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In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear

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rdfs:label	Tensorprodukt von Moduln (de) Produit tensoriel de deux modules (fr) 加群のテンソル積 (ja) Iloczyn tensorowy modułów (pl) Produto tensorial de módulos (pt) Tensor product of modules (en)
rdfs:comment	Das Tensorprodukt von Moduln über einem (beliebigen) Ring mit 1 ist eine Verallgemeinerung des Tensorprodukts von Vektorräumen über einem Körper.Es hat Bedeutung in der abstrakten Algebra und findet in der homologischen Algebra, in der algebraischen Topologie und in der algebraischen Geometrie Anwendung. (de) Le produit tensoriel de deux modules est une construction en théorie des modules qui, à deux modules sur un même anneau commutatif unifère A, assigne un module. Le produit tensoriel est très important dans les domaines de l'analyse fonctionnelle, de la topologie algébrique et de la géométrie algébrique. Le produit tensoriel permet en outre de ramener l'étude d'applications bilinéaires ou multilinéaires à des applications linéaires. (fr) 数学において、加群のテンソル積 (tensor product of modules) は双線型写像（例えば積）についての議論を線型写像（加群準同型）の言葉でできるようにする構成である。その加群の構成はベクトル空間のテンソル積の構成と類似であるが、可換環上の加群の組に対して実行して第三の加群を得ることができ、また任意の環上の左加群と右加群の組に対しても実行できてアーベル群が得られる。テンソル積は抽象代数学、ホモロジー代数学、代数トポロジー、代数幾何学の分野において重要である。ベクトル空間に関するテンソル積の普遍性は抽象代数学のより一般的な状況に拡張される。それによって線型演算を通じて双線型あるいは多重線型演算を研究することができる。代数と加群のテンソル積は係数拡大のために使うことができる。可換環の場合には、加群のテンソル積を繰り返して加群のテンソル代数を作ることができ、加群の積を普遍的な方法で定義することができる。 (ja) Iloczynem tensorowym modułów i nazywa się taki moduł, którego odwzorowania liniowe (homomorfizmy) w dowolny moduł są we wzajemnie jednoznacznej odpowiedniości z odwzorowaniami dwuliniowymi modułów i w moduł (pl) O produto tensorial de módulos é uma construção na teoria de módulos na qual a dois módulos correspondem um mesmo anel comutativo unitário. O produto tensor é muito importante nos campos da topologia algébrica e geometria algébrica. O produto tensorial também serve para trazer o estudo de ou multilinear para as aplicações lineares. (pt) In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear (en)
rdfs:seeAlso	Pure submodule Duality (mathematics) Tensor product Weil restriction Free product of associative algebras Tensor product bundle Tensor-hom adjunction
foaf:depiction
dcterms:subject	Module theory Multilinear algebra Operations on structures Homological algebra
Wikipage page ID	2993836 (xsd:integer)
Wikipage revision ID	1098970634 (xsd:integer)
Link from a Wikipage to another Wikipage	Cardinality of the continuum Module (mathematics) Representation theory Bifunctor Algebraic topology Module theory Right exact functor Universal coefficient theorem Vector space Derived tensor product Index set Commutator Mathematics Torsion abelian group Localization of a module Profinite integer Annihilator (ring theory) Linear operator Commutative ring Frobenius reciprocity Functor Ordered pair Trace (linear algebra) Distributive property Dual module Covariant functor Sheaf of modules Additive map Algebraic geometry Cyclic group Multilinear algebra Extension of scalars Fibered coproduct Field of p-adic numbers Flat module Operations on structures Cardinality Center (ring theory) Direct product Direct sum of modules Tensor contraction Tensor field Ring (mathematics) Inverse limit Tensor product Associative property Homological algebra Abelian group Abstract algebra Chinese remainder theorem Bilinear map Bimodule Coequalizer Homological algebra

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