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rdf:type: yago:Content105809192 yago:Abstraction100002137 yago:WikicatPrimeIdeals yago:Ideal105923696 yago:Idea105833840 yago:WikicatIdeals yago:PsychologicalFeature100023100 yago:Cognition100023271
rdfs:label: Maximální ideál (teorie okruhů) Максимальний ідеал Ideale massimale Ideał maksymalny Maximales Ideal 極大イデアル 극대 아이디얼 Maximal ideal Максимальный идеал Idéal maximal Maksimuma idealo Ideal maximal
rdfs:comment: Максимальным идеалом коммутативного кольца называется всякий собственный идеал кольца, не содержащийся ни в каком другом собственном идеале. En ringo-teorio, maksimuma idealo estas idealo de ringo, kiu estas maksimuma inter la propraj (t.e. kiuj ne koincidas kun la tuta ringo) idealoj. In matematica, in particolare nella teoria degli anelli, un ideale massimale è un ideale che risulta essere un elemento massimale (rispetto all'inclusione insiemistica) dell'insieme degli ideali propri di un anello, ovvero tale che non sia contenuto propriamente in nessun altro ideale proprio dell'anello. Un ideal maximal és un concepte matemàtic provinent de la teoria d'anells que és usat en diversos camps de l'àlgebra. Com el mateix nom indica, es tracta d'un ideal que és (respecte de la inclusió) d'entre tots els ideals propis d'un anell. En altres paraules, si I és un ideal maximal de l'anell A, aleshores tindrem que tot ideal J que contingui I dins seu haurà de ser igual a I o bé haurà de ser J = A. Per tant, estem dient que no hi haurà cap ideal J propi que faci I ⊊ J ⊊ A i sigui «més gran» que I. 環 R の極大左イデアル（きょくだいひだりいである、英: maximal left ideal）とは、R 以外の左イデアルの中で（集合の包含関係に関して）極大なもののことである。すなわち、左イデアル I を真に含む左イデアルが R しかないときに I を R の極大左イデアルという。極大右イデアルおよび極大両側イデアルも同様に定義される。これらのイデアルは（環が 0 でなく単位元をもつとき）ツォルンの補題によって存在が保証される。可換環においては、左・右・両側の区別はない。唯一の極大左イデアルをもつ環は局所環と呼ばれる。 In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. 환론에서 극대 아이디얼(極大ideal, 영어: maximal ideal)은 환 전체가 아닌 아이디얼들의 극대 원소이다. Максимальним ідеалом кільця в абстрактній алгебрі називається всякий власний ідеал кільця, що не міститься в жодному іншому власному ідеалі. Maximales Ideal ist ein Begriff aus der Algebra. Maximální ideál je v algebře takový ideál, který je v daném okruhu mezi vlastními ideály maximální vzhledem k inkluzi. Jinými slovy, I je maximální ideál okruhu R, pokud I je vlastní ideál a pro každý ideál J⊇I platí, že buď J = I, nebo J=R. Un idéal maximal est un concept associé à la théorie des anneaux en mathématiques et plus précisément en algèbre. Un idéal d'un anneau commutatif est dit maximal lorsqu’il est contenu dans exactement deux idéaux, lui-même et l'anneau tout entier. L'existence d'idéaux maximaux est assurée par le théorème de Krull. Cette définition permet de généraliser la notion d’élément irréductible à des anneaux différents de celui des entiers relatifs. Certains de ces anneaux ont un rôle important en théorie algébrique des nombres et en géométrie algébrique. Ideał maksymalny – ideał, który jest maksymalny (względem zawierania zbiorów) wśród wszystkich ideałów właściwych danego pierścienia; innymi słowy jest to taki ideał właściwy, który nie zawiera się w żadnym innym ideale danego pierścienia.
dcterms:subject: dbc:Ring_theory dbc:Ideals_(ring_theory) dbc:Prime_ideals
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dbo:abstract: 환론에서 극대 아이디얼(極大ideal, 영어: maximal ideal)은 환 전체가 아닌 아이디얼들의 극대 원소이다. Un ideal maximal és un concepte matemàtic provinent de la teoria d'anells que és usat en diversos camps de l'àlgebra. Com el mateix nom indica, es tracta d'un ideal que és (respecte de la inclusió) d'entre tots els ideals propis d'un anell. En altres paraules, si I és un ideal maximal de l'anell A, aleshores tindrem que tot ideal J que contingui I dins seu haurà de ser igual a I o bé haurà de ser J = A. Per tant, estem dient que no hi haurà cap ideal J propi que faci I ⊊ J ⊊ A i sigui «més gran» que I. En el cas d'anells commutatius (i amb unitat, és a dir, element neutre pel producte), podem formular una altra definició equivalent que és similar a la dels ideals primers: Un ideal I de l'anell A és maximal si i només si l'anell quocient A / I és un cos. 環 R の極大左イデアル（きょくだいひだりいである、英: maximal left ideal）とは、R 以外の左イデアルの中で（集合の包含関係に関して）極大なもののことである。すなわち、左イデアル I を真に含む左イデアルが R しかないときに I を R の極大左イデアルという。極大右イデアルおよび極大両側イデアルも同様に定義される。これらのイデアルは（環が 0 でなく単位元をもつとき）ツォルンの補題によって存在が保証される。可換環においては、左・右・両側の区別はない。唯一の極大左イデアルをもつ環は局所環と呼ばれる。 Maximální ideál je v algebře takový ideál, který je v daném okruhu mezi vlastními ideály maximální vzhledem k inkluzi. Jinými slovy, I je maximální ideál okruhu R, pokud I je vlastní ideál a pro každý ideál J⊇I platí, že buď J = I, nebo J=R. Максимальным идеалом коммутативного кольца называется всякий собственный идеал кольца, не содержащийся ни в каком другом собственном идеале. Максимальним ідеалом кільця в абстрактній алгебрі називається всякий власний ідеал кільця, що не міститься в жодному іншому власному ідеалі. En ringo-teorio, maksimuma idealo estas idealo de ringo, kiu estas maksimuma inter la propraj (t.e. kiuj ne koincidas kun la tuta ringo) idealoj. Maximales Ideal ist ein Begriff aus der Algebra. Un idéal maximal est un concept associé à la théorie des anneaux en mathématiques et plus précisément en algèbre. Un idéal d'un anneau commutatif est dit maximal lorsqu’il est contenu dans exactement deux idéaux, lui-même et l'anneau tout entier. L'existence d'idéaux maximaux est assurée par le théorème de Krull. Cette définition permet de généraliser la notion d’élément irréductible à des anneaux différents de celui des entiers relatifs. Certains de ces anneaux ont un rôle important en théorie algébrique des nombres et en géométrie algébrique. Ideał maksymalny – ideał, który jest maksymalny (względem zawierania zbiorów) wśród wszystkich ideałów właściwych danego pierścienia; innymi słowy jest to taki ideał właściwy, który nie zawiera się w żadnym innym ideale danego pierścienia. Istotność ideałów maksymalnych wynika zasadniczo z faktu, że pierścienie ilorazowe ideałów maksymalnych są , co w przypadku pierścieni przemiennych z jedynką oznacza, że są one także ciałami. Dla pierścieni nieprzemiennych definiuje się ideały maksymalne lewostronny i prawostronny jako maksymalne wśród częściowo uporządkowanego zbioru ideałów odpowiednio lewostronnych bądź prawostronnych. W tym przypadku iloraz jest tylko nad danym pierścieniem. Jeżeli pierścień ma dokładnie jeden prawostronny ideał maksymalny, to nazywa się go pierścieniem lokalnym; wówczas ideał ten jest równocześnie dokładnie jednym lewostronnym ideałem maksymalnym tego pierścienia, co oznacza, że jest on jego (obustronnym) ideałem maksymalnym – w istocie jest to radykał Jacobsona danego pierścienia. In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R). It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one sided ideals: for example, in the ring of 2 by 2 square matrices over a field, the zero ideal is a maximal two-sided ideal, but there are many maximal right ideals. In matematica, in particolare nella teoria degli anelli, un ideale massimale è un ideale che risulta essere un elemento massimale (rispetto all'inclusione insiemistica) dell'insieme degli ideali propri di un anello, ovvero tale che non sia contenuto propriamente in nessun altro ideale proprio dell'anello. Gli ideali massimali sono pertanto caratterizzati dalla proprietà di essere contenuti solamente in due ideali: l'intero anello e l'ideale massimale stesso. In un diagramma di Hasse questa proprietà è espressa dal fatto che gli ideali massimali sono sempre collegati direttamente al punto che rappresenta l'intero anello.
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Subject Item: dbr:Chinese_remainder_theorem
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Cohen_ring
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Cohen–Macaulay_ring
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Hensel's_lemma
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Henselian_ring
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Zorn's_lemma
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Module_homomorphism
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Differentiable_manifold
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Artinian_ring
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Axiom_of_choice
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Boolean_algebra_(structure)
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Boolean_ring
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Bézout_domain
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Polynomial_ring
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Splitting_field
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Field_extension
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Grothendieck's_connectedness_theorem
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:I-adic_topology
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Scheme_(mathematics)
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Semi-local_ring
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:V-ring_(ring_theory)
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Rigid_analytic_space
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Zariski_topology
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Nakayama's_lemma
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Splitting_of_prime_ideals_in_Galois_extensions
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Symbolic_power_of_an_ideal
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Uniform_algebra
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Maximal_left_ideal
dbo:wikiPageWikiLink: dbr:Maximal_ideal
dbo:wikiPageRedirects: dbr:Maximal_ideal

Subject Item: dbr:Maximal_right_ideal
dbo:wikiPageWikiLink: dbr:Maximal_ideal
dbo:wikiPageRedirects: dbr:Maximal_ideal

Subject Item: dbr:Residue_field
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Restricted_power_series
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:P-adically_closed_field
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Smooth_algebra
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Simple_module
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Tate_topology
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:T1_space
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Ring_homomorphism
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: dbr:Ring_of_mixed_characteristic
dbo:wikiPageWikiLink: dbr:Maximal_ideal

Subject Item: wikipedia-en:Maximal_ideal
foaf:primaryTopic: dbr:Maximal_ideal