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For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. Note: Some authors use the term module category for the category of modules. 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href="http://dbpedia.org/resource/Category_of_modules">Category of modules</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/ontology/TelevisionStation">television station</a>, from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, within Data Space: <a href="http://dbpedia.org">dbpedia.org</a> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. </div> </div> </div> </section>   <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract">dbo:abstract</a> </td><td class="col-10 text-break"><ul> <li>In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. (en)</li> <li style="display:none;">En mathématiques, la catégorie des modules sur un monoïde R est une construction qui rend compte abstraitement des propriétés observées dans l'étude des modules sur un anneau, en les généralisant. L'étude de catégories de modules apparaît naturellement en théorie des représentations et en géométrie algébrique. Puisqu'un R-module est un espace vectoriel lorsque R est un corps commutatif, on peut dans un tel cas identifier la catégorie des modules sur R à la (en) sur le corps R. D'autre part, tout groupe abélien a une structure naturelle de -module, ce qui permet d'identifier la catégorie des modules sur à la catégorie des groupes abéliens. (fr)</li> <li style="display:none;">Dalam aljabar, diberi gelanggang R, kategori modul kiri di atas R adalah kategori yang objek semuanya tersisa modul di atas R . Misalnya, jika R adalah ring integer s 'Z' , itu sama dengan kategori grup abelian. Kategori modul yang tepat didefinisikan dengan cara yang serupa. Catatan: Beberapa penulis menggunakan istilah kategori modul untuk kategori modul. Istilah ini bisa ambigu karena bisa juga merujuk ke kategori dengan . (in)</li> <li style="display:none;">数学の一分野である圏論において加群の圏（かぐんのけん、英: category of modules）Mod は、すべての加群を対象としすべての加群準同型を射とする圏である。 (ja)</li> <li style="display:none;">In de lineaire algebra en de categorietheorie, een deelgebied van de wiskunde, heeft de categorie K-Vect (sommige auteurs schrijven VectK) alle vectorruimten over een vast lichaam (Ned) / veld (Be) als objecten en -lineaire transformaties als morfismen. Als het lichaam van de reële getallen is, dan staat de categorie ook bekend als Vec. Aangezien vectorruimten over (als een lichaam/veld) hetzelfde zijn als modulen over de ring is K-Vect een speciaal geval van , de categorie van linker -modulen. K-Vect is een belangrijk voorbeeld van een abelse categorie. Een groot deel van de lineaire algebra heeft betrekking op de beschrijving van K-Vect. De dimensiestelling voor vectorruimten zegt bijvoorbeeld dat de in K-Vect exact overeenkomen met de kardinaalgetallen, en dat K-Vect is met de deelcategorie van K-Vect, als objecten dus de vrije vectorruimten heeft, waarin een willekeurig kardinaalgetal is. Er is een vergeetachtige functor van K-Vect naar Ab, de categorie van abelse groepen, die elke vectorruimte naar zijn additieve groep neemt. Dit kan worden samengesteld met de vergeetachtige functors uit Ab met als realutaat andere vergeetachtige functors, met als belangrijkste een naar de categorie van verzamelingen. K-Vect is een monoïdale categorie met (als een een-dimensionale vectorruimte over ) als de identiteit en het tensorproduct als het monoïdale product. (nl)</li> <li style="display:none;">Категория модулей ― категория, объекты которой ― правые (левые или двусторонние — по предварительной договорённости) унитарные модули над произвольным ассоциативным кольцом K с единицей, а морфизмы ― гомоморфизмы K-модулей. Эта категория является важнейшим примером абелевой категории.Более того, для всякой малой абелевой категории существует полное точное вложение в некоторую категорию модулей Свойства категории модулей отражают ряд важных свойств кольца , с этой категорией связан ряд важных свойств кольца, в частности, его и отчасти — внутреннюю структуру. Категория модулей над коммутативным конечнопорождённым кольцом содержит всю алгебро-геометрическую характеристику аффинной схемы спектра кольца (одна из теорем Серра). Категории модулей над разными кольцами могут быть эквивалентны (то есть, иметь одинаковый набор классов изоморфных объектов, находящихся в том же отношении между собой). В этом случае говорят, что соответствующие кольца . Например, эквивалентны между собой категории модулей над алгебрами матриц разного порядка, но общим полем. Все они эквивалентны категории пространств над тем же полем. 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href="http://dbpedia.org/ontology/TelevisionStation">dbo:TelevisionStation</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment">rdfs:comment</a> </td><td class="col-10 text-break"><ul> <li>In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. (en)</li> <li style="display:none;">Dalam aljabar, diberi gelanggang R, kategori modul kiri di atas R adalah kategori yang objek semuanya tersisa modul di atas R . Misalnya, jika R adalah ring integer s 'Z' , itu sama dengan kategori grup abelian. Kategori modul yang tepat didefinisikan dengan cara yang serupa. Catatan: Beberapa penulis menggunakan istilah kategori modul untuk kategori modul. Istilah ini bisa ambigu karena bisa juga merujuk ke kategori dengan . (in)</li> <li style="display:none;">数学の一分野である圏論において加群の圏（かぐんのけん、英: category of modules）Mod は、すべての加群を対象としすべての加群準同型を射とする圏である。 (ja)</li> <li style="display:none;">En mathématiques, la catégorie des modules sur un monoïde R est une construction qui rend compte abstraitement des propriétés observées dans l'étude des modules sur un anneau, en les généralisant. L'étude de catégories de modules apparaît naturellement en théorie des représentations et en géométrie algébrique. (fr)</li> <li style="display:none;">In de lineaire algebra en de categorietheorie, een deelgebied van de wiskunde, heeft de categorie K-Vect (sommige auteurs schrijven VectK) alle vectorruimten over een vast lichaam (Ned) / veld (Be) als objecten en -lineaire transformaties als morfismen. Als het lichaam van de reële getallen is, dan staat de categorie ook bekend als Vec. Aangezien vectorruimten over (als een lichaam/veld) hetzelfde zijn als modulen over de ring is K-Vect een speciaal geval van , de categorie van linker -modulen. K-Vect is een belangrijk voorbeeld van een abelse categorie. (nl)</li> <li style="display:none;">Категория модулей ― категория, объекты которой ― правые (левые или двусторонние — по предварительной договорённости) унитарные модули над произвольным ассоциативным кольцом K с единицей, а морфизмы ― гомоморфизмы K-модулей. 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