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The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A. If every element of B is integral over A, then we say that B is integral over A, or equivalently B is an integral extension of A. 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href="http://dbpedia.org/resource/Integral_element">Integral element</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/class/yago/WikicatAlgebraicStructures">WikicatAlgebraicStructures</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 commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj in A such that That is to say, b is a root of a monic polynomial over A. The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A. If every element of B is integral over A, then we say that B is integral over A, or equivalently B is an integral extension of A. In this article, the term ring will be understood to mean commutative ring with a multiplicative identity. </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;">Celistvý prvek je pojem z oboru . Je-li dán komutativní okruh a jeho , pak je prvek celistvý nad , je-li kořenem nějakého monického polynomu s koeficienty z , tedy pokud existují a taková, že . Definice celistvého prvku se liší od definice algebraického prvku pouze v přidaném požadavku, aby byl polynom monický, z čehož plyne, že každý celistvý prvek je algebraický. Množina prvků , které jsou celistvé nad , se nazývá celistvý uzávěr v . (cs)</li> <li style="display:none;">Im mathematischen Teilgebiet der kommutativen Algebra ist der Begriff eines ganzen Elementes in einer Ringerweiterung eine Verallgemeinerung des Begriffes eines algebraischen Elementes in einer Körpererweiterung. (de)</li> <li style="display:none;">En nombroteorio, entjera elemento estas ĝeneraligo de la koncepto de gaŭsaj entjeroj en la kampo de kompleksaj nombroj. (eo)</li> <li>In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj in A such that That is to say, b is a root of a monic polynomial over A. The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A. If every element of B is integral over A, then we say that B is integral over A, or equivalently B is an integral extension of A. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., or ); in this context, the integral elements are usually called algebraic integers. The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory. In this article, the term ring will be understood to mean commutative ring with a multiplicative identity. (en)</li> <li style="display:none;">En mathématiques, et plus particulièrement en algèbre commutative, les éléments entiers sur un anneau commutatif sont à la fois une généralisation des entiers algébriques (les éléments entiers sur l'anneau des entiers relatifs) et des éléments algébriques dans une extension de corps. C'est une notion très utile en théorie algébrique des nombres et en géométrie algébrique. Son émergence a commencé par l'étude des entiers quadratiques, en particulier les entiers de Gauss. (fr)</li> <li style="display:none;">가환대수학에서 정수적 원소(整數的元素, 영어: integral element)는 어떤 부분환에 계수를 갖는 일계수 다항식의 근으로 나타낼 수 있는 가환환 원소이다. (ko)</li> <li style="display:none;">可換環論において、可換環 B とその部分環 A について、B の元 b が A 係数のモニック多項式の根であるとき、b は A 上整である（integral over A）という。B のすべての元が A 上整であるとき、B は A 上整である、または、B は A の整拡大（integral extension）であるという。本記事において、環とは単位元をもつ可換環のこととする。 (ja)</li> <li style="display:none;">In de commutatieve algebra wordt een element van een commutatieve ring met eenheid geheel genoemd ten opzichte van een deelring (met eenheid) als dat element een nulpunt is van een monische polynoom met coëfficiënten in de deelring. De eigenschap 'geheel' generaliseert enerzijds algebraïsche gehele getallen, en anderzijds een algebraïsche uitbreiding van een commutatief lichaam. (nl)</li> <li style="display:none;">In algebra, un'estensione intera di un anello commutativo unitario è un'estensione di anelli tale che ogni elemento di B è intero su A, ovvero tale che ogni elemento di B è radice di un polinomio monico a coefficienti in A. Rappresenta una generalizzazione del concetto di estensione algebrica di campi: se A è un campo, le estensioni intere sono infatti le estensione algebriche (dal momento che ogni polinomio può essere reso monico moltiplicando per l'inverso del coefficiente direttore). (it)</li> <li style="display:none;">Ціле розширення кільця — розширення B комутативного кільця R з одиницею таке, що будь-який елемент є цілим над R, тобто задовольняє деякому рівнянню вигляду де . Дане рівняння називається рівнянням цілої залежності. Елемент x є цілим в R тоді і тільки тоді, коли виконується одна з двох еквівалентних умов: 1. * R[x] є скінченно породженим R-модулем ; 2. * існує точний R[x]-модуль, що є скінченно породженим R-модулем. (uk)</li> <li style="display:none;">Целый элемент — элемент заданного коммутативного кольца с единицей относительно подкольца , являющийся корнем приведённого многочлена с коэффициентами в , то есть такой , для которого существуют коэффициенты , такие что: . Если каждый элемент является целым над , кольцо называется целым расширением (или просто кольцом, целым над ). Если и — поля, терминам «цел над…» и «целое расширение» соответствуют термины «алгебраичен над…» и «алгебраическое расширение». Частный случай, особенно важный в теории чисел, — комплексные числа, являющиеся целыми над , называемые целыми алгебраическими числами. Множество всех элементов , целых над , образует кольцо; оно называется целым замыканием в . Целое замыкание рациональных чисел в некотором конечном расширении поля называется кольцом целых поля , этот объект является фундаментальным для алгебраической теории чисел. Целые числа — единственные элементы , являющиеся целыми над (что может служить объяснением использования термина «целый»). Гауссовы целые числа, как элементы поля комплексных чисел, являются целыми над . Целое замыкание в круговом поле — это . Если — алгебраическое замыкание поля , то цело над . Если конечная группа действует на кольце гомоморфизмами колец, то является целым над множеством элементов, являющихся неподвижными точками действия группы. (ru)</li> <li style="display:none;">整性是交換代數中的概念，用于描述在有理数域的某些扩域中，某些元素是否有类似于整数的性质。元素的整性（是否为整元素）本质上只依赖于環的概念。整性與環的整擴張推廣了代數數與代數擴張的概念。 (zh)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageExternalLink">dbo:wikiPageExternalLink</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="http://www.jmilne.org/math/" href="http://www.jmilne.org/math/">http://www.jmilne.org/math/</a></li> <li><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="https://mathoverflow.net/q/66445" href="https://mathoverflow.net/q/66445">https://mathoverflow.net/q/66445</a></li> <li><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="https://mathoverflow.net/q/7775" href="https://mathoverflow.net/q/7775">https://mathoverflow.net/q/7775</a></li> <li><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="http://people.reed.edu/~iswanson/trieste.pdf" href="http://people.reed.edu/~iswanson/trieste.pdf">http://people.reed.edu/~iswanson/trieste.pdf</a></li> <li><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="http://people.reed.edu/~iswanson/book/index.html" href="http://people.reed.edu/~iswanson/book/index.html">http://people.reed.edu/~iswanson/book/index.html</a></li> <li><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="https://archive.org/details/commutativerings00irvi" href="https://archive.org/details/commutativerings00irvi">https://archive.org/details/commutativerings00irvi</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageID">dbo:wikiPageID</a> </td><td class="col-10 text-break"><ul> <li>9478630 (xsd:integer)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageLength">dbo:wikiPageLength</a> </td><td class="col-10 text-break"><ul> <li>32308 (xsd:nonNegativeInteger)</li> 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Je-li dán komutativní okruh a jeho , pak je prvek celistvý nad , je-li kořenem nějakého monického polynomu s koeficienty z , tedy pokud existují a taková, že . Definice celistvého prvku se liší od definice algebraického prvku pouze v přidaném požadavku, aby byl polynom monický, z čehož plyne, že každý celistvý prvek je algebraický. Množina prvků , které jsou celistvé nad , se nazývá celistvý uzávěr v . (cs)</li> <li style="display:none;">Im mathematischen Teilgebiet der kommutativen Algebra ist der Begriff eines ganzen Elementes in einer Ringerweiterung eine Verallgemeinerung des Begriffes eines algebraischen Elementes in einer Körpererweiterung. (de)</li> <li style="display:none;">En nombroteorio, entjera elemento estas ĝeneraligo de la koncepto de gaŭsaj entjeroj en la kampo de kompleksaj nombroj. (eo)</li> <li style="display:none;">En mathématiques, et plus particulièrement en algèbre commutative, les éléments entiers sur un anneau commutatif sont à la fois une généralisation des entiers algébriques (les éléments entiers sur l'anneau des entiers relatifs) et des éléments algébriques dans une extension de corps. C'est une notion très utile en théorie algébrique des nombres et en géométrie algébrique. Son émergence a commencé par l'étude des entiers quadratiques, en particulier les entiers de Gauss. (fr)</li> <li style="display:none;">가환대수학에서 정수적 원소(整數的元素, 영어: integral element)는 어떤 부분환에 계수를 갖는 일계수 다항식의 근으로 나타낼 수 있는 가환환 원소이다. (ko)</li> <li style="display:none;">可換環論において、可換環 B とその部分環 A について、B の元 b が A 係数のモニック多項式の根であるとき、b は A 上整である（integral over A）という。B のすべての元が A 上整であるとき、B は A 上整である、または、B は A の整拡大（integral extension）であるという。本記事において、環とは単位元をもつ可換環のこととする。 (ja)</li> <li style="display:none;">In de commutatieve algebra wordt een element van een commutatieve ring met eenheid geheel genoemd ten opzichte van een deelring (met eenheid) als dat element een nulpunt is van een monische polynoom met coëfficiënten in de deelring. De eigenschap 'geheel' generaliseert enerzijds algebraïsche gehele getallen, en anderzijds een algebraïsche uitbreiding van een commutatief lichaam. (nl)</li> <li style="display:none;">In algebra, un'estensione intera di un anello commutativo unitario è un'estensione di anelli tale che ogni elemento di B è intero su A, ovvero tale che ogni elemento di B è radice di un polinomio monico a coefficienti in A. Rappresenta una generalizzazione del concetto di estensione algebrica di campi: se A è un campo, le estensioni intere sono infatti le estensione algebriche (dal momento che ogni polinomio può essere reso monico moltiplicando per l'inverso del coefficiente direttore). (it)</li> <li style="display:none;">Ціле розширення кільця — розширення B комутативного кільця R з одиницею таке, що будь-який елемент є цілим над R, тобто задовольняє деякому рівнянню вигляду де . Дане рівняння називається рівнянням цілої залежності. Елемент x є цілим в R тоді і тільки тоді, коли виконується одна з двох еквівалентних умов: 1. * R[x] є скінченно породженим R-модулем ; 2. * існує точний R[x]-модуль, що є скінченно породженим R-модулем. (uk)</li> <li style="display:none;">整性是交換代數中的概念，用于描述在有理数域的某些扩域中，某些元素是否有类似于整数的性质。元素的整性（是否为整元素）本质上只依赖于環的概念。整性與環的整擴張推廣了代數數與代數擴張的概念。 (zh)</li> <li>In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj in A such that That is to say, b is a root of a monic polynomial over A. The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A. If every element of B is integral over A, then we say that B is integral over A, or equivalently B is an integral extension of A. In this article, the term ring will be understood to mean commutative ring with a multiplicative identity. (en)</li> <li style="display:none;">Целый элемент — элемент заданного коммутативного кольца с единицей относительно подкольца , являющийся корнем приведённого многочлена с коэффициентами в , то есть такой , для которого существуют коэффициенты , такие что: . Если каждый элемент является целым над , кольцо называется целым расширением (или просто кольцом, целым над ). Целые числа — единственные элементы , являющиеся целыми над (что может служить объяснением использования термина «целый»). Гауссовы целые числа, как элементы поля комплексных чисел, являются целыми над . Целое замыкание в круговом поле — это . (ru)</li> </ul></td></tr><tr class="odd"><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;">Celistvý prvek (cs)</li> <li style="display:none;">Ganzes Element (de)</li> <li style="display:none;">Entjera elemento (eo)</li> <li style="display:none;">Estensione intera (it)</li> <li>Integral element (en)</li> <li style="display:none;">Élément entier (fr)</li> <li style="display:none;">정수적 원소 (ko)</li> <li style="display:none;">整拡大 (ja)</li> <li style="display:none;">Geheel element (nl)</li> <li style="display:none;">Целый элемент (ru)</li> <li style="display:none;">Ціле розширення кільця (uk)</li> <li style="display:none;">整性 (zh)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#seeAlso">rdfs:seeAlso</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="rdfs:seeAlso" 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