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For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals. Jacobson rings were introduced independently by Wolfgang Krull , who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by Oscar Goldman, who named them Hilbert rings after David Hilbert because of their relation to Hilbert's Nullstellensatz." /> <meta property="og:site_name" content="DBpedia" />  </head> <body about="http://dbpedia.org/resource/Jacobson_ring">  <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"> </button> <div class="collapse navbar-collapse" id="dbp-navbar"> <ul class="navbar-nav me-auto 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href="http://dbpedia.org/resource/Jacobson_ring">Jacobson ring</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://dbpedia.org/ontology/AnatomicalStructure">anatomical structure</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 algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals. Jacobson rings were introduced independently by Wolfgang Krull , who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by Oscar Goldman, who named them Hilbert rings after David Hilbert because of their relation to Hilbert's Nullstellensatz. </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>In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals. Jacobson rings were introduced independently by Wolfgang Krull , who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by Oscar Goldman, who named them Hilbert rings after David Hilbert because of their relation to Hilbert's Nullstellensatz. (en)</li> <li style="display:none;">Un anneau de Jacobson est un anneau commutatif dont tout idéal premier est intersection d'idéaux maximaux. Comme tout idéal radiciel est intersection des idéaux premiers qui le contiennent, un anneau de Jacobson est tel que tout idéal radiciel soit intersection d'idéaux maximaux. * Un anneau local artinien est de Jacobson car son idéal maximal est le seul idéal premier. Ce sont d'ailleurs les seuls anneaux locaux de Jacobson. * Une algèbre de type fini sur un corps est un anneau de Jacobson. Plus généralement, une algèbre de type fini sur un anneau de Jacobson est de Jacobson. * Un anneau de Dedekind ayant un nombre infini d'idéaux maximaux (par exemple l'anneau ℤ des entiers relatifs ou même un anneau d'entiers algébriques) est de Jacobson. * Si A est un anneau de Jacobson et si B est de type fini sur A alors, pour tout idéal maximal m de B, l'image réciproque p de m par le morphisme A → B est un idéal maximal de A.De plus, par le Nullstellensatz, le morphisme de corps A/p → B/m est une extension finie.Une conséquence est que les idéaux maximaux de sont nécessairement à corps résiduel fini. Certaines de ces propriétés peuvent se démontrer à l'aide du lemme de normalisation de Noether. * D'un point de vue géométrique, A est de Jacobson si et seulement si dans toute partie fermée non vide Z(I) du spectre de A, l'ensemble des points fermés est dense. Il en résulte qu'un élément vu comme fonction régulière sur Spec(A) s'annule aux points fermés de Z(I) si et seulement si appartient au radical de I. Ainsi les anneaux de Jacobson forment la classe d'anneaux parfaite pour fournir une version générale du Nullstellensatz. (fr)</li> <li style="display:none;">In de ringtheorie, een deelgebied van de abstracte algebra, zegt men dat een commutatieve ring met identiteit een Hilbert-ring of een Jacobson-ring is, als iedere priemideaal van de ring een doorsnede van is. In een commutatieve is elke radicale ideaal een doorsnede van priemidealen, en dus is een gelijkwaardige criterium voor een ring om een Hilbert-ring te zijn dat elke een doorsnede is van maximale idealen. De beroemde Nullstellensatz van Hilbert uit de algebraïsche meetkunde vertaalt naar de stelling dat de veeltermring in een eindig aantal variabelen over een veld een Hilbert-ring is. Een algemene vorm van Hilberts Nullstellensatz stelt dat als R een Jacobson-ring is, dat dan ook een eindig gegenereerde R-algebra S en Jacobson-ring is. Bovendien is de pullback van enige maximale ideaal J van S een maximaal ideaal I van R, en is S/J een eindige uitbreiding van het veld R/I. (nl)</li> <li style="display:none;">代数学において、ヒルベルト環 (Hilbert ring) あるいはジャコブソン環 (Jacobson ring) はすべての素イデアルが原始イデアルの共通部分であるような環である。可換環に対しては原始イデアルは極大イデアルと同じなのでこの場合ジャコブソン環はすべての素イデアルが極大イデアルの共通部分であるような環である。ジャコブソン環は Krull とによって独立に導入された。Krull はジャコブソン根基との関連からにちなんで名づけ、Goldman はヒルベルトの零点定理との関連から David Hilbert にちなんで名づけた。 (ja)</li> <li style="display:none;">У абстрактній алгебрі довільне кільце називається кільцем Джекобсона (іноді також кільцем Гільберта) якщо кожен його простий ідеал є рівним перетину примітивних ідеалів (тобто ідеалів, що є ануляторами простих модулів). Для комутативних кілець примітивні ідеали це те ж саме, що і максимальні і тому комутативне кільце з одиницею називається кільцем Джекобсона, якщо будь-який простий ідеал цього кільця є перетином максимальних ідеалів, що його містять. Інакше кажучи будь-яке цілісне фактор-кільце має нульовий радикал Джекобсона. 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resource="http://dbpedia.org/resource/Goldman_ideal" href="http://dbpedia.org/resource/Goldman_ideal">dbr:Goldman_ideal</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/authorlink">dbp:authorlink</a> </td><td class="col-10 text-break"><ul> <li>Oscar Goldman (en)</li> <li>Wolfgang Krull (en)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/book">dbp:book</a> </td><td class="col-10 text-break"><ul> <li>4 (xsd:integer)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/first">dbp:first</a> </td><td class="col-10 text-break"><ul> <li>Oscar (en)</li> <li>Wolfgang (en)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/last">dbp:last</a> </td><td class="col-10 text-break"><ul> <li>Goldman (en)</li> <li>Krull (en)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/pages">dbp:pages</a> </td><td class="col-10 text-break"><ul> <li>section 10 (en)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/wikiPageUsesTemplate">dbp:wikiPageUsesTemplate</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Citation" href="http://dbpedia.org/resource/Template:Citation">dbt:Citation</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Cite_book" href="http://dbpedia.org/resource/Template:Cite_book">dbt:Cite_book</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Distinguish" href="http://dbpedia.org/resource/Template:Distinguish">dbt:Distinguish</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Harv" href="http://dbpedia.org/resource/Template:Harv">dbt:Harv</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Reflist" href="http://dbpedia.org/resource/Template:Reflist">dbt:Reflist</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Harvs" href="http://dbpedia.org/resource/Template:Harvs">dbt:Harvs</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:EGA" href="http://dbpedia.org/resource/Template:EGA">dbt:EGA</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Eom" href="http://dbpedia.org/resource/Template:Eom">dbt:Eom</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/year">dbp:year</a> </td><td class="col-10 text-break"><ul> <li>1951 (xsd:integer)</li> <li>1952 (xsd:integer)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://purl.org/dc/terms/subject">dcterms:subject</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="dcterms:subject" resource="http://dbpedia.org/resource/Category:Ring_theory" prefix="dcterms: http://purl.org/dc/terms/" href="http://dbpedia.org/resource/Category:Ring_theory">dbc:Ring_theory</a></li> <li><a class="uri" rel="dcterms:subject" resource="http://dbpedia.org/resource/Category:Commutative_algebra" prefix="dcterms: http://purl.org/dc/terms/" href="http://dbpedia.org/resource/Category:Commutative_algebra">dbc:Commutative_algebra</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://purl.org/linguistics/gold/hypernym">gold:hypernym</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="gold:hypernym" resource="http://dbpedia.org/resource/Ring" prefix="gold: http://purl.org/linguistics/gold/" href="http://dbpedia.org/resource/Ring">dbr:Ring</a></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/1999/02/22-rdf-syntax-ns#type">rdf:type</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="rdf:type" resource="http://www.w3.org/2002/07/owl#Thing" href="http://www.w3.org/2002/07/owl#Thing">owl:Thing</a></li> <li><a class="uri" rel="rdf:type" resource="http://dbpedia.org/ontology/AnatomicalStructure" href="http://dbpedia.org/ontology/AnatomicalStructure">dbo:AnatomicalStructure</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment">rdfs:comment</a> </td><td class="col-10 text-break"><ul> <li>In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals. Jacobson rings were introduced independently by Wolfgang Krull , who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by Oscar Goldman, who named them Hilbert rings after David Hilbert because of their relation to Hilbert's Nullstellensatz. (en)</li> <li style="display:none;">代数学において、ヒルベルト環 (Hilbert ring) あるいはジャコブソン環 (Jacobson ring) はすべての素イデアルが原始イデアルの共通部分であるような環である。可換環に対しては原始イデアルは極大イデアルと同じなのでこの場合ジャコブソン環はすべての素イデアルが極大イデアルの共通部分であるような環である。ジャコブソン環は Krull とによって独立に導入された。Krull はジャコブソン根基との関連からにちなんで名づけ、Goldman はヒルベルトの零点定理との関連から David Hilbert にちなんで名づけた。 (ja)</li> <li style="display:none;">У абстрактній алгебрі довільне кільце називається кільцем Джекобсона (іноді також кільцем Гільберта) якщо кожен його простий ідеал є рівним перетину примітивних ідеалів (тобто ідеалів, що є ануляторами простих модулів). Для комутативних кілець примітивні ідеали це те ж саме, що і максимальні і тому комутативне кільце з одиницею називається кільцем Джекобсона, якщо будь-який простий ідеал цього кільця є перетином максимальних ідеалів, що його містять. Інакше кажучи будь-яке цілісне фактор-кільце має нульовий радикал Джекобсона. (uk)</li> <li style="display:none;">Un anneau de Jacobson est un anneau commutatif dont tout idéal premier est intersection d'idéaux maximaux. Comme tout idéal radiciel est intersection des idéaux premiers qui le contiennent, un anneau de Jacobson est tel que tout idéal radiciel soit intersection d'idéaux maximaux. Certaines de ces propriétés peuvent se démontrer à l'aide du lemme de normalisation de Noether. (fr)</li> <li style="display:none;">In de ringtheorie, een deelgebied van de abstracte algebra, zegt men dat een commutatieve ring met identiteit een Hilbert-ring of een Jacobson-ring is, als iedere priemideaal van de ring een doorsnede van is. In een commutatieve is elke radicale ideaal een doorsnede van priemidealen, en dus is een gelijkwaardige criterium voor een ring om een Hilbert-ring te zijn dat elke een doorsnede is van maximale idealen. (nl)</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;">Anneau de Jacobson (fr)</li> <li>Jacobson ring (en)</li> <li style="display:none;">ジャコブソン環 (ja)</li> <li style="display:none;">Jacobson-ring (nl)</li> <li style="display:none;">Кільце Джекобсона (uk)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#differentFrom">owl:differentFrom</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="owl:differentFrom" resource="http://dbpedia.org/resource/Jacobson_semisimple_ring" href="http://dbpedia.org/resource/Jacobson_semisimple_ring">dbr:Jacobson_semisimple_ring</a></li> </ul></td></tr><tr class="even"><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" 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href="http://dbpedia.org/resource/Jacobson_space">dbr:Jacobson_space</a></li> </ul></td></tr><tr class="odd"><td class="col-2">is <a class="uri" href="http://dbpedia.org/ontology/wikiPageWikiLink">dbo:wikiPageWikiLink</a> of</td><td class="col-10 text-break"><ul> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Witt_group" href="http://dbpedia.org/resource/Witt_group">dbr:Witt_group</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Nilradical_of_a_ring" href="http://dbpedia.org/resource/Nilradical_of_a_ring">dbr:Nilradical_of_a_ring</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Glossary_of_commutative_algebra" href="http://dbpedia.org/resource/Glossary_of_commutative_algebra">dbr:Glossary_of_commutative_algebra</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Glossary_of_ring_theory" 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href="http://dbpedia.org/resource/Hilbert_ring">dbr:Hilbert_ring</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Jacobson_scheme" href="http://dbpedia.org/resource/Jacobson_scheme">dbr:Jacobson_scheme</a></li> <li><a class="uri" rev="dbo:wikiPageWikiLink" resource="http://dbpedia.org/resource/Jacobson_space" href="http://dbpedia.org/resource/Jacobson_space">dbr:Jacobson_space</a></li> </ul></td></tr><tr class="even"><td class="col-2">is <a class="uri" href="http://xmlns.com/foaf/0.1/primaryTopic">foaf:primaryTopic</a> of</td><td class="col-10 text-break"><ul> <li><a class="uri" rev="foaf:primaryTopic" resource="http://en.wikipedia.org/wiki/Jacobson_ring" href="http://en.wikipedia.org/wiki/Jacobson_ring">wikipedia-en:Jacobson_ring</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 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