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About: Noetherian ring
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That is, every increasing sequence of left (or right) ideals has a largest element; that is, there exists an n such that: Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian." /> <meta property="og:site_name" content="DBpedia" /> <!-- /OpenGraph--> </head> <body about="http://dbpedia.org/resource/Noetherian_ring"> <!-- navbar --> <nav class="navbar navbar-expand-md navbar-light bg-light fixed-top align-items-center"> <div class="container-xl"> <a class="navbar-brand" href="http://wiki.dbpedia.org/about" title="About DBpedia" style="color: #2c5078"> <img class="img-fluid" src="/statics/images/dbpedia_logo_land_120.png" alt="About DBpedia" /> </a> <button class="navbar-toggler" type="button" data-bs-toggle="collapse" data-bs-target="#dbp-navbar" aria-controls="dbp-navbar" aria-expanded="false" aria-label="Toggle navigation"> <span class="navbar-toggler-icon"></span> </button> <div class="collapse navbar-collapse" id="dbp-navbar"> <ul class="navbar-nav me-auto mb-2 mb-lg-0"> <li class="nav-item dropdown"> <a class="nav-link dropdown-toggle" href="#" id="navbarDropdownBrowse" role="button" 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to /sparql endpoint"><i class="bi-box-arrow-up-right"></i> Sparql Endpoint </a> </li> </ul> </div> </div> </nav> <div style="margin-bottom: 60px"></div> <!-- /navbar --> <!-- page-header --> <section> <div class="container-xl"> <div class="row"> <div class="col"> <h1 id="title" class="display-6"><b>About:</b> <a href="http://dbpedia.org/resource/Noetherian_ring">Noetherian ring</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> <span class="text-nowrap">An Entity of Type: <a href="javascript:void()">Thing</a>, </span> <span class="text-nowrap">from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, </span> <span class="text-nowrap">within Data Space: <a href="http://dbpedia.org">dbpedia.org</a></span> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> <p class="lead">In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left (or right) ideals has a largest element; that is, there exists an n such that: Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian.</p> </div> </div> </div> </section> <!-- page-header --> <!-- property-table --> <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"><small>dbo:</small>abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ca" >En àlgebra abstracta, un anell noetherià és un anell commutatiu i unitari que satisfà que la cadena d'ideals és estacionària. És a dir, donada una cadena d'ideals: existeix un n tal que: Aquest tipus d'anells reben aquest nom en honor de la matemàtica alemanya Emmy Noether. La noció d'anell noetherià és d'una importància fonamental tant en teoria d'anells commutatius com no commutatius, a causa del rol que juga a l'hora de simplificar l'estructura d'ideals d'un anell. Per exemple l'anell dels enters i l'anell de polinomis sobre un cos són tots dos anells noetherians, i en conseqüència, hi són vàlids teoremes com el , el o el . Addicionalment, si un anell és noetherià, llavors satisfà la sobre ideals primers. Aquesta propietat suggereix una teoria profunda sobre la dimensió dels anells noetherians, començant amb la noció de la .</span><small> (ca)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="cs" >Noetherovský okruh je pojem z algebry, z teorie okruhů, pojmenovaný po Emmy Noetherové. Jedná se o takový okruh, který neobsahuje shora neomezený řetězec do sebe vnořených ideálů, tedy pro každý řetězec ideálů existuje nějaký index n, že: Alternativní definice říká, že noetherovský je takový okruh, ve kterém je každý ideál . Při změně směru inkluzí v definici (tj. požadavku, že každý klesající řetězec ideálů je konečný) získáme artinovský okruh. Všechny artinovské okruhy jsou zároveň noetherovské a jejich struktura je méně různorodá než okruhů noetherovských.</span><small> (cs)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="de" >In der Algebra werden bestimmte Strukturen (Ringe und Moduln) noethersch genannt, wenn sie keine unendliche Schachtelung von immer größeren Unterstrukturen enthalten können. Der Begriff ist nach der Mathematikerin Emmy Noether benannt.</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="es" >En álgebra abstracta, un anillo es Noetheriano por la izquierda si sus ideales por la izquierda satisfacen la condición de cadena ascendente. Diremos que un anillo es noetheriano si es noetheriano por la izquierda y por la derecha. En los anillos conmutativos no se utiliza esta distinción, pues noetheriano por un lado implica noetheriano por el otro. Los anillos Noetherianos son nombrados así en honor a Amalie Emmy Noether. Uno de los primeros resultados que puede demostrarse es que un anillo es noetheriano por la izquierda si, y sólo si, todo ideal por la izquierda del anillo es finitamente generado. Otra condición equivalente a las anteriores es que todo conjunto no vacío de ideales del anillo tiene un ideal maximal. También es conocido que si es un anillo noetheriano, entonces es un anillo noetheriano (teorema de la base de Hilbert)</span><small> (es)</small></span></li> <li><span class="literal"><span property="dbo:abstract" lang="en" >In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left (or right) ideals has a largest element; that is, there exists an n such that: Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Lasker–Noether theorem and the Krull intersection theorem). Noetherian rings are named after Emmy Noether, but the importance of the concept was recognized earlier by David Hilbert, with the proof of Hilbert's basis theorem (which asserts that polynomial rings are Noetherian) and Hilbert's syzygy theorem.</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="fr" >En mathématique, un anneau noethérien est un cas particulier d'anneau, c'est-à-dire d'un ensemble muni d'une addition et d'une multiplication compatible avec l'addition, au sens de la distributivité. De nombreuses questions mathématiques s'expriment dans un contexte d'anneau, les endomorphismes d'un espace vectoriel ou d'un module sur un anneau, les entiers algébriques de la théorie algébrique des nombres, ou encore les surfaces de la géométrie algébrique. Si les anneaux sont nombreux, rares sont ceux disposant des propriétés communes aux exemples les plus simples comme les entiers relatifs ou les polynômes à coefficients dans un corps. La division euclidienne n'existe en général plus, les idéaux, outils majeurs de la théorie des anneaux, ne sont plus toujours principaux et le théorème fondamental de l'arithmétique ne possède plus d'équivalent. L'approche consistant à étudier une question uniquement sous l'angle des propriétés spécifiques d'une structure d'anneau particulière s'est révélée fructueuse. Richard Dedekind l'a utilisée avec succès en arithmétique et David Hilbert en géométrie algébrique. En 1920-1921, Emmy Noether choisit un nombre plus limité de propriétés vérifiées par certains anneaux et démontre de nombreux résultats sur ceux-ci. Le terme d'« anneau noethérien » apparait en 1943 sous la plume de Claude Chevalley.</span><small> (fr)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ko" >환론에서 뇌터 환(Noether環, 영어: Noetherian ring)은 아이디얼들이 오름 사슬 조건을 만족하는 환이다. 대략, 체 위의 유한 개의 변수에 대한 다항식환처럼, "지나치게 크지 않은" 환을 뜻한다.</span><small> (ko)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ja" >数学においてネーター環（ネーターかん、英: Noetherian ring）は、イデアルの昇鎖条件などのある種の有限性を持つ環の一種。エミー・ネーターによって提唱された。すべてのイデアルは有限生成という条件から単項イデアル整域の一般化と見ることもできる。</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="it" >In algebra, un anello noetheriano è un anello i cui ideali sono finitamente generati. Questa proprietà per gli anelli costituisce un analogo della finitezza, e fu studiata per prima da Emmy Noether, che la rilevò sugli anelli di polinomi.</span><small> (it)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="nl" >In de abstracte algebra wordt een ring Noethers genoemd als zijn idealen aan een bepaalde voorwaarde van eindigheid voldoen. Men spreekt in dit verband ook uitdrukkelijk over ringen die voldoen aan de stijgende ketenvoorwaarde. Noetherse ringen zijn genoemd naar Emmy Noether.</span><small> (nl)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="pl" >Pierścień noetherowski – pierścień, w którym każdy ciąg wstępujący (w sensie inkluzji) jego ideałów stabilizuje się, tzn. istnieje dla którego mówi się też wtedy, że w pierścień spełnia warunek rosnących łańcuchów (ACC) dla ideałów; pojęcie nosi nazwisko Emmy Noether. Równoważnie pierścień jest noetherowski wtedy i tylko wtedy, gdy każdy ideał właściwy jest skończenie generowany, tzn.dla każdego ideału istnieją takie elementy dla których Można też powiedzieć, że pierścień jest noetherowski wtedy i tylko wtedy, gdy każdy ideał tego pierścienia można przedstawić w postaci skończonej sumy ideałów głównych pierścienia Prawdziwe jest również twierdzenie Hilberta o bazie: jeżeli pierścień jest noetherowski, to jego pierścień wielomianów również jest noetherowski.</span><small> (pl)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ru" >Нётерово кольцо́ — тип колец, обобщение кольца главных идеалов.Названы в честь Эмми Нётер.</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="pt" >Em álgebra abstracta, um anel noetheriano é um anel comutativo que satisfaz a para ideais.O termo noetheriano é uma homenagem à matemática alemã Emmy Noether. Anéis de polinômios sobre corpos possuem muitas propriedades especiais; propriedades que derivam do fato de que anéis polinomiais não são em certo sentido "grandes demais". Emmy Noether descobriu que uma propriedade fundamental dos anéis de polinômios é a propriedade da cadeia ascendente para ideais. Para anéis não-comutativos, devemos fazer algumas distinções entre conceitos similares: * Um anel é dito noetheriano à esquerda caso satisfaça a condição da cadeia ascendente para ideais à esquerda. * Um anel é dito noetheriano à direita caso satisfaça a condição da cadeia ascendente para ideais à direita. * Um anel é dito noetheriano caso seja noetheriano tanto à esquerda quanto à direita. Para anéis comutativos as três definições coincidem.</span><small> (pt)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="sv" >En noethersk ring är inom matematiken en speciell sorts ring, uppkallad efter Emmy Noether. En kommutativ ring med etta R kallas noethersk om varje ideal är , det vill säga att för varje ideal I finns en ändlig mängd av element i I så att varje element x i I kan skrivas som en linjärkombination av dessa element: där elementen är element i R. Att I genereras av skrivs vanligtvis . För okommutativa ringar ringar med etta måste man vara litet mer precis. Man får två inte helt ekvivalenta egenskaper: Ringen kan vara högernoethersk eller vänsternoethersk; se de formella definitionerna nedan.</span><small> (sv)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="uk" >Кільце Нетер — в абстрактній алгебрі це таке асоціативне кільце з одиницею для якого справджується наступне твердження: нехай маємо деяку зростаючу послідовність ідеалів кільця: тоді існує таке для якого: Якщо ідеали в означенні ліві, то кільце називається лівим кільцем Нетер, якщо праві - правим кільцем Нетер. Якщо твердження виконується і для лівих і для правих ідеалів то кільце просто називається кільцем Нетер.Дані кільця названі на честь німецького математика Еммі Нетер (нім. Emmy Noether).</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="zh" >諾特環是抽象代數中一類滿足升鏈條件的環。希爾伯特首先在研究不變量理論時證明了多項式環的每個理想都是有限生成的，隨後埃米·諾特從中提煉出升鏈條件，諾特環由此命名。</span><small> (zh)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageExternalLink"><small>dbo:</small>wikiPageExternalLink</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="https://www.ams.org/journals/proc/1974-046-02/S0002-9939-1974-0414625-5/home.html" href="https://www.ams.org/journals/proc/1974-046-02/S0002-9939-1974-0414625-5/home.html">https://www.ams.org/journals/proc/1974-046-02/S0002-9939-1974-0414625-5/home.html</a></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageID"><small>dbo:</small>wikiPageID</a> </td><td class="col-10 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ring</span><small> (en)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/wikiPageUsesTemplate"><small>dbp:</small>wikiPageUsesTemplate</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Springer" href="http://dbpedia.org/resource/Template:Springer"><small>dbt</small>:Springer</a></span></li> <li><span class="literal"><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Algebraic_structures" href="http://dbpedia.org/resource/Template:Algebraic_structures"><small>dbt</small>:Algebraic_structures</a></span></li> <li><span class="literal"><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Citation" href="http://dbpedia.org/resource/Template:Citation"><small>dbt</small>:Citation</a></span></li> <li><span class="literal"><a 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class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="dct:subject" resource="http://dbpedia.org/resource/Category:Ring_theory" href="http://dbpedia.org/resource/Category:Ring_theory"><small>dbc</small>:Ring_theory</a></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment"><small>rdfs:</small>comment</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="cs" >Noetherovský okruh je pojem z algebry, z teorie okruhů, pojmenovaný po Emmy Noetherové. Jedná se o takový okruh, který neobsahuje shora neomezený řetězec do sebe vnořených ideálů, tedy pro každý řetězec ideálů existuje nějaký index n, že: Alternativní definice říká, že noetherovský je takový okruh, ve kterém je každý ideál . Při změně směru inkluzí v definici (tj. požadavku, že každý klesající řetězec ideálů je konečný) získáme artinovský okruh. Všechny artinovské okruhy jsou zároveň noetherovské a jejich struktura je méně různorodá než okruhů noetherovských.</span><small> (cs)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="de" >In der Algebra werden bestimmte Strukturen (Ringe und Moduln) noethersch genannt, wenn sie keine unendliche Schachtelung von immer größeren Unterstrukturen enthalten können. Der Begriff ist nach der Mathematikerin Emmy Noether benannt.</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ko" >환론에서 뇌터 환(Noether環, 영어: Noetherian ring)은 아이디얼들이 오름 사슬 조건을 만족하는 환이다. 대략, 체 위의 유한 개의 변수에 대한 다항식환처럼, "지나치게 크지 않은" 환을 뜻한다.</span><small> (ko)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ja" >数学においてネーター環（ネーターかん、英: Noetherian ring）は、イデアルの昇鎖条件などのある種の有限性を持つ環の一種。エミー・ネーターによって提唱された。すべてのイデアルは有限生成という条件から単項イデアル整域の一般化と見ることもできる。</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="it" >In algebra, un anello noetheriano è un anello i cui ideali sono finitamente generati. Questa proprietà per gli anelli costituisce un analogo della finitezza, e fu studiata per prima da Emmy Noether, che la rilevò sugli anelli di polinomi.</span><small> (it)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="nl" >In de abstracte algebra wordt een ring Noethers genoemd als zijn idealen aan een bepaalde voorwaarde van eindigheid voldoen. Men spreekt in dit verband ook uitdrukkelijk over ringen die voldoen aan de stijgende ketenvoorwaarde. Noetherse ringen zijn genoemd naar Emmy Noether.</span><small> (nl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ru" >Нётерово кольцо́ — тип колец, обобщение кольца главных идеалов.Названы в честь Эмми Нётер.</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="sv" >En noethersk ring är inom matematiken en speciell sorts ring, uppkallad efter Emmy Noether. En kommutativ ring med etta R kallas noethersk om varje ideal är , det vill säga att för varje ideal I finns en ändlig mängd av element i I så att varje element x i I kan skrivas som en linjärkombination av dessa element: där elementen är element i R. Att I genereras av skrivs vanligtvis . För okommutativa ringar ringar med etta måste man vara litet mer precis. Man får två inte helt ekvivalenta egenskaper: Ringen kan vara högernoethersk eller vänsternoethersk; se de formella definitionerna nedan.</span><small> (sv)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="uk" >Кільце Нетер — в абстрактній алгебрі це таке асоціативне кільце з одиницею для якого справджується наступне твердження: нехай маємо деяку зростаючу послідовність ідеалів кільця: тоді існує таке для якого: Якщо ідеали в означенні ліві, то кільце називається лівим кільцем Нетер, якщо праві - правим кільцем Нетер. Якщо твердження виконується і для лівих і для правих ідеалів то кільце просто називається кільцем Нетер.Дані кільця названі на честь німецького математика Еммі Нетер (нім. Emmy Noether).</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="zh" >諾特環是抽象代數中一類滿足升鏈條件的環。希爾伯特首先在研究不變量理論時證明了多項式環的每個理想都是有限生成的，隨後埃米·諾特從中提煉出升鏈條件，諾特環由此命名。</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ca" >En àlgebra abstracta, un anell noetherià és un anell commutatiu i unitari que satisfà que la cadena d'ideals és estacionària. És a dir, donada una cadena d'ideals: existeix un n tal que: Aquest tipus d'anells reben aquest nom en honor de la matemàtica alemanya Emmy Noether.</span><small> (ca)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="es" >En álgebra abstracta, un anillo es Noetheriano por la izquierda si sus ideales por la izquierda satisfacen la condición de cadena ascendente. Diremos que un anillo es noetheriano si es noetheriano por la izquierda y por la derecha. En los anillos conmutativos no se utiliza esta distinción, pues noetheriano por un lado implica noetheriano por el otro. Los anillos Noetherianos son nombrados así en honor a Amalie Emmy Noether.</span><small> (es)</small></span></li> <li><span class="literal"><span property="rdfs:comment" lang="en" >In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left (or right) ideals has a largest element; that is, there exists an n such that: Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian.</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="fr" >En mathématique, un anneau noethérien est un cas particulier d'anneau, c'est-à-dire d'un ensemble muni d'une addition et d'une multiplication compatible avec l'addition, au sens de la distributivité. De nombreuses questions mathématiques s'expriment dans un contexte d'anneau, les endomorphismes d'un espace vectoriel ou d'un module sur un anneau, les entiers algébriques de la théorie algébrique des nombres, ou encore les surfaces de la géométrie algébrique. Si les anneaux sont nombreux, rares sont ceux disposant des propriétés communes aux exemples les plus simples comme les entiers relatifs ou les polynômes à coefficients dans un corps. La division euclidienne n'existe en général plus, les idéaux, outils majeurs de la théorie des anneaux, ne sont plus toujours principaux et le théorème fon</span><small> (fr)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="pl" >Pierścień noetherowski – pierścień, w którym każdy ciąg wstępujący (w sensie inkluzji) jego ideałów stabilizuje się, tzn. istnieje dla którego mówi się też wtedy, że w pierścień spełnia warunek rosnących łańcuchów (ACC) dla ideałów; pojęcie nosi nazwisko Emmy Noether. Równoważnie pierścień jest noetherowski wtedy i tylko wtedy, gdy każdy ideał właściwy jest skończenie generowany, tzn.dla każdego ideału istnieją takie elementy dla których Prawdziwe jest również twierdzenie Hilberta o bazie: jeżeli pierścień jest noetherowski, to jego pierścień wielomianów również jest noetherowski.</span><small> (pl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="pt" >Em álgebra abstracta, um anel noetheriano é um anel comutativo que satisfaz a para ideais.O termo noetheriano é uma homenagem à matemática alemã Emmy Noether. Anéis de polinômios sobre corpos possuem muitas propriedades especiais; propriedades que derivam do fato de que anéis polinomiais não são em certo sentido "grandes demais". Emmy Noether descobriu que uma propriedade fundamental dos anéis de polinômios é a propriedade da cadeia ascendente para ideais. Para anéis não-comutativos, devemos fazer algumas distinções entre conceitos similares: Para anéis comutativos as três definições coincidem.</span><small> (pt)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label"><small>rdfs:</small>label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ca" >Anell noetherià</span><small> (ca)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="cs" >Noetherovský okruh</span><small> (cs)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="de" >Noetherscher Ring</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="es" >Anillo noetheriano</span><small> (es)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="fr" >Anneau noethérien</span><small> (fr)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="it" >Anello noetheriano</span><small> (it)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ja" >ネーター環</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ko" >뇌터 환</span><small> (ko)</small></span></li> <li><span class="literal"><span property="rdfs:label" lang="en" >Noetherian ring</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="nl" >Noetherse ring</span><small> (nl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="pl" >Pierścień noetherowski</span><small> (pl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="pt" >Anel noetheriano</span><small> (pt)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ru" >Нётерово кольцо</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="sv" >Noethersk ring</span><small> (sv)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="zh" >諾特環</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="uk" >Кільце Нетер</span><small> (uk)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs"><small>owl:</small>sameAs</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="owl:sameAs" resource="http://rdf.freebase.com/ns/m.0fk5n" href="http://rdf.freebase.com/ns/m.0fk5n"><small>freebase</small>:Noetherian ring</a></span></li> <li><span 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