Editing Integral domain

{{Short description|Commutative ring with no zero divisors other than zero}}
{{Use American English|date = March 2019}}
{{distinguish |text= [[ Integral | domain of integration]]}}
{{Ring theory sidebar}}
In [[mathematics]], an '''integral domain''' is a [[Zero ring|nonzero]] [[commutative ring]] in which [[zero-product property|the product of any two nonzero elements is nonzero]].{{sfn|Bourbaki|1998|p=116|ps=none}}{{sfn|Dummit|Foote|2004|p=228|ps=none}} Integral domains are generalizations of the [[ring (mathematics)|ring]] of [[integer]]s and provide a natural setting for studying [[divisibility (ring theory)|divisibility]].  In an integral domain, every nonzero element ''a'' has the [[cancellation property]], that is, if {{nowrap|''a'' ≠ 0}}, an equality {{nowrap|''ab'' {{=}} ''ac''}} implies {{nowrap|''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.{{sfn|van der Waerden|1966|p=36|ps=none}}{{sfn|Herstein|1964|pp=88–90|ps=none}} Noncommutative integral domains are sometimes admitted.{{sfn|McConnell|Robson|ps=none}}  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.

Some sources, notably [[Serge Lang|Lang]], use the term '''entire ring''' for integral domain.{{sfn|Lang|1993|pp=91–92|ps=none}}

Some specific kinds of integral domains are given with the following chain of [[subclass (set theory)|class inclusions]]:

{{Commutative ring classes}}

{{Algebraic structures |Ring}}

== Definition ==
An ''integral domain'' is a [[zero ring|nonzero]] [[commutative ring]] in which the product of any two nonzero elements is nonzero.  Equivalently:
* An integral domain is a nonzero commutative ring with no nonzero [[zero divisor]]s.
* An integral domain is a commutative ring in which the [[zero ideal]] {0} is a [[prime ideal]].
* An integral domain is a nonzero commutative ring for which every nonzero element is [[cancellation property|cancellable]] under multiplication.
* An integral domain is a ring for which the set of nonzero elements is a commutative [[monoid]] under multiplication (because a monoid must be [[closure (mathematics)| closed]] under multiplication).
* An integral domain is a 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.
* An integral domain is a ring that is [[isomorphic]] to a [[subring]] of a [[field (mathematics)|field]].  (Given an integral domain, one can embed it in its [[field of fractions]].)

== Examples ==
* The archetypical example is the ring <math>\Z</math> of all [[integer]]s.
* Every [[field (mathematics)|field]] is an integral domain. For example, the field <math>\R</math> of all [[real number]]s 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 <math>\Z</math> provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:
*: <math>\Z \supset 2\Z \supset \cdots \supset 2^n\Z \supset 2^{n+1}\Z \supset \cdots</math>
* Rings of [[polynomial]]s are integral domains if the coefficients come from an integral domain. For instance, the ring <math>\Z[x]</math> of all polynomials in one variable with integer coefficients is an integral domain; so is the ring <math>\Complex[x_1,\ldots,x_n]</math> of all polynomials in ''n''-variables with [[Complex number|complex]] coefficients.
* The previous example can be further exploited by taking quotients from prime ideals. For example, the ring <math>\Complex[x,y]/(y^2 - x(x-1)(x-2))</math> corresponding to a plane [[elliptic curve]] is an integral domain. Integrality can be checked by showing <math>y^2 - x(x-1)(x-2)</math> is an [[irreducible polynomial]].
* The ring <math>\Z[x]/(x^2 - n) \cong \Z[\sqrt{n}]</math> is an integral domain for any non-square integer <math>n</math>. If <math>n > 0</math>, then this ring is always a subring of <math>\R</math>, otherwise, it is a subring of <math>\Complex.</math>
* The ring of [[p-adic number|''p''-adic integers]] <math>\Z_p</math> is an integral domain.
* The ring of [[formal power series]] of an integral domain is an integral domain.
* If <math>U</math> is a [[connectedness|connected]] [[open subset]] of the [[complex number|complex plane]] <math>\Complex</math>, then the ring <math>\mathcal{H}(U)</math> consisting of all [[holomorphic function]]s is an integral domain. The same is true for rings of [[analytic function]]s on connected open subsets of analytic [[manifold]]s.
* A [[regular local ring]] is an integral domain. In fact, a regular local ring is a [[unique factorization domain|UFD]].{{sfn|Auslander|Buchsbaum|1959|ps=none}}{{sfn|Nagata|1958|ps=none}}

== Non-examples ==
The following rings are ''not'' integral domains.
* The [[zero ring]] (the ring in which <math>0=1</math>).
* The quotient ring <math>\Z/m\Z</math> when ''m'' is a [[composite number]]. To show this, choose a proper factorization <math>m = xy</math> (meaning that <math>x</math> and <math>y</math> are not equal to <math>1</math> or <math>m</math>). Then <math>x \not\equiv 0 \bmod{m}</math> and <math>y \not\equiv 0 \bmod{m}</math>, but <math>xy \equiv 0 \bmod{m}</math>.
* A [[product ring|product]] of two nonzero commutative rings.  In such a product <math>R \times S</math>, one has <math>(1,0) \cdot (0,1) = (0,0)</math>.
* The quotient ring <math>\Z[x]/(x^2 - n^2)</math> for any <math>n \in \mathbb{Z}</math>.  The images of <math>x+n</math> and <math>x-n</math> are nonzero, while their product is 0 in this ring.
* The [[matrix ring|ring]] of ''n'' × ''n'' [[Matrix (mathematics)|matrices]] over any [[zero ring|nonzero ring]] when ''n'' ≥ 2. If <math>M</math> and <math>N</math> are matrices such that the image of <math>N</math> is contained in the kernel of <math>M</math>, then <math>MN = 0</math>.  For example, this happens for <math>M = N = (\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix})</math>.
* The quotient ring <math>k[x_1,\ldots,x_n]/(fg)</math> for any field <math>k</math> and any non-constant polynomials <math>f,g \in k[x_1,\ldots,x_n]</math>. The images of {{math|''f''}} and {{math|''g''}} in this quotient ring are nonzero elements whose product is 0. This argument shows, equivalently, that <math>(fg)</math> is not a [[prime ideal]]. The geometric interpretation of this result is that the [[zero of a function|zeros]] of {{math|''fg''}} form an [[affine algebraic set]] that is not irreducible (that is, not an [[algebraic variety]]) in general. The only case where this algebraic set may be irreducible is when {{math|''fg''}} is a power of an [[irreducible polynomial]], which defines the same algebraic set.
* The ring of [[continuous function]]s on the [[unit interval]].  Consider the functions
*: <math> f(x) = \begin{cases} 1-2x & x \in \left [0, \tfrac{1}{2} \right ] \\ 0 & x \in \left [\tfrac{1}{2}, 1 \right ] \end{cases} \qquad g(x) = \begin{cases} 0 & x \in \left [0, \tfrac{1}{2} \right ] \\ 2x-1 & x \in \left [\tfrac{1}{2}, 1 \right ] \end{cases}</math>
: Neither <math>f</math> nor <math>g</math> is everywhere zero, but <math>fg</math> is.
* The [[tensor product of algebras|tensor product]] <math>\Complex \otimes_{\R} \Complex</math>. This ring has two non-trivial [[idempotent (ring theory)|idempotent]]s, <math>e_1 = \tfrac{1}{2}(1 \otimes 1) - \tfrac{1}{2}(i \otimes i)</math> and <math>e_2 = \tfrac{1}{2}(1 \otimes 1) + \tfrac{1}{2}(i \otimes i)</math>. They are orthogonal, meaning that <math>e_1e_2 = 0</math>, and hence <math>\Complex \otimes_{\R} \Complex</math> is not a domain. In fact, there is an isomorphism <math>\Complex \times \Complex \to \Complex \otimes_{\R} \Complex</math> defined by <math>(z, w) \mapsto z \cdot e_1 + w \cdot e_2</math>. Its inverse is defined by <math>z \otimes w \mapsto (zw, z\overline{w})</math>. This example shows that a [[fiber product of schemes|fiber product]] of irreducible affine schemes need not be irreducible.

== Divisibility, prime elements, and irreducible elements ==
<!-- This section is redirected from [[Associate elements]] -->
{{see also|Divisibility (ring theory)}}
In this section, ''R'' is an integral domain.

Given elements ''a'' and ''b'' of ''R'', one says 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 {{nowrap|1=''ax'' = ''b''}}.

The ''[[unit (ring theory)|unit]]s'' of ''R'' are the elements that divide 1; these are precisely the invertible elements in ''R''. Units divide all other elements.

If ''a'' divides ''b'' and ''b'' divides ''a'', then ''a'' and ''b'' are '''associated elements''' or '''associates'''.{{sfn|Durbin|1993|loc=p. 224, "Elements ''a'' and ''b'' of [an integral domain] are called ''associates'' if ''a'' {{!}} ''b'' and ''b'' {{!}} ''a''."|ps=none}}  Equivalently, ''a'' and ''b'' are associates if {{nowrap|1=''a'' = ''ub''}} for some [[unit (ring theory)|unit]] ''u''.

An ''[[irreducible element]]'' is a nonzero non-unit that cannot be written as a product of two non-units.

A nonzero non-unit ''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]]. 

Both notions of irreducible elements and prime elements generalize the ordinary definition of [[prime number]]s in the ring <math>\Z,</math> if one considers as prime the negative primes.

Every prime element is irreducible. The converse is not true in general: for example, in the [[quadratic integer]] ring <math>\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 <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.

While [[Fundamental theorem of arithmetic|unique factorization]] does not hold in <math>\Z\left[\sqrt{-5}\right]</math>, there is unique factorization of [[Ideal (ring theory)|ideals]].  See [[Lasker–Noether theorem]].

== Properties ==
* A commutative ring ''R'' is an integral domain if and only if the ideal (0) of ''R'' is a prime ideal.
* 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]].
* Let ''R'' be an integral domain. Then the [[polynomial ring]]s over ''R'' (in any number of indeterminates) are integral domains. This is in particular the case if ''R'' is a [[field (mathematics)|field]].
* The cancellation property holds in any integral domain: for any ''a'', ''b'', and ''c'' in an integral domain, if {{nowrap|''a'' ≠ ''0''}} and {{nowrap|1=''ab'' = ''ac''}} then {{nowrap|1=''b'' = ''c''}}. Another way to state this is that the function {{nowrap|''x'' ↦ ''ax''}} is injective for any nonzero ''a'' in the domain.
* The cancellation property holds for ideals in any integral domain: if {{nowrap|1=''xI'' = ''xJ''}}, then either ''x'' is zero or {{nowrap|1=''I'' = ''J''}}.
* An integral domain is equal to the intersection of its [[localization of a ring|localizations]] at maximal ideals.
* An [[inductive limit]] of integral domains is an integral domain.
* If ''A'', ''B'' are integral domains over an algebraically closed field ''k'', then {{nowrap|''A'' ⊗<sub>''k''</sub> ''B''}} is an integral domain. This is a consequence of [[Hilbert's nullstellensatz]],{{efn|Proof: First assume ''A'' is finitely generated as a ''k''-algebra and pick a ''k''-basis <math>g_i</math> of ''B''. Suppose <math display="inline">\sum f_i \otimes g_i \sum h_j \otimes g_j = 0</math> (only finitely many <math>f_i, h_j</math> are nonzero). For each maximal ideal <math>\mathfrak{m}</math> of ''A'', consider the ring homomorphism <math>A \otimes_k B \to A/\mathfrak{m} \otimes_k B = k \otimes_k B \simeq B</math>. Then the image is <math display="inline">\sum \overline{f_i} g_i \sum \overline{h_i} g_i = 0</math> and thus either <math display="inline">\sum \overline{f_i} g_i = 0</math> or <math display="inline">\sum \overline{h_i} g_i = 0</math> and, by linear independence, <math>\overline{f_i} = 0</math> for all <math>i</math> or <math>\overline{h_i} = 0</math> for all <math>i</math>. Since <math>\mathfrak{m}</math> is arbitrary, we have <math display="inline">(\sum f_iA) (\sum h_iA) \subset \operatorname{Jac}(A) = </math> the intersection of all maximal ideals <math>= (0)</math> where the last equality is by the Nullstellensatz. Since <math>(0)</math> is a prime ideal, this implies either <math display="inline">\sum f_iA</math> or <math display="inline">\sum h_iA</math> is the zero ideal; i.e., either <math>f_i</math> are all zero or <math>h_i</math> are all zero. Finally, ''A'' is an inductive limit of finitely generated ''k''-algebras that are integral domains and thus, using the previous property, <math>A \otimes_k B = \varinjlim A_i \otimes_k B</math> is an integral domain. <math>\square</math>}} and, in algebraic geometry, it implies the statement that the coordinate ring of the product of two affine algebraic varieties over an algebraically closed field is again an integral domain.

== Field of fractions ==
{{main|Field of fractions}}

The [[field of fractions]] ''K'' of an integral domain ''R'' is the set of fractions ''a''/''b'' with ''a'' and ''b'' in ''R'' and {{nowrap|''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 {{nowrap|''R'' → ''K''}} such that any injective ring homomorphism from ''R'' to a field factors through ''K''. The field of fractions of the ring of integers <math>\Z</math> is the field of [[rational number]]s <math>\Q.</math> The field of fractions of a field is [[isomorphism|isomorphic]] to the field itself.

== Algebraic geometry ==

Integral domains are characterized by the condition that they are [[reduced ring|reduced]] (that is {{nowrap|1=''x''<sup>2</sup> = 0}} implies {{nowrap|1=''x'' = 0}}) and [[irreducible ring|irreducible]] (that is there is only one [[minimal prime ideal]]). The former condition ensures that the [[nilradical of a ring|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.

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]].

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]].

== Characteristic and homomorphisms ==

The [[characteristic (algebra)|characteristic]] of an integral domain is either 0 or a [[prime number]].

If ''R'' is an integral domain of prime characteristic ''p'', then the [[Frobenius endomorphism]] {{nowrap|''x'' ↦ ''x''<sup>''p''</sup>}} is [[injective]].

== See also ==
{{Wikibooks|Abstract algebra|Integral domains}}
* [[Dedekind–Hasse norm]] – the extra structure needed for an integral domain to be principal
* [[Zero-product property]]

== Notes ==
{{notelist}}

== Citations ==
{{reflist}}

== References ==
{{sfn whitelist |CITEREFLang1993}}
{{refbegin}}
* {{cite book |last=Adamson |first=Iain T. | title=Elementary rings and modules | series=University Mathematical Texts | publisher=Oliver and Boyd | year=1972 | isbn=0-05-002192-3 }}
* {{cite book | last=Bourbaki | first=Nicolas | author-link=Nicolas Bourbaki | title=Algebra, Chapters 1–3 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-64243-5 | year=1998}}
* {{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}}
* {{cite book | last1=Durbin | first1=John R. | year=1993 | title=Modern Algebra: An Introduction | edition=3rd | publisher=John Wiley and Sons | isbn=0-471-51001-7 }}
* {{citation |last1=Herstein |first1= I.N. |year=1964 |title=Topics in Algebra |publisher=Blaisdell Publishing Company |location=London}}
* {{cite book | last=Hungerford | first=Thomas W. | author-link=Thomas W. Hungerford | title=Abstract Algebra: An Introduction | publisher=Cengage Learning | edition=3rd | year=2013 | isbn= 978-1-111-56962-4 }}
* {{Lang Algebra |edition=3}}
* {{cite book | last=Lang | first=Serge | author-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 }}
* {{cite book | last1=Mac Lane | first1=Saunders | author-link1=Saunders Mac Lane | last2=Birkhoff | first2=Garrett | author-link2=Garrett Birkhoff | title=Algebra | publisher=The Macmillan Co. | location=New York | mr=0214415 | year=1967 | isbn=1-56881-068-7}}
* {{citation | last1=McConnell | first1=J.C. | last2=Robson | first2=J.C. | title=Noncommutative Noetherian Rings | series=[[Graduate Studies in Mathematics]] | volume=30 | publisher=AMS }}
* {{cite book | last1=Milies | first1=César Polcino |last2=Sehgal |first2=Sudarshan K. | title=An introduction to group rings | publisher=Springer | year=2002 | isbn=1-4020-0238-6 }}
* {{cite book | last=Lanski | first=Charles | title=Concepts in abstract algebra | publisher=AMS Bookstore | year=2005 | isbn=0-534-42323-X }}
* {{cite book | last=Rowen | first=Louis Halle | title=Algebra: groups, rings, and fields | publisher=[[A K Peters]] | year=1994 | isbn=1-56881-028-8 }}
* {{cite book | last=Sharpe | first=David | title=Rings and factorization | url=https://archive.org/details/ringsfactorizati0000shar | url-access=registration | publisher=[[Cambridge University Press]] | year=1987 | isbn=0-521-33718-6 }}
* {{citation | last1=van der Waerden | first1=Bartel Leendert |authorlink = Bartel Leendert van der Waerden| year=1966 | title=Algebra | volume=1 | publisher=Springer-Verlag | location=Berlin, Heidelberg }}
* {{cite Q |Q24655880 |last1=Auslander |first1=M| last2=Buchsbaum |first2=D A |year=1959}}
* {{cite Q |Q56049883 |last1=Nagata |first1=Masayoshi |authorlink = Masayoshi Nagata|year=1958}}
{{refend}}

== External links ==
* {{cite web |url=https://math.stackexchange.com/q/45945 |title=where does the term "integral domain" come from? }}

{{DEFAULTSORT:Integral Domain}}
[[Category:Commutative algebra]]
[[Category:Ring theory]]