"Le produit tensoriel de deux modules est une construction en th\u00E9orie des modules qui, \u00E0 deux modules sur un m\u00EAme anneau commutatif unif\u00E8re A, assigne un module. Le produit tensoriel est tr\u00E8s important dans les domaines de l'analyse fonctionnelle, de la topologie alg\u00E9brique et de la g\u00E9om\u00E9trie alg\u00E9brique. Le produit tensoriel permet en outre de ramener l'\u00E9tude d'applications bilin\u00E9aires ou multilin\u00E9aires \u00E0 des applications lin\u00E9aires."@fr . . . . . . . . . "Le produit tensoriel de deux modules est une construction en th\u00E9orie des modules qui, \u00E0 deux modules sur un m\u00EAme anneau commutatif unif\u00E8re A, assigne un module. Le produit tensoriel est tr\u00E8s important dans les domaines de l'analyse fonctionnelle, de la topologie alg\u00E9brique et de la g\u00E9om\u00E9trie alg\u00E9brique. Le produit tensoriel permet en outre de ramener l'\u00E9tude d'applications bilin\u00E9aires ou multilin\u00E9aires \u00E0 des applications lin\u00E9aires."@fr . . . . . . . . . . . . . . . . . . "Produto tensorial de m\u00F3dulos"@pt . . . . . . "48836"^^ . . . . "Iloczynem tensorowym modu\u0142\u00F3w i nazywa si\u0119 taki modu\u0142, kt\u00F3rego odwzorowania liniowe (homomorfizmy) w dowolny modu\u0142 s\u0105 we wzajemnie jednoznacznej odpowiednio\u015Bci z odwzorowaniami dwuliniowymi modu\u0142\u00F3w i w modu\u0142"@pl . . . "\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u52A0\u7FA4\u306E\u30C6\u30F3\u30BD\u30EB\u7A4D (tensor product of modules) \u306F\u53CC\u7DDA\u578B\u5199\u50CF\uFF08\u4F8B\u3048\u3070\u7A4D\uFF09\u306B\u3064\u3044\u3066\u306E\u8B70\u8AD6\u3092\u7DDA\u578B\u5199\u50CF\uFF08\u52A0\u7FA4\u6E96\u540C\u578B\uFF09\u306E\u8A00\u8449\u3067\u3067\u304D\u308B\u3088\u3046\u306B\u3059\u308B\u69CB\u6210\u3067\u3042\u308B\u3002\u305D\u306E\u52A0\u7FA4\u306E\u69CB\u6210\u306F\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306E\u30C6\u30F3\u30BD\u30EB\u7A4D\u306E\u69CB\u6210\u3068\u985E\u4F3C\u3067\u3042\u308B\u304C\u3001\u53EF\u63DB\u74B0\u4E0A\u306E\u52A0\u7FA4\u306E\u7D44\u306B\u5BFE\u3057\u3066\u5B9F\u884C\u3057\u3066\u7B2C\u4E09\u306E\u52A0\u7FA4\u3092\u5F97\u308B\u3053\u3068\u304C\u3067\u304D\u3001\u307E\u305F\u4EFB\u610F\u306E\u74B0\u4E0A\u306E\u5DE6\u52A0\u7FA4\u3068\u53F3\u52A0\u7FA4\u306E\u7D44\u306B\u5BFE\u3057\u3066\u3082\u5B9F\u884C\u3067\u304D\u3066\u30A2\u30FC\u30D9\u30EB\u7FA4\u304C\u5F97\u3089\u308C\u308B\u3002\u30C6\u30F3\u30BD\u30EB\u7A4D\u306F\u62BD\u8C61\u4EE3\u6570\u5B66\u3001\u30DB\u30E2\u30ED\u30B8\u30FC\u4EE3\u6570\u5B66\u3001\u4EE3\u6570\u30C8\u30DD\u30ED\u30B8\u30FC\u3001\u4EE3\u6570\u5E7E\u4F55\u5B66\u306E\u5206\u91CE\u306B\u304A\u3044\u3066\u91CD\u8981\u3067\u3042\u308B\u3002\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306B\u95A2\u3059\u308B\u30C6\u30F3\u30BD\u30EB\u7A4D\u306E\u666E\u904D\u6027\u306F\u62BD\u8C61\u4EE3\u6570\u5B66\u306E\u3088\u308A\u4E00\u822C\u7684\u306A\u72B6\u6CC1\u306B\u62E1\u5F35\u3055\u308C\u308B\u3002\u305D\u308C\u306B\u3088\u3063\u3066\u7DDA\u578B\u6F14\u7B97\u3092\u901A\u3058\u3066\u53CC\u7DDA\u578B\u3042\u308B\u3044\u306F\u591A\u91CD\u7DDA\u578B\u6F14\u7B97\u3092\u7814\u7A76\u3059\u308B\u3053\u3068\u304C\u3067\u304D\u308B\u3002\u4EE3\u6570\u3068\u52A0\u7FA4\u306E\u30C6\u30F3\u30BD\u30EB\u7A4D\u306F\u4FC2\u6570\u62E1\u5927\u306E\u305F\u3081\u306B\u4F7F\u3046\u3053\u3068\u304C\u3067\u304D\u308B\u3002\u53EF\u63DB\u74B0\u306E\u5834\u5408\u306B\u306F\u3001\u52A0\u7FA4\u306E\u30C6\u30F3\u30BD\u30EB\u7A4D\u3092\u7E70\u308A\u8FD4\u3057\u3066\u52A0\u7FA4\u306E\u30C6\u30F3\u30BD\u30EB\u4EE3\u6570\u3092\u4F5C\u308B\u3053\u3068\u304C\u3067\u304D\u3001\u52A0\u7FA4\u306E\u7A4D\u3092\u666E\u904D\u7684\u306A\u65B9\u6CD5\u3067\u5B9A\u7FA9\u3059\u308B\u3053\u3068\u304C\u3067\u304D\u308B\u3002"@ja . "M1 \u00D7 M2 \u00D7 M3 \u2192 Z"@en . "O produto tensorial de m\u00F3dulos \u00E9 uma constru\u00E7\u00E3o na teoria de m\u00F3dulos na qual a dois m\u00F3dulos correspondem um mesmo anel comutativo unit\u00E1rio. O produto tensor \u00E9 muito importante nos campos da topologia alg\u00E9brica e geometria alg\u00E9brica. O produto tensorial tamb\u00E9m serve para trazer o estudo de ou multilinear para as aplica\u00E7\u00F5es lineares."@pt . . . . . . . . . "O produto tensorial de m\u00F3dulos \u00E9 uma constru\u00E7\u00E3o na teoria de m\u00F3dulos na qual a dois m\u00F3dulos correspondem um mesmo anel comutativo unit\u00E1rio. O produto tensor \u00E9 muito importante nos campos da topologia alg\u00E9brica e geometria alg\u00E9brica. O produto tensorial tamb\u00E9m serve para trazer o estudo de ou multilinear para as aplica\u00E7\u00F5es lineares."@pt . . . . . . . "Das Tensorprodukt von Moduln \u00FCber einem (beliebigen) Ring mit 1 ist eine Verallgemeinerung des Tensorprodukts von Vektorr\u00E4umen \u00FCber einem K\u00F6rper.Es hat Bedeutung in der abstrakten Algebra und findet in der homologischen Algebra, in der algebraischen Topologie und in der algebraischen Geometrie Anwendung."@de . . . . . . "mr = rm."@en . . . . . . . . . "M1 \u2297 M2 \u2297 M3 \u2192 Z."@en . "x \u2297 y"@en . . "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 . . "\u52A0\u7FA4\u306E\u30C6\u30F3\u30BD\u30EB\u7A4D"@ja . . . . . . . . . . . "Tensorprodukt von Moduln"@de . . . . . "Das Tensorprodukt von Moduln \u00FCber einem (beliebigen) Ring mit 1 ist eine Verallgemeinerung des Tensorprodukts von Vektorr\u00E4umen \u00FCber einem K\u00F6rper.Es hat Bedeutung in der abstrakten Algebra und findet in der homologischen Algebra, in der algebraischen Topologie und in der algebraischen Geometrie Anwendung."@de . . . . . . . . . . . . "rs \u2212 sr"@en . "Tensor product of modules"@en . . . . . . . . "1.5"^^ . . "Iloczyn tensorowy modu\u0142\u00F3w"@pl . . . . . . . . . . . . . . . . . "Iloczynem tensorowym modu\u0142\u00F3w i nazywa si\u0119 taki modu\u0142, kt\u00F3rego odwzorowania liniowe (homomorfizmy) w dowolny modu\u0142 s\u0105 we wzajemnie jednoznacznej odpowiednio\u015Bci z odwzorowaniami dwuliniowymi modu\u0142\u00F3w i w modu\u0142"@pl . . . . . . . . . . . . . . "Produit tensoriel de deux modules"@fr . . . . . "2993836"^^ . . . "M1 \u2297 M2 \u2297 M3"@en . . . . . . . . . . . . . . . "1098970634"^^ . "\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u52A0\u7FA4\u306E\u30C6\u30F3\u30BD\u30EB\u7A4D (tensor product of modules) \u306F\u53CC\u7DDA\u578B\u5199\u50CF\uFF08\u4F8B\u3048\u3070\u7A4D\uFF09\u306B\u3064\u3044\u3066\u306E\u8B70\u8AD6\u3092\u7DDA\u578B\u5199\u50CF\uFF08\u52A0\u7FA4\u6E96\u540C\u578B\uFF09\u306E\u8A00\u8449\u3067\u3067\u304D\u308B\u3088\u3046\u306B\u3059\u308B\u69CB\u6210\u3067\u3042\u308B\u3002\u305D\u306E\u52A0\u7FA4\u306E\u69CB\u6210\u306F\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306E\u30C6\u30F3\u30BD\u30EB\u7A4D\u306E\u69CB\u6210\u3068\u985E\u4F3C\u3067\u3042\u308B\u304C\u3001\u53EF\u63DB\u74B0\u4E0A\u306E\u52A0\u7FA4\u306E\u7D44\u306B\u5BFE\u3057\u3066\u5B9F\u884C\u3057\u3066\u7B2C\u4E09\u306E\u52A0\u7FA4\u3092\u5F97\u308B\u3053\u3068\u304C\u3067\u304D\u3001\u307E\u305F\u4EFB\u610F\u306E\u74B0\u4E0A\u306E\u5DE6\u52A0\u7FA4\u3068\u53F3\u52A0\u7FA4\u306E\u7D44\u306B\u5BFE\u3057\u3066\u3082\u5B9F\u884C\u3067\u304D\u3066\u30A2\u30FC\u30D9\u30EB\u7FA4\u304C\u5F97\u3089\u308C\u308B\u3002\u30C6\u30F3\u30BD\u30EB\u7A4D\u306F\u62BD\u8C61\u4EE3\u6570\u5B66\u3001\u30DB\u30E2\u30ED\u30B8\u30FC\u4EE3\u6570\u5B66\u3001\u4EE3\u6570\u30C8\u30DD\u30ED\u30B8\u30FC\u3001\u4EE3\u6570\u5E7E\u4F55\u5B66\u306E\u5206\u91CE\u306B\u304A\u3044\u3066\u91CD\u8981\u3067\u3042\u308B\u3002\u30D9\u30AF\u30C8\u30EB\u7A7A\u9593\u306B\u95A2\u3059\u308B\u30C6\u30F3\u30BD\u30EB\u7A4D\u306E\u666E\u904D\u6027\u306F\u62BD\u8C61\u4EE3\u6570\u5B66\u306E\u3088\u308A\u4E00\u822C\u7684\u306A\u72B6\u6CC1\u306B\u62E1\u5F35\u3055\u308C\u308B\u3002\u305D\u308C\u306B\u3088\u3063\u3066\u7DDA\u578B\u6F14\u7B97\u3092\u901A\u3058\u3066\u53CC\u7DDA\u578B\u3042\u308B\u3044\u306F\u591A\u91CD\u7DDA\u578B\u6F14\u7B97\u3092\u7814\u7A76\u3059\u308B\u3053\u3068\u304C\u3067\u304D\u308B\u3002\u4EE3\u6570\u3068\u52A0\u7FA4\u306E\u30C6\u30F3\u30BD\u30EB\u7A4D\u306F\u4FC2\u6570\u62E1\u5927\u306E\u305F\u3081\u306B\u4F7F\u3046\u3053\u3068\u304C\u3067\u304D\u308B\u3002\u53EF\u63DB\u74B0\u306E\u5834\u5408\u306B\u306F\u3001\u52A0\u7FA4\u306E\u30C6\u30F3\u30BD\u30EB\u7A4D\u3092\u7E70\u308A\u8FD4\u3057\u3066\u52A0\u7FA4\u306E\u30C6\u30F3\u30BD\u30EB\u4EE3\u6570\u3092\u4F5C\u308B\u3053\u3068\u304C\u3067\u304D\u3001\u52A0\u7FA4\u306E\u7A4D\u3092\u666E\u904D\u7684\u306A\u65B9\u6CD5\u3067\u5B9A\u7FA9\u3059\u308B\u3053\u3068\u304C\u3067\u304D\u308B\u3002"@ja . . . . . . "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 operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way."@en .