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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, specifically in
ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
, a torsion element is an element of a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the
submodule In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
, that is, when the regular elements of the ring are all its nonzero elements. This terminology applies to
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s (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
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s (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 A group is a number of people 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 identi ...
that are noncommutative, a ''torsion element'' is an element of
finite order Finite is the opposite of infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinit ...
. Contrary to the commutative case, the torsion elements do not form a subgroup, in general.


Definition

An element ''m'' of a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
''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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
) that annihilates ''m'', i.e., In an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
(a
commutative ring In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...
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
NoetherianIn mathematics, the adjective In linguistics, an adjective (list of glossing abbreviations, abbreviated ) is a word that grammatical modifier, modifies a noun or noun phrase or describes its referent. Its Semantics, semantic role is to change inf ...
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
(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 A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
''G'' is called a ''torsion element'' of the group if it has finite
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
, i.e., if there is a positive
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
''m'' such that ''g''''m'' = ''e'', where ''e'' denotes the
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
may be viewed as a module over the ring Z of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, and in this case the two notions of torsion coincide.


Examples

# Let ''M'' be a
free module In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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 In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation that is associative, commutative, and invertible. A basis, also called ...
is torsion-free and any
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
over a field K is torsion-free when viewed as the module over K. # By contrast with example 1, any
finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
(abelian or not) is periodic and finitely generated.
Burnside's problem The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite Order (group theory), order must necessarily be ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of a field are its
roots of unity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
. # In the
modular group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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. # 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
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s 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. # The
torsion subgroup In the theory of abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
of (R/Z, +) is (Q/Z, +) while the groups (R, +) and (Z, +) are torsion-free. The quotient of a
torsion-free abelian group In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion (algebra), torsion elements; that is, a group (mathematics), group in which the group operation is commutative pro ...
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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...
), V is a torsion F ''Lmodule.


Case of a principal ideal domain

Suppose that ''R'' is a (commutative)
principal ideal domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
and ''M'' is a finitely-generated ''R''-module. Then the
structure theorem for finitely generated modules over a principal ideal domain In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely ...
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 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 : M_Q = M \otimes_R Q, 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 (mathematics), module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'', *f_! M = S\otimes_R M, the induced module. ...
. Since ''Q'' is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, a module over ''Q'' is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, possibly, infinite-dimensional. There is a canonical homomorphism 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 learnin ...
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 Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
of the ''R''-module ''M'', : M_S = M \otimes_R R_S, which is a module over the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
''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 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 ...
''S'' and right ''R''-module ''M''.


Torsion in homological algebra

The concept of torsion plays an important role in
homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contain ...
. If ''M'' and ''N'' are two modules over a commutative ring ''R'' (for example, two abelian groups, when ''R'' = Z),
Tor functor In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring (mathematics), ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology ...
s 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 0\to R\to R_S \to R_S/R \to 0 of ''R''-modules yields an exact sequence 0\to\operatorname^R_1(M, R_S/R)\to M\to M_S, hence \operatorname^R_1(M, R_S/R) 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 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 ...
.


Abelian varieties

The torsion elements of an
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
are ''torsion points'' or, in an older terminology, ''division points''. On
elliptic curve In mathematics, an elliptic curve is a Nonsingular variety, smooth, Projective variety, projective, algebraic curve of Genus of an algebraic curve, genus one, on which there is a specified point ''O''. An elliptic curve is defined over a field ...
s they may be computed in terms of
division polynomialsIn mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion points. They play a central role in the study of counting points on elliptic curves in Schoof's alg ...
.


See also

* Analytic torsion *
Arithmetic dynamics Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, str ...
*
Flat module In algebra, a flat module over a ring (mathematics), ring ''R'' is an ''R''-module (mathematics), module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor pro ...
*
Localization of a module In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring (mathematics), ring or module (mathematics), module. That is, it introduces a new ring/module out of an existing ring/module ...
*
Rank of an abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
* Ray–Singer torsion *
Torsion-free abelian group In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion (algebra), torsion elements; that is, a group (mathematics), group in which the group operation is commutative pro ...
*
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 musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews. htt ...
,
Infinite abelian groups
, University of Michigan, 1954. * * {{DEFAULTSORT:Torsion (Algebra) Abelian group theory Module theory Homological algebra