group Q / Z shows the orders of the elements in general are not limited; the p- primary subgroup of Qp / Zp has this property.
• Is the order of elements is limited, this does not mean that the group is finitely generated (and thus finite) is in an infinite direct product of cyclic groups of order 2, each element (except the neutral element ) order 2

Torsion-free abelian groups

• An abelian group is torsion-free if and only if there exists a total order that is compatible with the group structure.

Torsion-free modules

• If a finitely generated module over a principal ideal ring torsion-free, so it's free. This is especially true for abelian groups.
• If a finitely generated module over a Dedekind ring torsion-free, so it is projective.
• Flat modules are torsion-free. About Dedekind rings (especially so over principal ideal rings ) agree the terms " flat" and " torsion " even match.

The following diagram summarizes these implications for a module over a commutative integral domain together:

Torsion with respect to a ring member

Definitions

Let be a commutative ring with unit element and a module. In the simplest case; is then merely an abelian group.

For a ring member

A sub-module, which is referred to as the torsion of. ( The likelihood of confusion with the notation for localization is low. ) The notation is standard.

The module

Is called the Twist.

Properties

• A module in a natural way.
• The functor is left exact (as representable functor reversed even with any Limites ); more accurately applies: is

• The Torsionsuntermodul of is the union of all non- zero divisor.
• For ring elements.
• For an abelian group and a prime number is the proportion of primary - torsion of.

Is an abelian group and a prime, so is the projective limit

( the transition illustrations are given by the multiplication by ) a module (whole - adic numbers ), which is referred to as - adic Tate module of ( according to John Tate ). Due to the transition to

One obtains a vector space over a field of characteristic 0; This is particularly advantageous for imaging theoretical considerations.

The most important example of this construction is the Tate module of an elliptic curve over a non- algebraically closed field whose characteristic is not. The Tate module is a module isomorphic to and carries a natural operation of the Galois group. In the case of the multiplicative group of the associated Tate module of rank 1 is he is referred to, the operation of the Galois group is done by the cyclotomic character.

Generalizations

For Z -modules is the Torsionsuntermodul a module M is Tor1 (Q / Z, M). The functors Tor can thus be viewed as a generalization of the concept of Torsionsuntermoduls.