{ "@context": { "wikiPageLength": { "@id": "http://dbpedia.org/ontology/wikiPageLength", "@type": "http://www.w3.org/2001/XMLSchema#nonNegativeInteger" }, "isPrimaryTopicOf": { "@id": "http://xmlns.com/foaf/0.1/isPrimaryTopicOf", "@type": "@id" }, "wikiPageID": { "@id": "http://dbpedia.org/ontology/wikiPageID" }, "wikiPageDisambiguates": { "@id": "http://dbpedia.org/ontology/wikiPageDisambiguates", "@type": "@id" }, "seeAlso": { "@id": "http://www.w3.org/2000/01/rdf-schema#seeAlso", "@type": "@id" }, "primaryTopic": { "@id": "http://xmlns.com/foaf/0.1/primaryTopic", "@type": "@id" }, "sameAs": { "@id": "http://www.w3.org/2002/07/owl#sameAs", "@type": "@id" }, "wikiPageRedirects": { "@id": "http://dbpedia.org/ontology/wikiPageRedirects", "@type": "@id" }, "wasDerivedFrom": { "@id": "http://www.w3.org/ns/prov#wasDerivedFrom", "@type": "@id" }, "wikiPageExternalLink": { "@id": "http://dbpedia.org/ontology/wikiPageExternalLink", "@type": "@id" }, "wikiPageWikiLink": { "@id": "http://dbpedia.org/ontology/wikiPageWikiLink", "@type": "@id" }, "abstract": { "@id": "http://dbpedia.org/ontology/abstract" }, "subject": { "@id": "http://purl.org/dc/terms/subject", "@type": "@id" }, "wikiPageUsesTemplate": { "@id": "http://dbpedia.org/property/wikiPageUsesTemplate", "@type": "@id" }, "label": { "@id": "http://www.w3.org/2000/01/rdf-schema#label" }, "comment": { "@id": "http://www.w3.org/2000/01/rdf-schema#comment" }, "wikiPageRevisionID": { "@id": "http://dbpedia.org/ontology/wikiPageRevisionID" }}, "@graph": [ { "@id": "http://dbpedia.org/resource/Integral_element", "@type": [ "http://www.w3.org/2002/07/owl#Thing", "http://dbpedia.org/class/yago/Artifact100021939", "http://dbpedia.org/class/yago/Object100002684", "http://dbpedia.org/class/yago/PhysicalEntity100001930", "http://dbpedia.org/class/yago/YagoGeoEntity", "http://dbpedia.org/class/yago/YagoPermanentlyLocatedEntity", "http://dbpedia.org/class/yago/Structure104341686", "http://dbpedia.org/class/yago/Whole100003553", "http://dbpedia.org/class/yago/WikicatAlgebraicStructures" ], "label": [ { "@value" : "\u6574\u62E1\u5927" , "@language" : "ja" }, { "@value" : "\u0426\u0435\u043B\u044B\u0439 \u044D\u043B\u0435\u043C\u0435\u043D\u0442" , "@language" : "ru" }, { "@value" : "\uC815\uC218\uC801 \uC6D0\uC18C" , "@language" : "ko" }, { "@value" : "Estensione intera" , "@language" : "it" }, { "@value" : "Geheel element" , "@language" : "nl" }, { "@value" : "Celistv\u00FD prvek" , "@language" : "cs" }, { "@value" : "\u00C9l\u00E9ment entier" , "@language" : "fr" }, { "@value" : "Ganzes Element" , "@language" : "de" }, { "@value" : "Entjera elemento" , "@language" : "eo" }, { "@value" : "\u0426\u0456\u043B\u0435 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u043A\u0456\u043B\u044C\u0446\u044F" , "@language" : "uk" }, { "@value" : "Integral element" , "@language" : "en" }, { "@value" : "\u6574\u6027" , "@language" : "zh" } ], "comment": [ { "@value" : "In algebra, un'estensione intera di un anello commutativo unitario \u00E8 un'estensione di anelli tale che ogni elemento di B \u00E8 intero su A, ovvero tale che ogni elemento di B \u00E8 radice di un polinomio monico a coefficienti in A. Rappresenta una generalizzazione del concetto di estensione algebrica di campi: se A \u00E8 un campo, le estensioni intere sono infatti le estensione algebriche (dal momento che ogni polinomio pu\u00F2 essere reso monico moltiplicando per l'inverso del coefficiente direttore)." , "@language" : "it" }, { "@value" : "\u0426\u0435\u043B\u044B\u0439 \u044D\u043B\u0435\u043C\u0435\u043D\u0442 \u2014 \u044D\u043B\u0435\u043C\u0435\u043D\u0442 \u0437\u0430\u0434\u0430\u043D\u043D\u043E\u0433\u043E \u043A\u043E\u043C\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u043E\u0433\u043E \u043A\u043E\u043B\u044C\u0446\u0430 \u0441 \u0435\u0434\u0438\u043D\u0438\u0446\u0435\u0439 \u043E\u0442\u043D\u043E\u0441\u0438\u0442\u0435\u043B\u044C\u043D\u043E \u043F\u043E\u0434\u043A\u043E\u043B\u044C\u0446\u0430 , \u044F\u0432\u043B\u044F\u044E\u0449\u0438\u0439\u0441\u044F \u043A\u043E\u0440\u043D\u0435\u043C \u043F\u0440\u0438\u0432\u0435\u0434\u0451\u043D\u043D\u043E\u0433\u043E \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0430 \u0441 \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u0430\u043C\u0438 \u0432 , \u0442\u043E \u0435\u0441\u0442\u044C \u0442\u0430\u043A\u043E\u0439 , \u0434\u043B\u044F \u043A\u043E\u0442\u043E\u0440\u043E\u0433\u043E \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044E\u0442 \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u044B , \u0442\u0430\u043A\u0438\u0435 \u0447\u0442\u043E: . \u0415\u0441\u043B\u0438 \u043A\u0430\u0436\u0434\u044B\u0439 \u044D\u043B\u0435\u043C\u0435\u043D\u0442 \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u0446\u0435\u043B\u044B\u043C \u043D\u0430\u0434 , \u043A\u043E\u043B\u044C\u0446\u043E \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u0446\u0435\u043B\u044B\u043C \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043D\u0438\u0435\u043C (\u0438\u043B\u0438 \u043F\u0440\u043E\u0441\u0442\u043E \u043A\u043E\u043B\u044C\u0446\u043E\u043C, \u0446\u0435\u043B\u044B\u043C \u043D\u0430\u0434 ). \u0426\u0435\u043B\u044B\u0435 \u0447\u0438\u0441\u043B\u0430 \u2014 \u0435\u0434\u0438\u043D\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0435 \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u044B , \u044F\u0432\u043B\u044F\u044E\u0449\u0438\u0435\u0441\u044F \u0446\u0435\u043B\u044B\u043C\u0438 \u043D\u0430\u0434 (\u0447\u0442\u043E \u043C\u043E\u0436\u0435\u0442 \u0441\u043B\u0443\u0436\u0438\u0442\u044C \u043E\u0431\u044A\u044F\u0441\u043D\u0435\u043D\u0438\u0435\u043C \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u043E\u0432\u0430\u043D\u0438\u044F \u0442\u0435\u0440\u043C\u0438\u043D\u0430 \u00AB\u0446\u0435\u043B\u044B\u0439\u00BB). \u0413\u0430\u0443\u0441\u0441\u043E\u0432\u044B \u0446\u0435\u043B\u044B\u0435 \u0447\u0438\u0441\u043B\u0430, \u043A\u0430\u043A \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u044B \u043F\u043E\u043B\u044F \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u044B\u0445 \u0447\u0438\u0441\u0435\u043B, \u044F\u0432\u043B\u044F\u044E\u0442\u0441\u044F \u0446\u0435\u043B\u044B\u043C\u0438 \u043D\u0430\u0434 . \u0426\u0435\u043B\u043E\u0435 \u0437\u0430\u043C\u044B\u043A\u0430\u043D\u0438\u0435 \u0432 \u043A\u0440\u0443\u0433\u043E\u0432\u043E\u043C \u043F\u043E\u043B\u0435 \u2014 \u044D\u0442\u043E ." , "@language" : "ru" }, { "@value" : "\uAC00\uD658\uB300\uC218\uD559\uC5D0\uC11C \uC815\uC218\uC801 \uC6D0\uC18C(\u6574\u6578\u7684\u5143\u7D20, \uC601\uC5B4: integral element)\uB294 \uC5B4\uB5A4 \uBD80\uBD84\uD658\uC5D0 \uACC4\uC218\uB97C \uAC16\uB294 \uC77C\uACC4\uC218 \uB2E4\uD56D\uC2DD\uC758 \uADFC\uC73C\uB85C \uB098\uD0C0\uB0BC \uC218 \uC788\uB294 \uAC00\uD658\uD658 \uC6D0\uC18C\uC774\uB2E4." , "@language" : "ko" }, { "@value" : "\u6574\u6027\u662F\u4EA4\u63DB\u4EE3\u6578\u4E2D\u7684\u6982\u5FF5\uFF0C\u7528\u4E8E\u63CF\u8FF0\u5728\u6709\u7406\u6570\u57DF\u7684\u67D0\u4E9B\u6269\u57DF\u4E2D\uFF0C\u67D0\u4E9B\u5143\u7D20\u662F\u5426\u6709\u7C7B\u4F3C\u4E8E\u6574\u6570\u7684\u6027\u8D28\u3002\u5143\u7D20\u7684\u6574\u6027\uFF08\u662F\u5426\u4E3A\u6574\u5143\u7D20\uFF09\u672C\u8D28\u4E0A\u53EA\u4F9D\u8D56\u4E8E\u74B0\u7684\u6982\u5FF5\u3002\u6574\u6027\u8207\u74B0\u7684\u6574\u64F4\u5F35\u63A8\u5EE3\u4E86\u4EE3\u6578\u6578\u8207\u4EE3\u6578\u64F4\u5F35\u7684\u6982\u5FF5\u3002" , "@language" : "zh" }, { "@value" : "Im mathematischen Teilgebiet der kommutativen Algebra ist der Begriff eines ganzen Elementes in einer Ringerweiterung eine Verallgemeinerung des Begriffes eines algebraischen Elementes in einer K\u00F6rpererweiterung." , "@language" : "de" }, { "@value" : "\u53EF\u63DB\u74B0\u8AD6\u306B\u304A\u3044\u3066\u3001\u53EF\u63DB\u74B0 B \u3068\u305D\u306E\u90E8\u5206\u74B0 A \u306B\u3064\u3044\u3066\u3001B \u306E\u5143 b \u304C A \u4FC2\u6570\u306E\u30E2\u30CB\u30C3\u30AF\u591A\u9805\u5F0F\u306E\u6839\u3067\u3042\u308B\u3068\u304D\u3001b \u306F A \u4E0A\u6574\u3067\u3042\u308B\uFF08integral over A\uFF09\u3068\u3044\u3046\u3002B \u306E\u3059\u3079\u3066\u306E\u5143\u304C A \u4E0A\u6574\u3067\u3042\u308B\u3068\u304D\u3001B \u306F A \u4E0A\u6574\u3067\u3042\u308B\u3001\u307E\u305F\u306F\u3001B \u306F A \u306E\u6574\u62E1\u5927\uFF08integral extension\uFF09\u3067\u3042\u308B\u3068\u3044\u3046\u3002\u672C\u8A18\u4E8B\u306B\u304A\u3044\u3066\u3001\u74B0\u3068\u306F\u5358\u4F4D\u5143\u3092\u3082\u3064\u53EF\u63DB\u74B0\u306E\u3053\u3068\u3068\u3059\u308B\u3002" , "@language" : "ja" }, { "@value" : "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 \u2265 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." , "@language" : "en" }, { "@value" : "En nombroteorio, entjera elemento estas \u011Deneraligo de la koncepto de ga\u016Dsaj entjeroj en la kampo de kompleksaj nombroj." , "@language" : "eo" }, { "@value" : "Celistv\u00FD prvek je pojem z oboru . Je-li d\u00E1n komutativn\u00ED okruh a jeho , pak je prvek celistv\u00FD nad , je-li ko\u0159enem n\u011Bjak\u00E9ho monick\u00E9ho polynomu s koeficienty z , tedy pokud existuj\u00ED a takov\u00E1, \u017Ee . Definice celistv\u00E9ho prvku se li\u0161\u00ED od definice algebraick\u00E9ho prvku pouze v p\u0159idan\u00E9m po\u017Eadavku, aby byl polynom monick\u00FD, z \u010Deho\u017E plyne, \u017Ee ka\u017Ed\u00FD celistv\u00FD prvek je algebraick\u00FD. Mno\u017Eina prvk\u016F , kter\u00E9 jsou celistv\u00E9 nad , se naz\u00FDv\u00E1 celistv\u00FD uz\u00E1v\u011Br v ." , "@language" : "cs" }, { "@value" : "\u0426\u0456\u043B\u0435 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u043A\u0456\u043B\u044C\u0446\u044F \u2014 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F B \u043A\u043E\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u043E\u0433\u043E \u043A\u0456\u043B\u044C\u0446\u044F R \u0437 \u043E\u0434\u0438\u043D\u0438\u0446\u0435\u044E \u0442\u0430\u043A\u0435, \u0449\u043E \u0431\u0443\u0434\u044C-\u044F\u043A\u0438\u0439 \u0435\u043B\u0435\u043C\u0435\u043D\u0442 \u0454 \u0446\u0456\u043B\u0438\u043C \u043D\u0430\u0434 R, \u0442\u043E\u0431\u0442\u043E \u0437\u0430\u0434\u043E\u0432\u043E\u043B\u044C\u043D\u044F\u0454 \u0434\u0435\u044F\u043A\u043E\u043C\u0443 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044E \u0432\u0438\u0433\u043B\u044F\u0434\u0443 \u0434\u0435 . \u0414\u0430\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F\u043C \u0446\u0456\u043B\u043E\u0457 \u0437\u0430\u043B\u0435\u0436\u043D\u043E\u0441\u0442\u0456. \u0415\u043B\u0435\u043C\u0435\u043D\u0442 x \u0454 \u0446\u0456\u043B\u0438\u043C \u0432 R \u0442\u043E\u0434\u0456 \u0456 \u0442\u0456\u043B\u044C\u043A\u0438 \u0442\u043E\u0434\u0456, \u043A\u043E\u043B\u0438 \u0432\u0438\u043A\u043E\u043D\u0443\u0454\u0442\u044C\u0441\u044F \u043E\u0434\u043D\u0430 \u0437 \u0434\u0432\u043E\u0445 \u0435\u043A\u0432\u0456\u0432\u0430\u043B\u0435\u043D\u0442\u043D\u0438\u0445 \u0443\u043C\u043E\u0432: 1. \n* R[x] \u0454 \u0441\u043A\u0456\u043D\u0447\u0435\u043D\u043D\u043E \u043F\u043E\u0440\u043E\u0434\u0436\u0435\u043D\u0438\u043C R-\u043C\u043E\u0434\u0443\u043B\u0435\u043C ; 2. \n* \u0456\u0441\u043D\u0443\u0454 \u0442\u043E\u0447\u043D\u0438\u0439 R[x]-\u043C\u043E\u0434\u0443\u043B\u044C, \u0449\u043E \u0454 \u0441\u043A\u0456\u043D\u0447\u0435\u043D\u043D\u043E \u043F\u043E\u0440\u043E\u0434\u0436\u0435\u043D\u0438\u043C R-\u043C\u043E\u0434\u0443\u043B\u0435\u043C." , "@language" : "uk" }, { "@value" : "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 co\u00EBffici\u00EBnten in de deelring. De eigenschap 'geheel' generaliseert enerzijds algebra\u00EFsche gehele getallen, en anderzijds een algebra\u00EFsche uitbreiding van een commutatief lichaam." , "@language" : "nl" }, { "@value" : "En math\u00E9matiques, et plus particuli\u00E8rement en alg\u00E8bre commutative, les \u00E9l\u00E9ments entiers sur un anneau commutatif sont \u00E0 la fois une g\u00E9n\u00E9ralisation des entiers alg\u00E9briques (les \u00E9l\u00E9ments entiers sur l'anneau des entiers relatifs) et des \u00E9l\u00E9ments alg\u00E9briques dans une extension de corps. C'est une notion tr\u00E8s utile en th\u00E9orie alg\u00E9brique des nombres et en g\u00E9om\u00E9trie alg\u00E9brique. Son \u00E9mergence a commenc\u00E9 par l'\u00E9tude des entiers quadratiques, en particulier les entiers de Gauss." , "@language" : "fr" } ], "seeAlso": "http://dbpedia.org/resource/Integral_closure_of_an_ideal", "subject": [ "http://dbpedia.org/resource/Category:Ring_theory", "http://dbpedia.org/resource/Category:Algebraic_structures", "http://dbpedia.org/resource/Category:Commutative_algebra" ], "wikiPageID": 9478630, "wikiPageRevisionID": 1121506996, "wikiPageWikiLink": [ "http://dbpedia.org/resource/Cambridge_University_Press", "http://dbpedia.org/resource/Prime_ideal", "http://dbpedia.org/resource/Puiseux_series", "http://dbpedia.org/resource/Quadratic_integer", "http://dbpedia.org/resource/Root_of_unity", "http://dbpedia.org/resource/Minimal_prime_ideal", { "@id": "http://dbpedia.org/resource/Monic_polynomial", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, "http://dbpedia.org/resource/Noether_normalization_lemma", "http://dbpedia.org/resource/Mori\u2013Nagata_theorem", "http://dbpedia.org/resource/Determinant", "http://dbpedia.org/resource/Algebraic_closure", { "@id": "http://dbpedia.org/resource/Algebraic_extension", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, "http://dbpedia.org/resource/Algebraic_independence", "http://dbpedia.org/resource/Algebraically_closed_field", "http://dbpedia.org/resource/Resolution_of_singularities", { "@id": "http://dbpedia.org/resource/Ring_of_integers", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, "http://dbpedia.org/resource/Unit_(ring_theory)", "http://dbpedia.org/resource/Valuation_ring", "http://dbpedia.org/resource/Dedekind_domain", "http://dbpedia.org/resource/Integral_closure_of_an_ideal", "http://dbpedia.org/resource/Integral_domain", { "@id": "http://dbpedia.org/resource/Integrally_closed_domain", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, "http://dbpedia.org/resource/Category:Ring_theory", "http://dbpedia.org/resource/Complex_numbers", "http://dbpedia.org/resource/Noetherian_ring", "http://dbpedia.org/resource/Normal_scheme", "http://dbpedia.org/resource/Subring", "http://dbpedia.org/resource/Support_of_a_module", "http://dbpedia.org/resource/Closed_map", "http://dbpedia.org/resource/Eisenstein's_criterion", "http://dbpedia.org/resource/Endomorphism", "http://dbpedia.org/resource/Gaussian_integer", "http://dbpedia.org/resource/Geometry", "http://dbpedia.org/resource/Multiplicatively_closed_subset", "http://dbpedia.org/resource/Constructible_set_(topology)", "http://dbpedia.org/resource/Corollary", "http://dbpedia.org/resource/Roots_of_unity", "http://dbpedia.org/resource/Annihilator_(ring_theory)", "http://dbpedia.org/resource/Bijective", "http://dbpedia.org/resource/Commutative_algebra", { "@id": "http://dbpedia.org/resource/Commutative_ring", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, "http://dbpedia.org/resource/Ideal_(ring_theory)", "http://dbpedia.org/resource/Idealizer", "http://dbpedia.org/resource/Idempotent_(ring_theory)", "http://dbpedia.org/resource/Krull_dimension", "http://dbpedia.org/resource/Krull\u2013Akizuki_theorem", "http://dbpedia.org/resource/Maximal_ideal", "http://dbpedia.org/resource/Category:Algebraic_structures", "http://dbpedia.org/resource/Cayley\u2013Hamilton_theorem", "http://dbpedia.org/resource/Transitive_relation", "http://dbpedia.org/resource/James_Milne_(mathematician)", "http://dbpedia.org/resource/Local_ring", "http://dbpedia.org/resource/Minimal_polynomial_(field_theory)", "http://dbpedia.org/resource/Algebra_over_a_ring", "http://dbpedia.org/resource/Algebraic_element", "http://dbpedia.org/resource/Algebraic_geometry", { "@id": "http://dbpedia.org/resource/Algebraic_integer", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, "http://dbpedia.org/resource/Algebraic_number_theory", "http://dbpedia.org/resource/Analytically_unramified", "http://dbpedia.org/resource/Cyclotomic_field", "http://dbpedia.org/resource/Faithful_module", "http://dbpedia.org/resource/Field_(mathematics)", { "@id": "http://dbpedia.org/resource/Finitely_generated_module", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, "http://dbpedia.org/resource/Nicolas_Bourbaki", "http://dbpedia.org/resource/Nilpotent_element", "http://dbpedia.org/resource/Normal_extension", "http://dbpedia.org/resource/Number_theory", "http://dbpedia.org/resource/Going_up_and_going_down", "http://dbpedia.org/resource/Graded_ring", "http://dbpedia.org/resource/Isomorphism", "http://dbpedia.org/resource/Mathematical_proof", "http://dbpedia.org/resource/Projective_variety", "http://dbpedia.org/resource/Radical_of_an_ideal", "http://dbpedia.org/resource/Ring_(mathematics)", "http://dbpedia.org/resource/Group_(mathematics)", "http://dbpedia.org/resource/Group_action", "http://dbpedia.org/resource/Addison\u2013Wesley", "http://dbpedia.org/resource/Prime_number", "http://dbpedia.org/resource/Torsor_(algebraic_geometry)", "http://dbpedia.org/resource/Total_ring_of_fractions", "http://dbpedia.org/resource/Category:Commutative_algebra", { "@id": "http://dbpedia.org/resource/Henselian_ring", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, "http://dbpedia.org/resource/Homogeneous_coordinate_ring", "http://dbpedia.org/resource/Polynomial", "http://dbpedia.org/resource/Polynomial_ring", "http://dbpedia.org/resource/Field_extension", "http://dbpedia.org/resource/Field_of_fractions", "http://dbpedia.org/resource/Field_theory_(mathematics)", "http://dbpedia.org/resource/Coordinate_ring", "http://dbpedia.org/resource/Purely_inseparable", "http://dbpedia.org/resource/If_and_only_if", "http://dbpedia.org/resource/Integer", "http://dbpedia.org/resource/Michael_Atiyah", "http://dbpedia.org/resource/Miles_Reid", "http://dbpedia.org/resource/Open_set", "http://dbpedia.org/resource/Rational_number", "http://dbpedia.org/resource/Root_of_a_polynomial", "http://dbpedia.org/resource/Scheme_(mathematics)", "http://dbpedia.org/resource/University_of_Chicago_Press", "http://dbpedia.org/resource/Ian_G._Macdonald", "http://dbpedia.org/resource/Localization_of_a_ring", "http://dbpedia.org/resource/Finite_group", "http://dbpedia.org/resource/Finite_morphism", { "@id": "http://dbpedia.org/resource/Fixed-point_subring", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, "http://dbpedia.org/resource/Nakayama's_lemma", "http://dbpedia.org/resource/Splitting_of_prime_ideals_in_Galois_extensions", "http://dbpedia.org/resource/Nagata_ring", "http://dbpedia.org/resource/Topological_space", "http://dbpedia.org/resource/Sheaf_of_algebras", "http://dbpedia.org/resource/Unibranch_local_ring", "http://dbpedia.org/resource/Ring_homomorphism", "http://dbpedia.org/resource/Affine_curve", "http://dbpedia.org/resource/Krull_domain", "http://dbpedia.org/resource/Ring_of_all_algebraic_integers", "http://dbpedia.org/resource/Ring_of_sections", "http://dbpedia.org/resource/Springer-Verlag", "http://dbpedia.org/resource/Inductive_limit", "http://dbpedia.org/resource/Quotient_topology", "http://dbpedia.org/resource/Prime_avoidance", "http://dbpedia.org/resource/Injective_map", "http://dbpedia.org/resource/Alg\u00E8bre_commutative", "http://dbpedia.org/resource/Lying_over", "http://dbpedia.org/resource/Surjective_map", "http://dbpedia.org/resource/Submersion_(algebra)", "http://dbpedia.org/resource/%3C/nowiki%3E''x''%3Cnowiki%3E" ], "wikiPageExternalLink": [ "http://www.jmilne.org/math/", "https://mathoverflow.net/q/66445", "https://mathoverflow.net/q/7775", "http://people.reed.edu/~iswanson/trieste.pdf", "http://people.reed.edu/~iswanson/book/index.html", "https://archive.org/details/commutativerings00irvi" ], "sameAs": [ "http://rdf.freebase.com/ns/m.028bgtz", "http://yago-knowledge.org/resource/Integral_element", "http://www.wikidata.org/entity/Q1493740", "http://cs.dbpedia.org/resource/Celistv\u00FD_prvek", "http://de.dbpedia.org/resource/Ganzes_Element", "http://eo.dbpedia.org/resource/Entjera_elemento", "http://fi.dbpedia.org/resource/Kokonaissulkeuma", "http://fr.dbpedia.org/resource/\u00C9l\u00E9ment_entier", "http://it.dbpedia.org/resource/Estensione_intera", "http://ja.dbpedia.org/resource/\u6574\u62E1\u5927", "http://ko.dbpedia.org/resource/\uC815\uC218\uC801_\uC6D0\uC18C", "http://nl.dbpedia.org/resource/Geheel_element", "http://ru.dbpedia.org/resource/\u0426\u0435\u043B\u044B\u0439_\u044D\u043B\u0435\u043C\u0435\u043D\u0442", "http://uk.dbpedia.org/resource/\u0426\u0456\u043B\u0435_\u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F_\u043A\u0456\u043B\u044C\u0446\u044F", "http://zh.dbpedia.org/resource/\u6574\u6027", "https://global.dbpedia.org/id/VYux" ], "wikiPageUsesTemplate": [ "http://dbpedia.org/resource/Template:Citation", "http://dbpedia.org/resource/Template:Citation_needed", "http://dbpedia.org/resource/Template:Cite_book", "http://dbpedia.org/resource/Template:ISBN", "http://dbpedia.org/resource/Template:Main", "http://dbpedia.org/resource/Template:See_also", "http://dbpedia.org/resource/Template:Hartshorne_AG" ], "abstract": [ { "@value" : "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 \u2265 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." , "@language" : "en" }, { "@value" : "In algebra, un'estensione intera di un anello commutativo unitario \u00E8 un'estensione di anelli tale che ogni elemento di B \u00E8 intero su A, ovvero tale che ogni elemento di B \u00E8 radice di un polinomio monico a coefficienti in A. Rappresenta una generalizzazione del concetto di estensione algebrica di campi: se A \u00E8 un campo, le estensioni intere sono infatti le estensione algebriche (dal momento che ogni polinomio pu\u00F2 essere reso monico moltiplicando per l'inverso del coefficiente direttore)." , "@language" : "it" }, { "@value" : "\uAC00\uD658\uB300\uC218\uD559\uC5D0\uC11C \uC815\uC218\uC801 \uC6D0\uC18C(\u6574\u6578\u7684\u5143\u7D20, \uC601\uC5B4: integral element)\uB294 \uC5B4\uB5A4 \uBD80\uBD84\uD658\uC5D0 \uACC4\uC218\uB97C \uAC16\uB294 \uC77C\uACC4\uC218 \uB2E4\uD56D\uC2DD\uC758 \uADFC\uC73C\uB85C \uB098\uD0C0\uB0BC \uC218 \uC788\uB294 \uAC00\uD658\uD658 \uC6D0\uC18C\uC774\uB2E4." , "@language" : "ko" }, { "@value" : "\u6574\u6027\u662F\u4EA4\u63DB\u4EE3\u6578\u4E2D\u7684\u6982\u5FF5\uFF0C\u7528\u4E8E\u63CF\u8FF0\u5728\u6709\u7406\u6570\u57DF\u7684\u67D0\u4E9B\u6269\u57DF\u4E2D\uFF0C\u67D0\u4E9B\u5143\u7D20\u662F\u5426\u6709\u7C7B\u4F3C\u4E8E\u6574\u6570\u7684\u6027\u8D28\u3002\u5143\u7D20\u7684\u6574\u6027\uFF08\u662F\u5426\u4E3A\u6574\u5143\u7D20\uFF09\u672C\u8D28\u4E0A\u53EA\u4F9D\u8D56\u4E8E\u74B0\u7684\u6982\u5FF5\u3002\u6574\u6027\u8207\u74B0\u7684\u6574\u64F4\u5F35\u63A8\u5EE3\u4E86\u4EE3\u6578\u6578\u8207\u4EE3\u6578\u64F4\u5F35\u7684\u6982\u5FF5\u3002" , "@language" : "zh" }, { "@value" : "\u0426\u0435\u043B\u044B\u0439 \u044D\u043B\u0435\u043C\u0435\u043D\u0442 \u2014 \u044D\u043B\u0435\u043C\u0435\u043D\u0442 \u0437\u0430\u0434\u0430\u043D\u043D\u043E\u0433\u043E \u043A\u043E\u043C\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u043E\u0433\u043E \u043A\u043E\u043B\u044C\u0446\u0430 \u0441 \u0435\u0434\u0438\u043D\u0438\u0446\u0435\u0439 \u043E\u0442\u043D\u043E\u0441\u0438\u0442\u0435\u043B\u044C\u043D\u043E \u043F\u043E\u0434\u043A\u043E\u043B\u044C\u0446\u0430 , \u044F\u0432\u043B\u044F\u044E\u0449\u0438\u0439\u0441\u044F \u043A\u043E\u0440\u043D\u0435\u043C \u043F\u0440\u0438\u0432\u0435\u0434\u0451\u043D\u043D\u043E\u0433\u043E \u043C\u043D\u043E\u0433\u043E\u0447\u043B\u0435\u043D\u0430 \u0441 \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u0430\u043C\u0438 \u0432 , \u0442\u043E \u0435\u0441\u0442\u044C \u0442\u0430\u043A\u043E\u0439 , \u0434\u043B\u044F \u043A\u043E\u0442\u043E\u0440\u043E\u0433\u043E \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044E\u0442 \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u044B , \u0442\u0430\u043A\u0438\u0435 \u0447\u0442\u043E: . \u0415\u0441\u043B\u0438 \u043A\u0430\u0436\u0434\u044B\u0439 \u044D\u043B\u0435\u043C\u0435\u043D\u0442 \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u0446\u0435\u043B\u044B\u043C \u043D\u0430\u0434 , \u043A\u043E\u043B\u044C\u0446\u043E \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u0446\u0435\u043B\u044B\u043C \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043D\u0438\u0435\u043C (\u0438\u043B\u0438 \u043F\u0440\u043E\u0441\u0442\u043E \u043A\u043E\u043B\u044C\u0446\u043E\u043C, \u0446\u0435\u043B\u044B\u043C \u043D\u0430\u0434 ). \u0415\u0441\u043B\u0438 \u0438 \u2014 \u043F\u043E\u043B\u044F, \u0442\u0435\u0440\u043C\u0438\u043D\u0430\u043C \u00AB\u0446\u0435\u043B \u043D\u0430\u0434\u2026\u00BB \u0438 \u00AB\u0446\u0435\u043B\u043E\u0435 \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043D\u0438\u0435\u00BB \u0441\u043E\u043E\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044E\u0442 \u0442\u0435\u0440\u043C\u0438\u043D\u044B \u00AB\u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u043D \u043D\u0430\u0434\u2026\u00BB \u0438 \u00AB\u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u043E\u0435 \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043D\u0438\u0435\u00BB. \u0427\u0430\u0441\u0442\u043D\u044B\u0439 \u0441\u043B\u0443\u0447\u0430\u0439, \u043E\u0441\u043E\u0431\u0435\u043D\u043D\u043E \u0432\u0430\u0436\u043D\u044B\u0439 \u0432 \u0442\u0435\u043E\u0440\u0438\u0438 \u0447\u0438\u0441\u0435\u043B, \u2014 \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u044B\u0435 \u0447\u0438\u0441\u043B\u0430, \u044F\u0432\u043B\u044F\u044E\u0449\u0438\u0435\u0441\u044F \u0446\u0435\u043B\u044B\u043C\u0438 \u043D\u0430\u0434 , \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u043C\u044B\u0435 \u0446\u0435\u043B\u044B\u043C\u0438 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u0438\u043C\u0438 \u0447\u0438\u0441\u043B\u0430\u043C\u0438. \u041C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u043E \u0432\u0441\u0435\u0445 \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u043E\u0432 , \u0446\u0435\u043B\u044B\u0445 \u043D\u0430\u0434 , \u043E\u0431\u0440\u0430\u0437\u0443\u0435\u0442 \u043A\u043E\u043B\u044C\u0446\u043E; \u043E\u043D\u043E \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u0446\u0435\u043B\u044B\u043C \u0437\u0430\u043C\u044B\u043A\u0430\u043D\u0438\u0435\u043C \u0432 . \u0426\u0435\u043B\u043E\u0435 \u0437\u0430\u043C\u044B\u043A\u0430\u043D\u0438\u0435 \u0440\u0430\u0446\u0438\u043E\u043D\u0430\u043B\u044C\u043D\u044B\u0445 \u0447\u0438\u0441\u0435\u043B \u0432 \u043D\u0435\u043A\u043E\u0442\u043E\u0440\u043E\u043C \u043A\u043E\u043D\u0435\u0447\u043D\u043E\u043C \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043D\u0438\u0438 \u043F\u043E\u043B\u044F \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043A\u043E\u043B\u044C\u0446\u043E\u043C \u0446\u0435\u043B\u044B\u0445 \u043F\u043E\u043B\u044F , \u044D\u0442\u043E\u0442 \u043E\u0431\u044A\u0435\u043A\u0442 \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u0444\u0443\u043D\u0434\u0430\u043C\u0435\u043D\u0442\u0430\u043B\u044C\u043D\u044B\u043C \u0434\u043B\u044F \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0442\u0435\u043E\u0440\u0438\u0438 \u0447\u0438\u0441\u0435\u043B. \u0426\u0435\u043B\u044B\u0435 \u0447\u0438\u0441\u043B\u0430 \u2014 \u0435\u0434\u0438\u043D\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0435 \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u044B , \u044F\u0432\u043B\u044F\u044E\u0449\u0438\u0435\u0441\u044F \u0446\u0435\u043B\u044B\u043C\u0438 \u043D\u0430\u0434 (\u0447\u0442\u043E \u043C\u043E\u0436\u0435\u0442 \u0441\u043B\u0443\u0436\u0438\u0442\u044C \u043E\u0431\u044A\u044F\u0441\u043D\u0435\u043D\u0438\u0435\u043C \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u043E\u0432\u0430\u043D\u0438\u044F \u0442\u0435\u0440\u043C\u0438\u043D\u0430 \u00AB\u0446\u0435\u043B\u044B\u0439\u00BB). \u0413\u0430\u0443\u0441\u0441\u043E\u0432\u044B \u0446\u0435\u043B\u044B\u0435 \u0447\u0438\u0441\u043B\u0430, \u043A\u0430\u043A \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u044B \u043F\u043E\u043B\u044F \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u044B\u0445 \u0447\u0438\u0441\u0435\u043B, \u044F\u0432\u043B\u044F\u044E\u0442\u0441\u044F \u0446\u0435\u043B\u044B\u043C\u0438 \u043D\u0430\u0434 . \u0426\u0435\u043B\u043E\u0435 \u0437\u0430\u043C\u044B\u043A\u0430\u043D\u0438\u0435 \u0432 \u043A\u0440\u0443\u0433\u043E\u0432\u043E\u043C \u043F\u043E\u043B\u0435 \u2014 \u044D\u0442\u043E . \u0415\u0441\u043B\u0438 \u2014 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u043E\u0435 \u0437\u0430\u043C\u044B\u043A\u0430\u043D\u0438\u0435 \u043F\u043E\u043B\u044F , \u0442\u043E \u0446\u0435\u043B\u043E \u043D\u0430\u0434 . \u0415\u0441\u043B\u0438 \u043A\u043E\u043D\u0435\u0447\u043D\u0430\u044F \u0433\u0440\u0443\u043F\u043F\u0430 \u0434\u0435\u0439\u0441\u0442\u0432\u0443\u0435\u0442 \u043D\u0430 \u043A\u043E\u043B\u044C\u0446\u0435 \u0433\u043E\u043C\u043E\u043C\u043E\u0440\u0444\u0438\u0437\u043C\u0430\u043C\u0438 \u043A\u043E\u043B\u0435\u0446, \u0442\u043E \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u0446\u0435\u043B\u044B\u043C \u043D\u0430\u0434 \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u043E\u043C \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u043E\u0432, \u044F\u0432\u043B\u044F\u044E\u0449\u0438\u0445\u0441\u044F \u043D\u0435\u043F\u043E\u0434\u0432\u0438\u0436\u043D\u044B\u043C\u0438 \u0442\u043E\u0447\u043A\u0430\u043C\u0438 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044F \u0433\u0440\u0443\u043F\u043F\u044B." , "@language" : "ru" }, { "@value" : "Im mathematischen Teilgebiet der kommutativen Algebra ist der Begriff eines ganzen Elementes in einer Ringerweiterung eine Verallgemeinerung des Begriffes eines algebraischen Elementes in einer K\u00F6rpererweiterung." , "@language" : "de" }, { "@value" : "\u53EF\u63DB\u74B0\u8AD6\u306B\u304A\u3044\u3066\u3001\u53EF\u63DB\u74B0 B \u3068\u305D\u306E\u90E8\u5206\u74B0 A \u306B\u3064\u3044\u3066\u3001B \u306E\u5143 b \u304C A \u4FC2\u6570\u306E\u30E2\u30CB\u30C3\u30AF\u591A\u9805\u5F0F\u306E\u6839\u3067\u3042\u308B\u3068\u304D\u3001b \u306F A \u4E0A\u6574\u3067\u3042\u308B\uFF08integral over A\uFF09\u3068\u3044\u3046\u3002B \u306E\u3059\u3079\u3066\u306E\u5143\u304C A \u4E0A\u6574\u3067\u3042\u308B\u3068\u304D\u3001B \u306F A \u4E0A\u6574\u3067\u3042\u308B\u3001\u307E\u305F\u306F\u3001B \u306F A \u306E\u6574\u62E1\u5927\uFF08integral extension\uFF09\u3067\u3042\u308B\u3068\u3044\u3046\u3002\u672C\u8A18\u4E8B\u306B\u304A\u3044\u3066\u3001\u74B0\u3068\u306F\u5358\u4F4D\u5143\u3092\u3082\u3064\u53EF\u63DB\u74B0\u306E\u3053\u3068\u3068\u3059\u308B\u3002" , "@language" : "ja" }, { "@value" : "En nombroteorio, entjera elemento estas \u011Deneraligo de la koncepto de ga\u016Dsaj entjeroj en la kampo de kompleksaj nombroj." , "@language" : "eo" }, { "@value" : "Celistv\u00FD prvek je pojem z oboru . Je-li d\u00E1n komutativn\u00ED okruh a jeho , pak je prvek celistv\u00FD nad , je-li ko\u0159enem n\u011Bjak\u00E9ho monick\u00E9ho polynomu s koeficienty z , tedy pokud existuj\u00ED a takov\u00E1, \u017Ee . Definice celistv\u00E9ho prvku se li\u0161\u00ED od definice algebraick\u00E9ho prvku pouze v p\u0159idan\u00E9m po\u017Eadavku, aby byl polynom monick\u00FD, z \u010Deho\u017E plyne, \u017Ee ka\u017Ed\u00FD celistv\u00FD prvek je algebraick\u00FD. Mno\u017Eina prvk\u016F , kter\u00E9 jsou celistv\u00E9 nad , se naz\u00FDv\u00E1 celistv\u00FD uz\u00E1v\u011Br v ." , "@language" : "cs" }, { "@value" : "\u0426\u0456\u043B\u0435 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u043A\u0456\u043B\u044C\u0446\u044F \u2014 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F B \u043A\u043E\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u043E\u0433\u043E \u043A\u0456\u043B\u044C\u0446\u044F R \u0437 \u043E\u0434\u0438\u043D\u0438\u0446\u0435\u044E \u0442\u0430\u043A\u0435, \u0449\u043E \u0431\u0443\u0434\u044C-\u044F\u043A\u0438\u0439 \u0435\u043B\u0435\u043C\u0435\u043D\u0442 \u0454 \u0446\u0456\u043B\u0438\u043C \u043D\u0430\u0434 R, \u0442\u043E\u0431\u0442\u043E \u0437\u0430\u0434\u043E\u0432\u043E\u043B\u044C\u043D\u044F\u0454 \u0434\u0435\u044F\u043A\u043E\u043C\u0443 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044E \u0432\u0438\u0433\u043B\u044F\u0434\u0443 \u0434\u0435 . \u0414\u0430\u043D\u0435 \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F\u043C \u0446\u0456\u043B\u043E\u0457 \u0437\u0430\u043B\u0435\u0436\u043D\u043E\u0441\u0442\u0456. \u0415\u043B\u0435\u043C\u0435\u043D\u0442 x \u0454 \u0446\u0456\u043B\u0438\u043C \u0432 R \u0442\u043E\u0434\u0456 \u0456 \u0442\u0456\u043B\u044C\u043A\u0438 \u0442\u043E\u0434\u0456, \u043A\u043E\u043B\u0438 \u0432\u0438\u043A\u043E\u043D\u0443\u0454\u0442\u044C\u0441\u044F \u043E\u0434\u043D\u0430 \u0437 \u0434\u0432\u043E\u0445 \u0435\u043A\u0432\u0456\u0432\u0430\u043B\u0435\u043D\u0442\u043D\u0438\u0445 \u0443\u043C\u043E\u0432: 1. \n* R[x] \u0454 \u0441\u043A\u0456\u043D\u0447\u0435\u043D\u043D\u043E \u043F\u043E\u0440\u043E\u0434\u0436\u0435\u043D\u0438\u043C R-\u043C\u043E\u0434\u0443\u043B\u0435\u043C ; 2. \n* \u0456\u0441\u043D\u0443\u0454 \u0442\u043E\u0447\u043D\u0438\u0439 R[x]-\u043C\u043E\u0434\u0443\u043B\u044C, \u0449\u043E \u0454 \u0441\u043A\u0456\u043D\u0447\u0435\u043D\u043D\u043E \u043F\u043E\u0440\u043E\u0434\u0436\u0435\u043D\u0438\u043C R-\u043C\u043E\u0434\u0443\u043B\u0435\u043C." , "@language" : "uk" }, { "@value" : "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 co\u00EBffici\u00EBnten in de deelring. De eigenschap 'geheel' generaliseert enerzijds algebra\u00EFsche gehele getallen, en anderzijds een algebra\u00EFsche uitbreiding van een commutatief lichaam." , "@language" : "nl" }, { "@value" : "En math\u00E9matiques, et plus particuli\u00E8rement en alg\u00E8bre commutative, les \u00E9l\u00E9ments entiers sur un anneau commutatif sont \u00E0 la fois une g\u00E9n\u00E9ralisation des entiers alg\u00E9briques (les \u00E9l\u00E9ments entiers sur l'anneau des entiers relatifs) et des \u00E9l\u00E9ments alg\u00E9briques dans une extension de corps. C'est une notion tr\u00E8s utile en th\u00E9orie alg\u00E9brique des nombres et en g\u00E9om\u00E9trie alg\u00E9brique. Son \u00E9mergence a commenc\u00E9 par l'\u00E9tude des entiers quadratiques, en particulier les entiers de Gauss." , "@language" : "fr" } ], "wasDerivedFrom": "http://en.wikipedia.org/wiki/Integral_element?oldid=1121506996&ns=0", "wikiPageLength": { "@value" : "32308" , "@type" : "http://www.w3.org/2001/XMLSchema#nonNegativeInteger" }, "isPrimaryTopicOf": { "@id": "http://en.wikipedia.org/wiki/Integral_element", "primaryTopic": "http://dbpedia.org/resource/Integral_element" } }, { "@id": "http://dbpedia.org/resource/Almost_ring", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integral_(disambiguation)", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageDisambiguates": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integrally_closed", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Conductor_(ring_theory)", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Order_(ring_theory)", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Absolute_integral_closure", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integral_closure", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integral_extension", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integral_ring_extension", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integrality", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Unramified_morphism", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Universal_homeomorphism", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integral_(ring_theory)", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integral_dependence", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integral_extension_of_a_ring", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Integral_over", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" }, { "@id": "http://dbpedia.org/resource/Complete_integral_closure", "wikiPageWikiLink": "http://dbpedia.org/resource/Integral_element", "wikiPageRedirects": "http://dbpedia.org/resource/Integral_element" } ] }