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 comprises only the zero element.

- Definition
- Examples
- Case of a principal ideal domain
- Torsion and localization
- Torsion in homological algebra
- Abelian varieties
- See also
- References
- Sources

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. Contrary to the commutative case, the torsion elements do not form a subgroup, in general.

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., *r* *m* = 0. 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.^{ [1] } 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 ( Lam 2007 ) 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., *s* *m* = 0. 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 **torsion-free group** if its 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.

- 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*. - 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 of a field are its roots of unity.
- In the modular group,
**Γ**obtained from the group SL(2,**Z**) of 2×2 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 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 polynomials 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, 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**[**L**]-module 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),**V**is a torsion**F**[**L**]-module.

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.

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*.

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 of Tor^{R}_{*}: The short exact sequence of *R*-modules yields an exact sequence , hence 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.

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.

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 commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In mathematics, a **commutative ring** is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In mathematics, in particular abstract algebra, a **graded ring** is a ring such that the underlying additive group is a direct sum of abelian groups such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as **gradation** or **grading**.

In abstract algebra, a **Dedekind domain** or **Dedekind ring**, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

In mathematics, the **rank**, **Prüfer rank**, or **torsion-free rank** of an abelian group *A* is the cardinality of a maximal linearly independent subset. The rank of *A* determines the size of the largest free abelian group contained in *A*. If *A* is torsion-free then it embeds into a vector space over the rational numbers of dimension rank *A*. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved.

In mathematics, a **free abelian group** is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an **integral basis**, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called **free****-modules**, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

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 modules over the ring of integers.

In mathematics, a **finitely generated module** is a module that has a finite generating set. A finitely generated module over a ring *R* may also be called a **finite R-module**,

In algebra, a **group ring** is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

In mathematics, in particular commutative algebra, the concept of **fractional ideal** is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed * integral ideals* for clarity.

In mathematics, especially in the field of group theory, a **divisible group** is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an *n*th multiple for each positive integer *n*. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups.

In algebra, a **domain** is a nonzero ring in which *ab* = 0 implies *a* = 0 or *b* = 0. Equivalently, a domain is a ring in which 0 is the only left zero divisor. A commutative domain is called an integral domain. Mathematical literature contains multiple variants of the definition of "domain".

In mathematics, the **Tor functors** are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor_{1} and the torsion subgroup of an abelian group.

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 generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.

In mathematics, a **Prüfer domain** is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.

This is a **glossary of commutative algebra**.

In algebra, a **torsion-free module** is a module over a ring such that zero is the only element annihilated by a regular element of the ring. In other words, a module is *torsion free* if its torsion submodule is reduced to its zero element.

- ↑ Roman 2008 , p. 115, §4

- Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
- Irving Kaplansky, "Infinite abelian groups", University of Michigan, 1954.
- Michiel Hazewinkel (2001) [1994], "Torsion submodule",
*Encyclopedia of Mathematics*, EMS Press - Lam, Tsit Yuen (2007),
*Exercises in modules and rings*, Problem Books in Mathematics, New York: Springer, pp. xviii+412, doi:10.1007/978-0-387-48899-8, ISBN 978-0-387-98850-4, MR 2278849 - Roman, Stephen (2008),
*Advanced Linear Algebra*, Graduate Texts in Mathematics (Third ed.), Springer, p. 446, ISBN 978-0-387-72828-5 .

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