Integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c. Some sources, notably Lang, use the term entire ring for integral domain. Some specific kinds of integral domains are given with the following chain of class inclusions:
Absolute value (algebra)AlgebraAlgebraic equationAlgebraic geometryAlgebraic number fieldAlgebraic varietyAlmost ringAnalytically normal ringAnnihilator (ring theory)Anton SushkevichArtinian ringArtin–Rees lemmaAscending chain condition on principal idealsAssociateAssociate (ring theory)Associate elementsAssociated elementAssociated elementsAssociated graded ringAssociatednessAtomic domainBoolean ringBézout's identityBézout domainCancellation propertyCancellation ringCategory of ringsCellular algebraCharacteristic (algebra)Chinese remainder theoremCluster algebraCohen–Macaulay ringCommutative domainCommutative ringCompletion of a ringComplex-base systemConnected ringDedekind domainDedekind–Hasse normDegree of a polynomial
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Integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c. Some sources, notably Lang, use the term entire ring for integral domain. Some specific kinds of integral domains are given with the following chain of class inclusions:
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(Artikel ini bukan mengenai ra ...... omutatif ⊃ ⊃ ⊃ ⊃ ⊃ ⊃ ⊃ medan ⊃
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Dziedzina całkowitości – nieze ...... nga, jest pierścień całkowity.
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En àlgebra abstracta, un anell ...... egritat o domini d'integritat.
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Ett integritetsområde är inom ...... integritetsområde är en kropp.
@sv
In algebra, un dominio d'integ ...... o banale con un solo elemento.
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In de commutatieve algebra, ee ...... commutatieve ringen ⊂ ringen.
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In mathematics, specifically a ...... ⊃ algebraically closed fields
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Integreca ringo aŭ integreca d ...... kaj sen nuldivizoroj, do aŭ .
@eo
Obor integrity je komutativní ...... sou netriviální dělitelé nuly.
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Um domínio de integridade (ou ...... não existem divisores de zero)
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(Artikel ini bukan mengenai ra ...... erikan dengan rantai berikut :
@in
Dziedzina całkowitości – nieze ...... cią jest tzw. prawo skracania.
@pl
En àlgebra abstracta, un anell ...... res, però exclou altres casos.
@ca
Ett integritetsområde är inom ...... integritetsområde är en kropp.
@sv
In algebra, un dominio d'integ ...... o banale con un solo elemento.
@it
In de commutatieve algebra, ee ...... e keten van deelverzamelingen:
@nl
In mathematics, specifically a ...... ing chain of class inclusions:
@en
Integreca ringo aŭ integreca d ...... kaj sen nuldivizoroj, do aŭ .
@eo
Obor integrity je komutativní ...... sou netriviální dělitelé nuly.
@cs
Um domínio de integridade (ou ...... não existem divisores de zero)
@pt
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Anell íntegre
@ca
Anneau intègre
@fr
Dominio d'integrità
@it
Dominio de integridad
@es
Domínio de integridade
@pt
Dziedzina całkowitości
@pl
Integral domain
@en
Integreca ringo
@eo
Integriteitsdomein
@nl
Integritetsområde
@sv