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• Integral_domain abstract "In mathematics, and specifically in 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 the cancellation property holds for multiplication by a nonzero element a, that is, if a ≠ 0, an equality ab = ac implies b = c.\"Integral domain\" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term \"integral domain\" for the commutative case and using \"domain\" for the general case including noncommutative rings.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: commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields".
• Integral_domain wikiPageID "15462".
• Integral_domain wikiPageLength "14604".
• Integral_domain wikiPageOutDegree "97".
• Integral_domain wikiPageRevisionID "686171515".
• Integral_domain wikiPageWikiLink 1_(number).
• Integral_domain wikiPageWikiLink A_K_Peters,_Ltd..
• Integral_domain wikiPageWikiLink Abstract_algebra.
• Integral_domain wikiPageWikiLink Affine_variety.
• Integral_domain wikiPageWikiLink Algebraic_geometry.
• Integral_domain wikiPageWikiLink Algebraic_variety.
• Integral_domain wikiPageWikiLink Analytic_function.
• Integral_domain wikiPageWikiLink Artinian_ring.
• Integral_domain wikiPageWikiLink Cambridge_University_Press.
• Integral_domain wikiPageWikiLink Cancellation_property.
• Integral_domain wikiPageWikiLink Category:Commutative_algebra.
• Integral_domain wikiPageWikiLink Category:Ring_theory.
• Integral_domain wikiPageWikiLink Characteristic_(algebra).
• Integral_domain wikiPageWikiLink Commutative_ring.
• Integral_domain wikiPageWikiLink Complex_number.
• Integral_domain wikiPageWikiLink Composite_number.
• Integral_domain wikiPageWikiLink Connectedness.
• Integral_domain wikiPageWikiLink Continuous_function.
• Integral_domain wikiPageWikiLink Dedekind–Hasse_norm.
• Integral_domain wikiPageWikiLink Direct_limit.
• Integral_domain wikiPageWikiLink Divisibility_(ring_theory).
• Integral_domain wikiPageWikiLink Domain_(ring_theory).
• Integral_domain wikiPageWikiLink Field_(mathematics).
• Integral_domain wikiPageWikiLink Field_of_fractions.
• Integral_domain wikiPageWikiLink Finite_field.
• Integral_domain wikiPageWikiLink Frobenius_endomorphism.
• Integral_domain wikiPageWikiLink Fundamental_theorem_of_arithmetic.
• Integral_domain wikiPageWikiLink GCD_domain.
• Integral_domain wikiPageWikiLink Gaussian_integer.
• Integral_domain wikiPageWikiLink Glossary_of_algebraic_geometry.
• Integral_domain wikiPageWikiLink Holomorphic_function.
• Integral_domain wikiPageWikiLink Ideal_(ring_theory).
• Integral_domain wikiPageWikiLink Injective_function.
• Integral_domain wikiPageWikiLink Integer.
• Integral_domain wikiPageWikiLink Irreducible_element.
• Integral_domain wikiPageWikiLink Irreducible_ring.
• Integral_domain wikiPageWikiLink Isomorphism.
• Integral_domain wikiPageWikiLink John_Wiley_&_Sons.
• Integral_domain wikiPageWikiLink Localization_of_a_ring.
• Integral_domain wikiPageWikiLink Manifold.
• Integral_domain wikiPageWikiLink Mathematics.
• Integral_domain wikiPageWikiLink Matrix_(mathematics).
• Integral_domain wikiPageWikiLink Minimal_prime_ideal.
• Integral_domain wikiPageWikiLink Monoid.
• Integral_domain wikiPageWikiLink Open_set.
• Integral_domain wikiPageWikiLink P-adic_number.
• Integral_domain wikiPageWikiLink Polynomial.
• Integral_domain wikiPageWikiLink Primary_decomposition.
• Integral_domain wikiPageWikiLink Prime_element.
• Integral_domain wikiPageWikiLink Prime_ideal.
• Integral_domain wikiPageWikiLink Prime_number.
• Integral_domain wikiPageWikiLink Principal_ideal.
• Integral_domain wikiPageWikiLink Product_of_rings.
• Integral_domain wikiPageWikiLink Quadratic_integer.
• Integral_domain wikiPageWikiLink Quotient_ring.
• Integral_domain wikiPageWikiLink Rational_number.
• Integral_domain wikiPageWikiLink Real_number.
• Integral_domain wikiPageWikiLink Reduced_ring.
• Integral_domain wikiPageWikiLink Regular_local_ring.
• Integral_domain wikiPageWikiLink Ring_of_integers.
• Integral_domain wikiPageWikiLink Serge_Lang.
• Integral_domain wikiPageWikiLink Spectrum_of_a_ring.
• Integral_domain wikiPageWikiLink Springer_Science+Business_Media.
• Integral_domain wikiPageWikiLink Subclass_(set_theory).
• Integral_domain wikiPageWikiLink Tensor_product_of_algebras.
• Integral_domain wikiPageWikiLink Unique_factorization_domain.
• Integral_domain wikiPageWikiLink Unit_(ring_theory).
• Integral_domain wikiPageWikiLink Unit_interval.
• Integral_domain wikiPageWikiLink Wedderburns_little_theorem.
• Integral_domain wikiPageWikiLink Integral_domains.
• Integral_domain wikiPageWikiLink Zero-product_property.
• Integral_domain wikiPageWikiLink Zero_divisor.
• Integral_domain wikiPageWikiLink Zero_ring.
• Integral_domain wikiPageWikiLinkText "Integral domain".
• Integral_domain wikiPageWikiLinkText "Integral domain#Divisibility, prime and irreducible elements".
• Integral_domain wikiPageWikiLinkText "associate".
• Integral_domain wikiPageWikiLinkText "associated".
• Integral_domain wikiPageWikiLinkText "divisibility".
• Integral_domain wikiPageWikiLinkText "divisible".
• Integral_domain wikiPageWikiLinkText "domain".
• Integral_domain wikiPageWikiLinkText "integral domain".
• Integral_domain wikiPageWikiLinkText "integral domain#Divisibility, prime elements, and irreducible elements".
• Integral_domain wikiPageWikiLinkText "integral".
• Integral_domain wikiPageWikiLinkText "prime elements".
• Integral_domain wikiPageWikiLinkText "prime".
• Integral_domain wikiPageUsesTemplate Template:=.
• Integral_domain wikiPageUsesTemplate Template:Algebraic_structures.
• Integral_domain wikiPageUsesTemplate Template:Cite_book.
• Integral_domain wikiPageUsesTemplate Template:Commutative_ring_classes.
• Integral_domain wikiPageUsesTemplate Template:Main.
• Integral_domain wikiPageUsesTemplate Template:Mapsto.
• Integral_domain wikiPageUsesTemplate Template:Reflist.
• Integral_domain wikiPageUsesTemplate Template:See_also.
• Integral_domain subject Category:Commutative_algebra.
• Integral_domain subject Category:Ring_theory.

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