In
mathematics, specifically in
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, 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
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
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 comprises only 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 group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s (with "module" and "submodule" replaced by "
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
" and "
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
"). This is allowed by the fact that the abelian groups are the modules over the ring of
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
(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. Contrary to the
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
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
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
(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
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, o ...
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 In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
domain (which might not be commutative).
More generally, let ''M'' be a module over a ring ''R'' and ''S'' be a
multiplicatively closed subset In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold:
* 1 \in S,
* xy \in S for all x, y \in S.
In other words, ''S'' is closed under taking finite ...
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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''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
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
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 its only torsion element is the identity element. Any
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
may be viewed as a module over the ring Z of integers, and in this case the two notions of torsion coincide.
Examples
# 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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a
field ''K'' is torsion-free when viewed as the module over ''K''.
# By contrast with example 1, any
finite group (abelian or not) is periodic and
finitely generated.
Burnside's problem, conversely, asks whether any finitely generated periodic group must be finite? The answer is "no" in general, even if the period is fixed.
# The torsion elements of the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of a field are its
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
.
# In the
modular group, Γ obtained from the group SL(2, Z) of 2×2 integer
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with unit
determinant by factoring out its
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
, 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.
# The abelian group Q/Z, consisting of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s modulo 1, is periodic, i.e. every element has finite order. Analogously, the module K(''t'')/K
't''over the ring ''R'' = K
't''of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s in one variable is pure torsion. Both these examples can be generalized as follows: if ''R'' is an integral domain and ''Q'' is its
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
, then ''Q''/''R'' is a torsion ''R''-module.
# The
torsion subgroup of (R/Z, +) is (Q/Z, +) while the groups (R, +) and (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.
# Consider a
linear operator L acting on a
finite-dimensional vector space V. If we view V as an F
''Lmodule in the natural way, then (as a result of many things, either simply by finite-dimensionality or as a consequence of the
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...
), V is a torsion F
''Lmodule.
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
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
(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
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of it.
Torsion and localization
Assume that ''R'' is a commutative domain and ''M'' is an ''R''-module. Let ''Q'' be the
quotient field
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of the ring ''R''. Then one can consider the ''Q''-module
:
obtained from ''M'' by
extension of scalars
In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'',
*f_! M = S\otimes_R M, the induced module.
*f_* M = \operatorn ...
. Since ''Q'' is a field, a module over ''Q'' is a vector space, possibly infinite-dimensional. There is a canonical
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
of abelian groups from ''M'' to ''M''
''Q'', and the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
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
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or ...
, 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 domain ''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 Tor
''R''1(''M'', ''R''
''S''/''R'') by the
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
of Tor
''R''*: The
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
of ''R''-modules yields an exact sequence
, hence
is the kernel of the localisation map of ''M''. The symbol denoting the
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s 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.
See also
*
Analytic torsion
*
Arithmetic dynamics Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is ...
*
Flat module
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact se ...
*
Annihilator (ring theory)
In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of .
Over an integral domain, a module that has a nonzero annihilator is a ...
*
Localization of a module
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions ...
*
Rank of an abelian group
*
Ray–Singer torsion
In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and .
Analytic torsion (or Ray– ...
*
Torsion-free abelian group
*
Universal coefficient theorem
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'':
:
completely ...
References
*Ernst Kunz,
Introduction to Commutative algebra and algebraic geometry, Birkhauser 1985,
*
Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr ...
,
Infinite abelian groups, University of Michigan, 1954.
*
*
{{DEFAULTSORT:Torsion (Algebra)
Abelian group theory
Module theory
Homological algebra