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class="display-6">About: <a href="http://dbpedia.org/resource/Ideal_(ring_theory)">Ideal (ring theory)</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/ontology/ProgrammingLanguage">programming language</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 ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. </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;">Ideál je matematický pojem z oblasti algebry označující podmnožinu nějakého okruhu s jistými „dobrými“ vlastnostmi. * Tak jako normální podgrupy jsou speciálními případy podgrup, jsou rovněž ideály jisté podmnožiny daného okruhu. (cs)</li> <li style="display:none;">Un ideal d'un anell A és un subconjunt I d'elements de A que és tancat respecte a i que compleix una sèrie de condicions que detallarem a continuació. Per permetre l'aplicació a anells no commutatius, es defineixen ideals per l'esquerra i ideals per la dreta. Els ideals per les dues bandes (per exemple els d'anells commutatius) s'anomenen ideals bilàters o senzillament ideals. El concepte d'ideal fou proposat per primera vegada per Richard Dedekind el 1876 a la tercera edició del seu llibre Vorlesungen über Zahlentheorie ("Lliçons sobre teoria dels nombres"). Era una generalització del concepte de desenvolupat per Ernst Kummer. Més endavant la idea fou ampliada per David Hilbert i especialment Emmy Noether. La principal utilitat dels ideals que en motiva el seu ús és que permeten definir una relació d'equivalència que dona lloc al concepte d'anell quocient. (ca)</li> <li style="display:none;">في نظرية الحلقات، و هي فرع من الجبر التجريدي، المثالي (بالإنجليزية: Ideal)‏ مجموعة جزئية خاصة من حلقة تحقق عددا من الشروط. و يعمم مفهوم المثالي مفهوم بعض المجموعات الجزئية من مجموعة الأعداد الصحيحة كمجموعة الأعداد الزوجية أو مجموعة مضاعفات العدد 3. جمع وطرح الأعداد الزوجية يعطيان دائما عددا زوجيا، وضرب عدد زوجي في عدد صحيح ما يعطي دائما عددا زوجيا. هذان الخاصيتان المتمثلتان في الانغلاق والمص هما اللتان تعرفان مفهوم المجموعة المثالية. (ar)</li> <li style="display:none;">Στη Θεωρία δακτυλίων ιδεώδες είναι ένα ειδικό υποσύνολο του δακτυλίου. (el)</li> <li style="display:none;">En abstrakta algebro, idealo de ringo estas tia adicia subgrupo de , ke al ĝi apartenas la produtoj (maldekstra idealo), (dekstra idealo), aŭ kaj (ambaŭflanka aŭ duflanka idealo) elementoj kaj . La rolo de idealoj en la ringo-teorio estas simila al la rolo de normalaj subgrupoj en la grupo-teorio. Specife, la de ringa homomorfio estas idealo, kaj se estas subringo de oni povas krei la se kaj nur se estas idealo. Simile oni difinas la idealojn en semigrupoj. (eo)</li> <li style="display:none;">In der abstrakten Algebra ist ein Ideal eine Teilmenge eines Rings, die das Nullelement enthält und abgeschlossen gegenüber Addition und Subtraktion von Elementen des Ideals sowie abgeschlossen gegenüber Multiplikation mit beliebigen Ringelementen ist. Beispielsweise sind Summe und Differenz zweier gerader Zahlen wieder gerade und zudem ist das Produkt einer geraden Zahl mit einer beliebigen ganzen Zahl ebenfalls gerade. Zudem ist die 0 als additiv Neutrales gerade. Das heißt, die Menge der geraden Zahlen ist ein Ideal im Ring der ganzen Zahlen. Die Bezeichnung „Ideal“ ist abgeleitet aus dem Begriff „ideale Zahl“: Ideale können als Verallgemeinerung von Zahlen angesehen werden. Das Konzept der Ideale hat seinen Ursprung in der algebraischen Zahlentheorie des 19. Jahrhunderts bei Ernst Eduard Kummer und wurde weiterentwickelt von Richard Dedekind und Leopold Kronecker. Bei David Hilbert war ein Ideal ein System von unendlich vielen ganzen algebraischen Zahlen eines Rationalitätsbereiches (algebraischer Zahlkörper), mit der Eigenschaft, dass auch sämtliche Linearkombinationen dieser (mit ganzen algebraischen Zahlen als Koeffizienten) darin enthalten sind. Diese Definition entspricht dem heutigen Begriff des gebrochenen Ideals. In der Literatur findet man häufig die Begriffe Linksideal, Rechtsideal und zweiseitiges Ideal. Siehe dazu unten bei den . (de)</li> <li>In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory). The related, but distinct, concept of an ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity. (en)</li> <li style="display:none;">En álgebra moderna, un ideal es una subestructura algebraica definida en la teoría de anillos. Los ideales generalizan, de manera fecunda, el estudio de la divisibilidad entre los números enteros hacia otros objetos matemáticos. De este modo, es posible enunciar versiones muy generales de teoremas de la aritmética elemental, tales como el teorema chino del resto o el teorema fundamental de la aritmética, válidos para los ideales. Se puede comparar también esta noción con la de subgrupo normal para la estructura algebraica de grupo en el sentido de que facilita definir la noción de anillo cociente como una extensión natural de la noción de grupo cociente. (es)</li> <li style="display:none;">Dalam teori gelanggang, sebuah cabang dari aljabar abstrak, ideal dari gelanggang adalah himpunan bagian khusus dari elemen. Ideal menggeneralisasi himpunan bagian tertentu dari bilangan bulat, seperti bilangan genap atau kelipatan 3. Penambahan dan pengurangan bilangan genap mempertahankan kemerataan, dan mengalikan bilangan genap dengan bilangan bulat lainnya menghasilkan bilangan genap lainnya; dan sifat absorpsi adalah sifat yang menentukan dari suatu ideal. Ideal dapat digunakan untuk gelanggang hasil bagi dengan cara yang sama di teori grup, subgrup normal dapat digunakan untuk grup hasil bagi. Di antara bilangan bulat, yang ideal sesuai satu-untuk-satu dengan : dalam gelanggang, ideal adalah yang terdiri dari kelipatan satu bilangan non-negatif. Namun, dalam gelanggang lain, ideal mungkin tidak sesuai langsung dengan elemen gelanggang, dan sifat bilangan bulat tertentu, ketika digeneralisasikan ke gelanggang, lebih alami melekat pada ideal dari elemen. Misalnya, gelanggang dianalogikan dengan bilangan prima, dan dapat digeneralisasikan menjadi ideal. Ada versi untuk ideal (jenis gelanggang yang penting dalam teori bilangan). Terkait, tetapi berbeda, konsep dari dalam diturunkan dari gagasan ideal dalam teori gelanggang. Sebuah adalah generalisasi dari suatu ideal, dan ideal biasa kadang-kadang disebut ideal integral untuk kejelasan. (in)</li> <li style="display:none;">抽象代数学の分野である環論におけるイデアル（英: ideal, 独: Ideal）は環の特別な部分集合である。整数全体の成す環における、偶数全体の成す集合や 3 の倍数全体の成す集合などの持つ性質を一般化したもので、その部分集合に属する任意の元の和と差に関して閉じていて、なおかつ環の任意の元を掛けることについても閉じているものをイデアルという。整数の場合であれば、イデアルと非負整数とは一対一に対応する。即ち整数環 Z の任意のイデアルは、それぞれただ一つの整数の倍数すべてからなる主イデアルになる。しかしそれ以外の一般の環においてはイデアルと環の元とは全く異なるものを指しうるもので、整数のある種の性質を一般の環に対して一般化する際に、環の元を考えるよりもそのイデアルを考えるほうが自然であるということがある。例えば、環の素イデアルは素数の環における対応物であり、中国の剰余定理もイデアルに対するものに一般化することができる。素因数分解の一意性もデデキント環のイデアルに対応するものが存在し、数論において重要な役割を持つ。イデアルは整数の算術から定義される合同算術の方法と同様の剰余環（商環）の構成にも用いられる、この点において群論で剰余群（商群）の構成に用いられる正規部分群と同様のものと理解することができる。順序集合に対するの概念は環論におけるこのイデアルの概念に由来する。またイデアルの概念を一般化して分数イデアルの概念を考えることもでき、それとの区別のためここで扱う通常のイデアルは整イデアルと呼ばれることもある。 (ja)</li> <li style="display:none;">En mathématiques, et plus particulièrement en algèbre, un idéal est un sous-ensemble remarquable d'un anneau : c'est un sous-groupe du groupe additif de l'anneau qui est, de plus, stable par multiplication par les éléments de l'anneau. À certains égards, les idéaux s'apparentent donc aux sous-espaces vectoriels — qui sont des sous-groupes additifs stables par une multiplication externe ; à d'autres égards, ils se comportent comme les sous-groupes distingués — ce sont des sous-groupes additifs à partir desquels on peut construire une structure d'anneau quotient. Apparus à la fin du XIXe siècle en théorie algébrique des nombres pour généraliser à des entiers algébriques la décomposition en facteurs premiers des entiers, les idéaux ont rapidement joué un rôle central en algèbre et en géométrie algébrique, en particulier à la suite des travaux d'Emmy Noether isolant l'importance des conditions de chaîne. Au-delà de l'algèbre, ils interviennent de façon centrale dans les développements du XXe siècle de certains chapitres d'analyse fonctionnelle, notamment l'étude des algèbres de Banach et l'analyse harmonique commutative. En algèbre commutative, deux types d'idéaux sont omniprésents : les idéaux maximaux et, sans doute encore davantage, les idéaux premiers. Dans l'anneau des entiers relatifs, tant les idéaux maximaux que les idéaux premiers (non nuls) sont les pZ, où p est un nombre premier ; dans les anneaux commutatifs plus abstraits ces familles d'idéaux généralisent la notion de nombre premier. En théorie des anneaux non commutatifs, il faut prendre garde à l'existence juxtaposée de deux concepts distincts d'idéaux : les idéaux à gauche (ou à droite), qui sont des sous-modules, et les idéaux bilatères, ceux par lesquels on peut quotienter. Alors que la structure des anneaux non commutatifs les plus généraux peut être extrêmement complexe, on a plus de prise sur ceux vérifiant les conditions de finitude découvertes par Emmy Noether et Emil Artin, à savoir des conditions de chaîne sur leurs idéaux à gauche ou à droite. (fr)</li> <li style="display:none;">( 다른 뜻에 대해서는 아이디얼 (동음이의) 문서를 참고하십시오.) 환론에서 아이디얼(영어: ideal) 또는 이데알(독일어: Ideal)은 특정한 조건을 만족시키는 환의 부분집합이다. 이에 대하여 몫환을 취할 수 있으며, 군론에서 정규 부분군에 대하여 몫군을 취하는 것과 유사한 개념이다. 아이디얼을 사용하여 수론적 개념을 보다 일반적인 환들에 대하여 확장시킬 수 있다. 예를 들어, 소수의 개념을 확장한 소 아이디얼 및 서로소인 수의 개념을 확장한 서로소 아이디얼을 정의하면, 일반화된 중국인의 나머지 정리를 증명할 수 있다. 수론에서 중요한 개념인 데데킨트 정역의 경우, 아이디얼에 대해 산술의 기본정리까지도 성립함을 보일 수 있다. (즉, 임의의 0이 아닌 아이디얼은 소 아이디얼들의 곱으로 유일하게 표현할 수 있다.) (ko)</li> <li style="display:none;">Een ideaal is in de abstracte algebra, specifiek in de ringtheorie, een deelgebied van de wiskunde, een deelverzameling van een ring, die gesloten is ten aanzien van lineaire combinaties met coëfficiënten uit de ring. Dat houdt in dat een ideaal ten aanzien van de optelling een ondergroep is en dat de vermenigvuldiging, zowel links als rechts, van een element uit het ideaal met een element van de ring een resultaat geeft dat binnen het ideaal ligt. De term 'ideaal' verwijst naar het begrip ideaal getal, waarvan idealen een generalisatie vormen in verband met deelbaarheidseigenschappen. De specifieke studie van idealen in commutatieve ringen met eenheidselement heette aanvankelijk ideaaltheorie, maar tegenwoordig is commutatieve algebra gebruikelijker. (nl)</li> <li style="display:none;">In matematica, e più precisamente in algebra, un ideale è un sottoinsieme di un anello chiuso rispetto alla somma interna e al prodotto con qualsiasi elemento dell'anello. (it)</li> <li style="display:none;">Em teoria dos anéis, um ramo da álgebra abstrata, um ideal é um subconjunto especial de um anel. O conceito generaliza de uma maneira apropriada algumas importantes propriedades dos inteiros como "número par" e "múltiplo de 3". Por exemplo, em anéis estuda-se ideais primos ao invés de números primos, define-se ideais coprimos como generalizações dos números coprimos e pode-se provar um teorema do resto chinês para ideais. Nos domínios de Dedekind, importante classe de anéis para a teoria dos números, pode-se inclusive imitar-se uma versão do teorema fundamental da aritmética. Nesses anéis, todo ideal não-zero pode ser escrito como um produto único de ideais primos. Um ideal pode ser usado para a construção de um anel quociente da mesma forma que um subgrupo normal pode ser usado para a construção de um grupo quociente. (pt)</li> <li style="display:none;">Ideał – podzbiór pierścienia o własnościach pozwalających na konstrukcję pierścienia ilorazowego. Pojęcie ideału zostało wprowadzone przez Dedekinda jako uogólnienie pojęcia liczby idealnej, rozważanego przez Kummera. Badania Dedekinda były kontynuowane przez Hilberta i, szczególnie, przez Emmy Noether. Ideały odgrywają w teorii pierścieni rolę analogiczną do podgrup normalnych w teorii grup. (pl)</li> <li style="display:none;">Идеал — одно из основных понятий общей алгебры. Наибольшее значение идеалы имеют в теории колец, но также определяются и для полугрупп, алгебр и некоторых других алгебраических структур. Название «идеал» ведёт своё происхождение от «идеальных чисел», которые были введены в 1847 году немецким математиком Э. Э. Куммером. Простейшим примером идеала может служить подкольцо чётных чисел в кольце целых чисел. Идеалы дают удобный язык для обобщения результатов теории чисел на общие кольца. Например, в кольцах вместо простых чисел изучаются простые идеалы, как обобщение взаимно простых чисел вводятся взаимно простые идеалы, можно доказать аналог китайской теоремы об остатках для идеалов. В некотором важном классе колец (т. н. дедекиндовых) можно даже получить аналог основной теоремы арифметики: в этих кольцах каждый ненулевой идеал можно единственным образом представить как произведение простых идеалов. Примером идеала может служить множество целых чисел, которые делятся на 6: , если рассматривать их в кольце . Это множество является идеалом потому, что и сумма любых двух таких чисел, и произведение любого из них на любое целое число сами входят в это множество. При этом то же самое множество не будет идеалом в кольце вещественных чисел, так как результат умножения какого-либо из этих чисел на произвольное вещественное число в общем случае не входит в это множество. (ru)</li> <li style="display:none;">En icke-tom delmängd I till ringen R kallas för ett ideal om: 1. I är en additiv delgrupp till R.2. Om för varje element i som tillhör I och r som tillhör R följer, att både i·r och r·i tillhör I. Den icke-tomma delmängden I av de hela talen Z, är ett ideal om för alla x och y i I följer att x - y tillhör I. Inom ringteorin, är ett ideal ett av Richard Dedekind infört begrepp i anslutning till ett uppslag av Ernst Kummer, kallat "ideala tal". Detta begrepp var tänkt för att bevara den entydiga faktoriseringen för algebraiska heltal (motsvarande heltalens primtalsfaktoriseringar). Begreppet ideal är en generalisering av detta begrepp inom den abstrakta algebran. Emmy Noether byggde senare ut definitionen till det axiomatiska ringteorin. (sv)</li> <li style="display:none;">理想（Ideal）是一个环论中的概念。若某环的子集为在原环加法的定义下的子群，且其中的元素在原环乘法下与任意原环中的元素结果都在该子群中，则称其为原环的理想。通俗地说，一环的理想在加法上成群且在乘法上表现如同一个黑洞。理想把整数的某些子集，例如偶数或3的倍数组成的集合给一般化了。两个偶数相加或相减结果仍是偶数，偶数与任意整数相乘的结果也仍是偶数；这些闭包和吸收的性质正是理想的定义。理想可以被用来构造商环，这类似于在群论里，正规子群可以被用来构造商群。 (zh)</li> <li style="display:none;">Ідеал — підструктура з певними властивостями в абстрактній алгебрі. Спочатку виникло поняття ідеал кільця, пізніше було узагальнено для інших алгебраїчних структур. Найважливішу роль ідеали відіграють при вивченні кілець, напівгруп, алгебр над кільцем та ін. Назва «ідеал» веде своє походження від . Ідеали дають зручну мову для узагальнення результатів теорії чисел на загальні кільця. Прикладом ідеала може служити підкільце парних чисел в кільці цілих чисел, позначають 2Z. 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(cs)</li> <li style="display:none;">في نظرية الحلقات، و هي فرع من الجبر التجريدي، المثالي (بالإنجليزية: Ideal)‏ مجموعة جزئية خاصة من حلقة تحقق عددا من الشروط. و يعمم مفهوم المثالي مفهوم بعض المجموعات الجزئية من مجموعة الأعداد الصحيحة كمجموعة الأعداد الزوجية أو مجموعة مضاعفات العدد 3. جمع وطرح الأعداد الزوجية يعطيان دائما عددا زوجيا، وضرب عدد زوجي في عدد صحيح ما يعطي دائما عددا زوجيا. هذان الخاصيتان المتمثلتان في الانغلاق والمص هما اللتان تعرفان مفهوم المجموعة المثالية. (ar)</li> <li style="display:none;">Στη Θεωρία δακτυλίων ιδεώδες είναι ένα ειδικό υποσύνολο του δακτυλίου. (el)</li> <li style="display:none;">En abstrakta algebro, idealo de ringo estas tia adicia subgrupo de , ke al ĝi apartenas la produtoj (maldekstra idealo), (dekstra idealo), aŭ kaj (ambaŭflanka aŭ duflanka idealo) elementoj kaj . La rolo de idealoj en la ringo-teorio estas simila al la rolo de normalaj subgrupoj en la grupo-teorio. Specife, la de ringa homomorfio estas idealo, kaj se estas subringo de oni povas krei la se kaj nur se estas idealo. Simile oni difinas la idealojn en semigrupoj. (eo)</li> <li style="display:none;">En álgebra moderna, un ideal es una subestructura algebraica definida en la teoría de anillos. Los ideales generalizan, de manera fecunda, el estudio de la divisibilidad entre los números enteros hacia otros objetos matemáticos. De este modo, es posible enunciar versiones muy generales de teoremas de la aritmética elemental, tales como el teorema chino del resto o el teorema fundamental de la aritmética, válidos para los ideales. Se puede comparar también esta noción con la de subgrupo normal para la estructura algebraica de grupo en el sentido de que facilita definir la noción de anillo cociente como una extensión natural de la noción de grupo cociente. (es)</li> <li style="display:none;">( 다른 뜻에 대해서는 아이디얼 (동음이의) 문서를 참고하십시오.) 환론에서 아이디얼(영어: ideal) 또는 이데알(독일어: Ideal)은 특정한 조건을 만족시키는 환의 부분집합이다. 이에 대하여 몫환을 취할 수 있으며, 군론에서 정규 부분군에 대하여 몫군을 취하는 것과 유사한 개념이다. 아이디얼을 사용하여 수론적 개념을 보다 일반적인 환들에 대하여 확장시킬 수 있다. 예를 들어, 소수의 개념을 확장한 소 아이디얼 및 서로소인 수의 개념을 확장한 서로소 아이디얼을 정의하면, 일반화된 중국인의 나머지 정리를 증명할 수 있다. 수론에서 중요한 개념인 데데킨트 정역의 경우, 아이디얼에 대해 산술의 기본정리까지도 성립함을 보일 수 있다. (즉, 임의의 0이 아닌 아이디얼은 소 아이디얼들의 곱으로 유일하게 표현할 수 있다.) (ko)</li> <li style="display:none;">Een ideaal is in de abstracte algebra, specifiek in de ringtheorie, een deelgebied van de wiskunde, een deelverzameling van een ring, die gesloten is ten aanzien van lineaire combinaties met coëfficiënten uit de ring. Dat houdt in dat een ideaal ten aanzien van de optelling een ondergroep is en dat de vermenigvuldiging, zowel links als rechts, van een element uit het ideaal met een element van de ring een resultaat geeft dat binnen het ideaal ligt. De term 'ideaal' verwijst naar het begrip ideaal getal, waarvan idealen een generalisatie vormen in verband met deelbaarheidseigenschappen. De specifieke studie van idealen in commutatieve ringen met eenheidselement heette aanvankelijk ideaaltheorie, maar tegenwoordig is commutatieve algebra gebruikelijker. (nl)</li> <li style="display:none;">In matematica, e più precisamente in algebra, un ideale è un sottoinsieme di un anello chiuso rispetto alla somma interna e al prodotto con qualsiasi elemento dell'anello. (it)</li> <li style="display:none;">Ideał – podzbiór pierścienia o własnościach pozwalających na konstrukcję pierścienia ilorazowego. Pojęcie ideału zostało wprowadzone przez Dedekinda jako uogólnienie pojęcia liczby idealnej, rozważanego przez Kummera. Badania Dedekinda były kontynuowane przez Hilberta i, szczególnie, przez Emmy Noether. Ideały odgrywają w teorii pierścieni rolę analogiczną do podgrup normalnych w teorii grup. (pl)</li> <li style="display:none;">理想（Ideal）是一个环论中的概念。若某环的子集为在原环加法的定义下的子群，且其中的元素在原环乘法下与任意原环中的元素结果都在该子群中，则称其为原环的理想。通俗地说，一环的理想在加法上成群且在乘法上表现如同一个黑洞。理想把整数的某些子集，例如偶数或3的倍数组成的集合给一般化了。两个偶数相加或相减结果仍是偶数，偶数与任意整数相乘的结果也仍是偶数；这些闭包和吸收的性质正是理想的定义。理想可以被用来构造商环，这类似于在群论里，正规子群可以被用来构造商群。 (zh)</li> <li style="display:none;">Ідеал — підструктура з певними властивостями в абстрактній алгебрі. Спочатку виникло поняття ідеал кільця, пізніше було узагальнено для інших алгебраїчних структур. Найважливішу роль ідеали відіграють при вивченні кілець, напівгруп, алгебр над кільцем та ін. Назва «ідеал» веде своє походження від . Ідеали дають зручну мову для узагальнення результатів теорії чисел на загальні кільця. Прикладом ідеала може служити підкільце парних чисел в кільці цілих чисел, позначають 2Z. (uk)</li> <li style="display:none;">Un ideal d'un anell A és un subconjunt I d'elements de A que és tancat respecte a i que compleix una sèrie de condicions que detallarem a continuació. Per permetre l'aplicació a anells no commutatius, es defineixen ideals per l'esquerra i ideals per la dreta. Els ideals per les dues bandes (per exemple els d'anells commutatius) s'anomenen ideals bilàters o senzillament ideals. (ca)</li> <li style="display:none;">In der abstrakten Algebra ist ein Ideal eine Teilmenge eines Rings, die das Nullelement enthält und abgeschlossen gegenüber Addition und Subtraktion von Elementen des Ideals sowie abgeschlossen gegenüber Multiplikation mit beliebigen Ringelementen ist. Beispielsweise sind Summe und Differenz zweier gerader Zahlen wieder gerade und zudem ist das Produkt einer geraden Zahl mit einer beliebigen ganzen Zahl ebenfalls gerade. Zudem ist die 0 als additiv Neutrales gerade. Das heißt, die Menge der geraden Zahlen ist ein Ideal im Ring der ganzen Zahlen. (de)</li> <li>In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. (en)</li> <li style="display:none;">Dalam teori gelanggang, sebuah cabang dari aljabar abstrak, ideal dari gelanggang adalah himpunan bagian khusus dari elemen. Ideal menggeneralisasi himpunan bagian tertentu dari bilangan bulat, seperti bilangan genap atau kelipatan 3. Penambahan dan pengurangan bilangan genap mempertahankan kemerataan, dan mengalikan bilangan genap dengan bilangan bulat lainnya menghasilkan bilangan genap lainnya; dan sifat absorpsi adalah sifat yang menentukan dari suatu ideal. Ideal dapat digunakan untuk gelanggang hasil bagi dengan cara yang sama di teori grup, subgrup normal dapat digunakan untuk grup hasil bagi. (in)</li> <li style="display:none;">En mathématiques, et plus particulièrement en algèbre, un idéal est un sous-ensemble remarquable d'un anneau : c'est un sous-groupe du groupe additif de l'anneau qui est, de plus, stable par multiplication par les éléments de l'anneau. À certains égards, les idéaux s'apparentent donc aux sous-espaces vectoriels — qui sont des sous-groupes additifs stables par une multiplication externe ; à d'autres égards, ils se comportent comme les sous-groupes distingués — ce sont des sous-groupes additifs à partir desquels on peut construire une structure d'anneau quotient. (fr)</li> <li style="display:none;">抽象代数学の分野である環論におけるイデアル（英: ideal, 独: Ideal）は環の特別な部分集合である。整数全体の成す環における、偶数全体の成す集合や 3 の倍数全体の成す集合などの持つ性質を一般化したもので、その部分集合に属する任意の元の和と差に関して閉じていて、なおかつ環の任意の元を掛けることについても閉じているものをイデアルという。整数の場合であれば、イデアルと非負整数とは一対一に対応する。即ち整数環 Z の任意のイデアルは、それぞれただ一つの整数の倍数すべてからなる主イデアルになる。しかしそれ以外の一般の環においてはイデアルと環の元とは全く異なるものを指しうるもので、整数のある種の性質を一般の環に対して一般化する際に、環の元を考えるよりもそのイデアルを考えるほうが自然であるということがある。例えば、環の素イデアルは素数の環における対応物であり、中国の剰余定理もイデアルに対するものに一般化することができる。素因数分解の一意性もデデキント環のイデアルに対応するものが存在し、数論において重要な役割を持つ。イデアルは整数の算術から定義される合同算術の方法と同様の剰余環（商環）の構成にも用いられる、この点において群論で剰余群（商群）の構成に用いられる正規部分群と同様のものと理解することができる。 (ja)</li> <li style="display:none;">Em teoria dos anéis, um ramo da álgebra abstrata, um ideal é um subconjunto especial de um anel. O conceito generaliza de uma maneira apropriada algumas importantes propriedades dos inteiros como "número par" e "múltiplo de 3". Um ideal pode ser usado para a construção de um anel quociente da mesma forma que um subgrupo normal pode ser usado para a construção de um grupo quociente. (pt)</li> <li style="display:none;">Идеал — одно из основных понятий общей алгебры. Наибольшее значение идеалы имеют в теории колец, но также определяются и для полугрупп, алгебр и некоторых других алгебраических структур. Название «идеал» ведёт своё происхождение от «идеальных чисел», которые были введены в 1847 году немецким математиком Э. Э. Куммером. Простейшим примером идеала может служить подкольцо чётных чисел в кольце целых чисел. Идеалы дают удобный язык для обобщения результатов теории чисел на общие кольца. (ru)</li> <li style="display:none;">En icke-tom delmängd I till ringen R kallas för ett ideal om: 1. I är en additiv delgrupp till R.2. Om för varje element i som tillhör I och r som tillhör R följer, att både i·r och r·i tillhör I. Den icke-tomma delmängden I av de hela talen Z, är ett ideal om för alla x och y i I följer att x - y tillhör I. Inom ringteorin, är ett ideal ett av Richard Dedekind infört begrepp i anslutning till ett uppslag av Ernst Kummer, kallat "ideala tal". Detta begrepp var tänkt för att bevara den entydiga faktoriseringen för algebraiska heltal (motsvarande heltalens primtalsfaktoriseringar). (sv)</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;">مثالي (نظرية الحلقات) (ar)</li> <li style="display:none;">Ideal (matemàtiques) (ca)</li> <li style="display:none;">Ideál (teorie okruhů) (cs)</li> <li style="display:none;">Ideal (Ringtheorie) (de)</li> <li style="display:none;">Ιδεώδες (μαθηματικά) (el)</li> <li style="display:none;">Idealo (algebro) (eo)</li> <li style="display:none;">Ideal (teoría de anillos) (es)</li> <li style="display:none;">Ideal (teori gelanggang) (in)</li> <li style="display:none;">Ideale (matematica) (it)</li> <li>Ideal (ring theory) (en)</li> <li style="display:none;">Idéal (fr)</li> <li style="display:none;">아이디얼 (ko)</li> <li style="display:none;">イデアル (環論) (ja)</li> <li style="display:none;">Ideaal (ringtheorie) (nl)</li> <li style="display:none;">Ideal (teoria dos anéis) (pt)</li> <li style="display:none;">Ideał (teoria pierścieni) (pl)</li> <li style="display:none;">Идеал (алгебра) (ru)</li> <li style="display:none;">Ideal (ringteori) (sv)</li> <li style="display:none;">Ідеал (алгебра) (uk)</li> <li style="display:none;">理想 (环论) (zh)</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.06gw1" href="http://rdf.freebase.com/ns/m.06gw1">freebase:Ideal (ring theory)</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://d-nb.info/gnd/4161198-6" href="http://d-nb.info/gnd/4161198-6">http://d-nb.info/gnd/4161198-6</a></li> <li><a class="uri" rel="owl:sameAs" resource="http://www.wikidata.org/entity/Q44649" href="http://www.wikidata.org/entity/Q44649">wikidata:Ideal (ring theory)</a></li> <li><a class="uri" rel="owl:sameAs" 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resource="http://en.wikipedia.org/wiki/Ideal_(ring_theory)" href="http://en.wikipedia.org/wiki/Ideal_(ring_theory)">wikipedia-en:Ideal_(ring_theory)</a></li> </ul></td></tr> </tbody> </table> </div> </div> </div> </section>   <section> <div class="container-xl"> <div class="text-center p-4 bg-light"> <a href="https://virtuoso.openlinksw.com/" title="OpenLink Virtuoso"><img class="powered_by" src="/statics/images/virt_power_no_border.png" alt="Powered by OpenLink Virtuoso"/></a>    <a href="http://linkeddata.org/"><img alt="This material is Open Knowledge" src="/statics/images/LoDLogo.gif"/></a>     <a href="http://dbpedia.org/sparql"><img alt="W3C Semantic Web Technology" src="/statics/images/sw-sparql-blue.png"/></a>     <a href="https://opendefinition.org/"><img alt="This material is Open Knowledge" src="/statics/images/od_80x15_red_green.png"/></a>    <a href="https://validator.w3.org/check?uri=referer"> <img src="https://www.w3.org/Icons/valid-xhtml-rdfa" alt="Valid XHTML + RDFa" 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