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/sparql endpoint"> Sparql Endpoint </a> </li> </ul> </div> </div> </nav> <div style="margin-bottom: 60px"></div>   <section> <div class="container-xl"> <div class="row"> <div class="col"> <h1 id="title" class="display-6">About: <a href="http://dbpedia.org/resource/Finitely_generated_module">Finitely generated module</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/ontology/Software">software</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 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, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. </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 style="display:none;">Die endliche Präsentierbarkeit ist ein Konzept aus der mathematischen Theorie der Moduln. Ein Modul ist endlich präsentierbar, wenn er ein endliches Erzeugendensystem besitzt, für das die Relationen, die zwischen dessen Elementen bestehen dürfen, einer Endlichkeitsbedingung unterworfen sind. (de)</li> <li>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, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group. (en)</li> <li style="display:none;">환론에서 유한 생성 가군(有限生成加群, 영어: finitely generated module)은 유한 계수의 자유 가군의 몫가군이다. 즉, 유한 개의 생성원과 (유한 또는 무한 개의) 관계로 나타내어지는 가군이다. (ko)</li> <li style="display:none;">数学において、有限生成加群（ゆうげんせいせいかぐん、英: finitely generated module）とは、有限な生成集合をもつ加群のことである。有限生成 R-加群はまた有限 R-加群 (finite R-module, module of finite type) や R 上有限 (finite over R) とも呼ばれる。関連した概念に、有限余生成加群 (finitely cogenerated module)、有限表示加群 (finitely presented module)、有限関係加群 (finitely related module)、連接加群 (coherent module) があり、これらはすべてあとで定義される。ネーター環上では、有限生成、有限表示、連接加群の概念は一致する。たとえば体上の有限生成加群とは単に有限次元ベクトル空間であり、有理整数環上の有限生成加群とは単に有限生成アーベル群である。 (ja)</li> <li style="display:none;">Inom ringteorin, en del av matematiken, är en ändligtgenererad modul en vänster- eller högermodul över någon unitär ring, sådan att den genereras av någon ändlig delmängd. Således är vänstermodulen M över ringen R ändligtgenererad, om det finns någon ändlig delmängd G = {g1,...,gr} av M, sådan att varje element i M kan framställas som en linjärkombination , där och . Motsvarande gäller om M är en högermodul; en allmän linjärkombination skrivs då dock . Ett linjärt rum är ändligtgenererat (som modul över sin kropp av skalärer) precis om rummet har ändlig dimension. En abelsk grupp är ändligtgenererad (som modul över Z) precis om den är en . Om generatormängden bara innehåller ett enda element, så kallas modulen cyklisk. (sv)</li> <li style="display:none;">Конечнопорождённым мо́дулем над ассоциативным кольцом называется такой модуль, который порождается конечным числом своих элементов. Например, для правого модуля это означает, что существует конечное множество элементов таких, что любой элемент из представим в виде суммы , где — какие-то элементы кольца . В числе свойств, тесно связанных с конечнопорождённостью — конечнопредставленность, конечносвязанность и когерентность модуля. Над нётеровым кольцом все четыре свойства эквивалентны. Конечнопорождённые модули над полем — это в точности конечномерные векторные пространства. (ru)</li> <li style="display:none;">Скінченнопородженим модулем над асоціативним кільцем називається такий модуль, який породжується скінченною кількістю своїх елементів. Наприклад, для правого модуля це означає, що існує скінченна множина елементів таких, що будь-який елемент з рівний сумі , де — елементи кільця . Еквівалентно скінченнопороджені модулі можна визначити такими умовами: * Для будь-якої сім'ї підмодулів {Ni | i ∈ I} модуля M, якщо , то для деякої скінченної підмножини F множиниI. * Для будь-якої лінійно впорядкованої множини підмодулів {Ni | i ∈ I} вM, якщо , тоді Ni = M для деякого i в I. * Якщо є епіморфізмом, тоді для деякої скінченної підмножини F множини I теж є епіморфізмом. Серед властивостей, тісно пов'язаних з скінченною породженістю — скінченне представлення, скінченна зв'язність і когерентність модуля. Над нетеровим кільцем всі чотири властивості є еквівалентними. Скінченнопороджені модулі над полем є скінченновимірними . 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text-break"><ul> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/ontology/Software" href="http://dbpedia.org/ontology/Software">dbo:Software</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 style="display:none;">Die endliche Präsentierbarkeit ist ein Konzept aus der mathematischen Theorie der Moduln. Ein Modul ist endlich präsentierbar, wenn er ein endliches Erzeugendensystem besitzt, für das die Relationen, die zwischen dessen Elementen bestehen dürfen, einer Endlichkeitsbedingung unterworfen sind. (de)</li> <li style="display:none;">환론에서 유한 생성 가군(有限生成加群, 영어: finitely generated module)은 유한 계수의 자유 가군의 몫가군이다. 즉, 유한 개의 생성원과 (유한 또는 무한 개의) 관계로 나타내어지는 가군이다. (ko)</li> <li style="display:none;">数学において、有限生成加群（ゆうげんせいせいかぐん、英: finitely generated module）とは、有限な生成集合をもつ加群のことである。有限生成 R-加群はまた有限 R-加群 (finite R-module, module of finite type) や R 上有限 (finite over R) とも呼ばれる。関連した概念に、有限余生成加群 (finitely cogenerated module)、有限表示加群 (finitely presented module)、有限関係加群 (finitely related module)、連接加群 (coherent module) があり、これらはすべてあとで定義される。ネーター環上では、有限生成、有限表示、連接加群の概念は一致する。たとえば体上の有限生成加群とは単に有限次元ベクトル空間であり、有理整数環上の有限生成加群とは単に有限生成アーベル群である。 (ja)</li> <li style="display:none;">Конечнопорождённым мо́дулем над ассоциативным кольцом называется такой модуль, который порождается конечным числом своих элементов. Например, для правого модуля это означает, что существует конечное множество элементов таких, что любой элемент из представим в виде суммы , где — какие-то элементы кольца . В числе свойств, тесно связанных с конечнопорождённостью — конечнопредставленность, конечносвязанность и когерентность модуля. Над нётеровым кольцом все четыре свойства эквивалентны. Конечнопорождённые модули над полем — это в точности конечномерные векторные пространства. (ru)</li> <li>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, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. (en)</li> <li style="display:none;">Inom ringteorin, en del av matematiken, är en ändligtgenererad modul en vänster- eller högermodul över någon unitär ring, sådan att den genereras av någon ändlig delmängd. Således är vänstermodulen M över ringen R ändligtgenererad, om det finns någon ändlig delmängd G = {g1,...,gr} av M, sådan att varje element i M kan framställas som en linjärkombination , där och . Motsvarande gäller om M är en högermodul; en allmän linjärkombination skrivs då dock . (sv)</li> <li style="display:none;">Скінченнопородженим модулем над асоціативним кільцем називається такий модуль, який породжується скінченною кількістю своїх елементів. Наприклад, для правого модуля це означає, що існує скінченна множина елементів таких, що будь-який елемент з рівний сумі , де — елементи кільця . Еквівалентно скінченнопороджені модулі можна визначити такими умовами: Серед властивостей, тісно пов'язаних з скінченною породженістю — скінченне представлення, скінченна зв'язність і когерентність модуля. Над нетеровим кільцем всі чотири властивості є еквівалентними. (uk)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label">rdfs:label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;">Endliche Präsentierbarkeit (Modul) (de)</li> <li>Finitely generated module (en)</li> <li style="display:none;">有限生成加群 (ja)</li> <li style="display:none;">유한 생성 가군 (ko)</li> <li style="display:none;">Конечнопорождённый модуль (ru)</li> <li style="display:none;">Ändligtgenererad modul (sv)</li> <li style="display:none;">Скінченнопороджений модуль (uk)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs">owl:sameAs</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="owl:sameAs" resource="http://rdf.freebase.com/ns/m.01w60f" href="http://rdf.freebase.com/ns/m.01w60f">freebase:Finitely generated module</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://www.wikidata.org/entity/Q1340572" 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