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| 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.<ref>Bourbaki, p. 116.</ref><ref>Dummit and Foote, p. 228.</ref> Integral domains are generalizations of the [[ring of integers]] and provide a natural setting for studying [[divisibility (ring theory)|divisibility]].
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| "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a 1, but some authors who do not follow this also do not require integral domains to have a 1.<ref>B.L. van der Waerden, Algebra Erster Teil, p. 36, Springer-Verlag, Berlin, Heidelberg 1966.</ref><ref>I.N. Herstein, Topics in Algebra, p. 88-90, Blaisdell Publishing Company, London 1964.</ref> Noncommutative integral domains are sometimes admitted.<ref>J.C. McConnel and J.C. Robson "Noncommutative Noetherian Rings" (Graduate studies in Mathematics Vol. 30, AMS)</ref> This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "[[domain (ring theory)|domain]]" for the general case including noncommutative rings.
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| Some sources, notably [[Serge Lang|Lang]], use the term '''entire ring''' for integral domain.<ref>Pages 91–92 of {{Lang Algebra|edition=3}}</ref>
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| Some specific kinds of integral domains are given with the following chain of [[subclass (set theory)|class inclusions]]:
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| : '''[[Commutative ring]]s''' ⊃ '''integral domains''' ⊃ '''[[integrally closed domain]]s''' ⊃ '''[[unique factorization domain]]s''' ⊃ '''[[principal ideal domain]]s''' ⊃ '''[[Euclidean domain]]s''' ⊃ '''[[field (mathematics)|field]]s'''
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| The absence of nonzero [[zero divisor]]s implies that in an integral domain the [[cancellation property]] holds for multiplication by any nonzero element ''a'': an equality {{nowrap| ''ab'' {{=}} ''ac''}} implies {{nowrap| ''b'' {{=}} ''c''}}.
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| {{Algebraic structures |Ring}}
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| == Definitions ==
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| There are a number of equivalent definitions of integral domain:
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| * An integral domain is a [[zero ring|nonzero]] commutative ring in which the product of any two nonzero elements is nonzero.
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| * An integral domain is a [[zero ring|nonzero]] commutative ring with no nonzero zero divisors.
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| * An integral domain is a commutative ring in which the zero [[ideal (ring theory)|ideal]] {0} is a [[prime ideal]].
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| * An integral domain is a ring for which the set of nonzero elements is a commutative [[monoid]] under multiplication.
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| * An integral domain is a ring that is (isomorphic to) a subring of a field. (This implies it is a [[zero ring|nonzero]] commutative ring.)
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| * An integral domain is a [[zero ring|nonzero]] commutative ring in which for every nonzero element ''r'', the function that maps each element ''x'' of the ring to the product ''xr'' is [[injective]]. Elements ''r'' with this property are called ''regular'', so it is equivalent to require that every nonzero element of the ring be regular.
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| == Examples ==
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| * The archetypical example is the ring '''Z''' of all [[integer]]s.
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| * Every [[field (mathematics)|field]] is an integral domain. Conversely, every [[artinian ring|Artinian]] integral domain is a field. In particular, all finite integral domains are [[finite field]]s (more generally, by [[Wedderburn's little theorem]], finite [[Domain (ring theory)|domains]] are [[finite field]]s). The ring of integers '''Z''' provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:
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| :: <math>\mathbf{Z}\;\supset\;2\mathbf{Z}\;\supset\;\cdots\;\supset\;2^n\mathbf{Z}\;\supset\;2^{n+1}\mathbf{Z}\;\supset\;\cdots</math>
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| * Rings of [[polynomial]]s are integral domains if the coefficients come from an integral domain. For instance, the ring '''Z'''[''X''] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring '''R'''[''X'',''Y''] of all polynomials in two variables with [[real number|real]] coefficients.
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| * For each integer ''n'' > 1, the set of all [[real number]]s of the form ''a'' + ''b''√''n'' with ''a'' and ''b'' [[integer]]s is a subring of '''R''' and hence an integral domain.
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| * For each integer ''n'' > 0 the set of all [[complex number]]s of the form ''a'' + ''bi''√''n'' with ''a'' and ''b'' integers is a subring of '''C''' and hence an integral domain. In the case ''n'' = 1 this integral domain is called the [[Gaussian integer]]s.
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| * The ring of [[p-adic number|p-adic integers]] is an integral domain.
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| * If ''U'' is a [[connectedness|connected]] [[open subset]] of the [[complex number|complex plane]] '''C''', then the ring H(''U'') consisting of all [[holomorphic function]]s ''f'' : ''U'' → '''C''' is an integral domain. The same is true for rings of [[analytic function]]s on connected open subsets of analytic [[manifold]]s.
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| * A [[regular local ring]] is an integral domain. In fact, a regular local ring is a [[unique factorization domain|UFD]].<ref>{{cite journal | author= Maurice Auslander | coauthors=D.A. Buchsbaum | title=Unique factorization in regular local rings | journal=Proc. Natl. Acad. Sci. USA | volume=45 | pages=733–734 | year=1959 | doi= 10.1073/pnas.45.5.733 | pmid= 16590434 | issue= 5 | pmc= 222624 }}</ref><ref>{{cite journal | author=Masayoshi Nagata | authorlink=Masayoshi Nagata | title=A general theory of algebraic geometry over Dedekind domains. II | journal=Amer. J. Math. | volume=80 | year=1958 | pages=382–420 | doi=10.2307/2372791 | jstor=2372791 | issue=2 | publisher=The Johns Hopkins University Press }}</ref>
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| == Non-examples ==
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| The following rings are ''not'' integral domains.
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| * The ring of ''n'' × ''n'' [[Matrix (mathematics)|matrices]] over any [[zero ring|nonzero ring]] when ''n'' ≥ 2.
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| * The ring of [[continuous function]]s on the [[unit interval]].
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| * The [[quotient ring]] '''Z'''/''m'''''Z''' when ''m'' is a [[composite number]].
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| * The [[product ring]] '''Z''' × '''Z'''.
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| * The [[zero ring]] in which 0=1.
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| == Divisibility, prime elements, and irreducible elements ==<!-- This section is redirected from [[Associate elements]] -->
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| {{see also|Divisibility (ring theory)}}
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| In this section, ''R'' is an integral domain.
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| Given elements ''a'' and ''b'' of ''R'', we say that ''a'' '''divides''' ''b'', or that ''a'' is a '''[[Divisibility (ring theory)|divisor]]''' of ''b'', or that ''b'' is a '''multiple''' of ''a'', if there exists an element ''x'' in ''R'' such that ''ax'' = ''b''.
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| The elements that divide 1 are called the '''[[unit (ring theory)|unit]]s''' of ''R''; these are precisely the invertible elements in ''R''. Units divide all other elements.
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| If ''a'' divides ''b'' and ''b'' divides ''a'', then we say ''a'' and ''b'' are '''associated elements''' or '''associates'''. Equivalently, ''a'' and ''b'' are associates if ''a''=''ub'' for some [[unit (ring theory)|unit]] ''u''.
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| If ''q'' is a non-unit, we say that ''q'' is an '''[[irreducible element]]''' if ''q'' cannot be written as a product of two non-units.
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| If ''p'' is a nonzero non-unit, we say that ''p'' is a '''prime element''' if, whenever ''p'' divides a product ''ab'', then ''p'' divides ''a'' or ''p'' divides ''b''. Equivalently, an element ''p'' is prime if and only if the [[principal ideal]] (''p'') is a nonzero prime ideal. The notion of '''[[prime element]]''' generalizes the ordinary definition of [[prime number]] in the ring '''Z''', except that it allows for negative prime elements.
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| Every prime element is irreducible. The converse is not true in general: for example, in the [[quadratic integer]] ring <math>\mathbb{Z}\left[\sqrt{-5}\right]</math> the element 3 is irreducible (if it factored nontrivially, the factors would each have to have norm 3, but there are no norm 3 elements since <math>a^2+5b^2=3</math> has no integer solutions), but not prime (since 3 divides since <math>\left(2 + \sqrt{-5}\right)\left(2 - \sqrt{-5}\right)</math> without dividing either factor). In a unique factorization domain (or more generally, a [[GCD domain]]), an irreducible element is a prime element.
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| While [[Fundamental theorem of arithmetic|unique factorization]] does not hold in <math>\mathbb{Z}\left[\sqrt{-5}\right]</math>, there is unique factorization of [[Ideal (ring theory)|ideals]]. See [[Lasker–Noether theorem]].
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| == Properties ==
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| * A commutative ring ''R'' is an integral domain if and only if the ideal (0) of ''R'' is a prime ideal.
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| * If ''R'' is a commutative ring and ''P'' is an [[ideal (ring theory)|ideal]] in ''R'', then the [[quotient ring]] ''R/P'' is an integral domain if and only if ''P'' is a [[prime ideal]].
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| * Let ''R'' be an integral domain. Then there is an integral domain ''S'' such that ''R'' ⊂ ''S'' and ''S'' has an element which is transcendental over ''R''.
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| * The cancellation property holds in any integral domain: for any ''a'', ''b'', and ''c'' in an integral domain, if ''a'' ≠ ''0'' and ''ab'' = ''ac'' then ''b'' = ''c''. Another way to state this is that the function ''x'' {{mapsto}} ''ax'' is injective for any nonzero ''a'' in the domain.
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| * An integral domain is equal to the intersection of its [[localization of a ring|localizations]] at maximal ideals.
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| * An [[inductive limit]] of integral domains is an integral domain.
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| == Field of fractions ==
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| {{Main|Field of fractions}}
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| The [[field of fractions]] ''K'' of an integral domain ''R'' is the set of fractions ''a''/''b'' with ''a'' and ''b'' in ''R'' and ''b'' ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is "the smallest field containing ''R''" in the sense that there is an injective ring homomorphism ''R''→''K'' such that any injective ring homomorphism from ''R'' to a field factors through ''K''.
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| The field of fractions of the ring of integers '''Z''' is the field of [[rational number]]s '''Q'''. The field of fractions of a field is [[isomorphism|isomorphic]] to the field itself.
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| == Algebraic geometry ==
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| Integral domains are characterized by the condition that they are [[reduced ring|reduced]] (that is ''x''<sup>2</sup> = 0 implies ''x'' = 0) and [[irreducible ring|irreducible]] (that is there is only one [[minimal prime ideal]]). The former condition ensures that the nilradical of the ring is zero, so that the intersection of all the ring's minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains. The converse is clear: an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal.
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| This translates, in [[algebraic geometry]], into the fact that the [[coordinate ring]] of an [[affine algebraic set]] is an integral domain if and only if the algebraic set is an [[algebraic variety]]. | |
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| More generally, a commutative ring is an integral domain if and only if its [[spectrum of a ring|spectrum]] is an [[integral scheme|integral]] [[affine scheme]].
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| == Characteristic and homomorphisms ==
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| The [[characteristic (algebra)|characteristic]] of an integral domain is either 0 or a [[prime number]].
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| If ''R'' is an integral domain of prime characteristic ''p'', then the [[Frobenius endomorphism]] ''f''(''x'') = ''x''<sup> ''p''</sup> is [[injective]].
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| == See also ==
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| * [[wikibooks:Abstract algebra/Integral domains|Integral domains]] – wikibook link
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| * [[Dedekind–Hasse norm]] – the extra structure needed for an integral domain to be principal
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| * [[Zero-product property]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{cite book | author= Iain T. Adamson | title=Elementary rings and modules | series=University Mathematical Texts | publisher=Oliver and Boyd | year=1972 | isbn=0-05-002192-3 }}
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| * {{Cite book | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Algebra, Chapters 1–3 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-64243-5 | year=1998}}
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| * {{Cite book | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | last2=Birkhoff | first2=Garrett | author2-link=Garrett Birkhoff | title=Algebra | publisher=The Macmillan Co. | location=New York | mr=0214415 | year=1967 | isbn=1-56881-068-7}}
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| * {{Cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons|Wiley]] | location=New York | edition=3rd | isbn=978-0-471-43334-7 | year=2004}}
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| * {{Cite book | last1=Hungerford | first1=Thomas W. | author1-link=Thomas W. Hungerford | title=Algebra | publisher=Holt, Rinehart and Winston, Inc. | location=New York | year=1974 | isbn=0-03-030558-6}}
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| * {{Cite book | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-95385-4 | mr=1878556 | year=2002 | volume=211}}
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| * {{cite book | author=David Sharpe | title=Rings and factorization | publisher=[[Cambridge University Press]] | year=1987 | isbn=0-521-33718-6 }}
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| * {{cite book | author=Louis Halle Rowen | title=Algebra: groups, rings, and fields | publisher=[[A K Peters]] | year=1994 | isbn=1-56881-028-8 }}
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| * {{cite book | author=Charles Lanski | title=Concepts in abstract algebra | publisher=AMS Bookstore | year=2005 | isbn=0-534-42323-X }}
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| * {{cite book | author=César Polcino Milies | author2=Sudarshan K. Sehgal | title=An introduction to group rings | publisher=Springer | year=2002 | isbn=1-4020-0238-6 }}
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| * B.L. van der Waerden, Algebra, Springer-Verlag, Berlin Heidelberg, 1966.
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| {{DEFAULTSORT:Integral Domain}}
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| [[Category:Commutative algebra]]
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| [[Category:Ring theory]]
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