About: Torsion (algebra)

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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. 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.

Attributes	Values
rdfs:label	Torsion (Algebra) (de) Torsión (álgebra) (es) Torsion (algèbre) (fr) 捩れ (代数学) (ja) Torsion (algebra) (en) Кручение (алгебра) (ru) Скрут (алгебра) (uk) 扭化 (zh)
rdfs:comment	En álgebra abstracta el término torsión se refiere a los elementos de orden finito de un grupo y a los elementos de módulos anulados por elementos regulares de un anillo. (es) Torsion ist das Phänomen der kommutativen Algebra, also der Theorie der Moduln über kommutativen Ringen, das sie fundamental von der (einfacheren) Theorie der Vektorräume unterscheidet. Torsion ist verwandt mit dem Begriff des Nullteilers. (de) 抽象代数学において、捩れ（ねじれ、英: torsion）は、群の場合は、有限位数の元を言い、また環上の加群の場合は、環のある正則元によって零化される加群の元を言う。捩れという言葉は、捩れた図形のホモロジー群に有限位数の元が現れることに由来する。 (ja) У загальній алгебрі, термін скрут (іноді кручення, за́крут) стосується елементів групи, що має скінченний порядок, або елементів модуля, що анулюються регулярним елементом кільця. (uk) В общей алгебре, термин кручение относится к элементам группы, имеющим конечный порядок, или к элементам модуля, аннулируемым регулярным элементом кольца. (ru) 在抽象代数，扭化是指那些在群中為有限階的中的元素，並且那些元素在模中被任意的環的元素給(annihilated) (zh) En algèbre, dans un groupe, un élément est dit de torsion s'il est d'ordre fini, c'est-à-dire si l'une de ses puissances non nulle est l'élément neutre. La torsion d'un groupe est l'ensemble de ses éléments de torsion. Un groupe est dit sans torsion si sa torsion ne contient que le neutre, c'est-à-dire si tout élément différent du neutre est d'ordre infini. Si le groupe est abélien, sa torsion est un sous-groupe. Par exemple, le sous-groupe de torsion du groupe abélien est . Un groupe abélien est sans torsion si et seulement s'il est plat en tant que ℤ-module. (fr) 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. 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. (en)
foaf:depiction
dcterms:subject	Module theory Homological algebra Abelian group theory
Wikipage page ID	3039330 (xsd:integer)
Wikipage revision ID	1116954425 (xsd:integer)
Link from a Wikipage to another Wikipage	Principal ideal domain Root of unity Module (mathematics) Multiplicative group Noetherian domain Non-zero-divisor Determinant Module theory Burnside's problem Universal coefficient theorem Vector space Integral domain Analytic torsion Mathematics Matrix (mathematics) Noetherian ring Elliptic curve Modular group Multiplicatively closed subset Corollary Localization of a module Ore condition Annihilator (ring theory) Linear operator Long exact sequence Short exact sequence Commutative ring Zero divisor Functor Identity element Kernel (algebra) Structure theorem for finitely generated modules over a principal ideal domain Subgroup Cayley–Hamilton theorem Center (group theory) Division polynomials Domain (ring theory) Irving Kaplansky Extension of scalars Field (mathematics) Finitely generated module Flat module Dimension (vector space) Isomorphism Conjugation (group theory) Rank of an abelian group Ring (mathematics) Ring theory Group (mathematics) Group homomorphism Arithmetic dynamics Homological algebra Abelian group Abelian variety Abelian group theory Homological algebra Tor functor Torsion-free abelian group Torsion-free module Torsion group Direct summand Polynomial Field of fractions Free abelian group Free module Integer Michiel Hazewinkel Order (group theory) Rational number Field of fractions Finite group Finitely generated group Pure subgroup Torsion subgroup Module (algebra) Ray–Singer torsion Submodule Quotient field

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