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Integral domain

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Commutative ring with no zero divisors other than zero

Not to be confused with domain of integration.

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Inmathematics, anintegral domain is anonzero commutative ring in whichthe product of any two nonzero elements is nonzero.^[1]^[2] In an integral domain, every nonzero elementa has thecancellation property, that is, ifa ≠ 0,ab =ac impliesb =c. Integral domains are generalizations of thering ofintegers and provide a setting that is useful for studyingdivisibility.

"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have amultiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity.^[3]^[4] Noncommutative integral domains are sometimes admitted.^[5] 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, notablyLang, use the termentire ring for integral domain.^[6]

Some specific kinds of integral domains are given with the following chain ofclass inclusions:

rngs ⊃rings ⊃commutative rings ⊃integral domains ⊃integrally closed domains ⊃GCD domains ⊃unique factorization domains ⊃principal ideal domains ⊃Euclidean domains ⊃fields ⊃algebraically closed fields

Algebraic structures
Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Rng Semiring Near-ring Commutative ring Domain Integral domain Field Division ring Lie ring Ring theory
Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map of lattices Lattice theory
Module-like Module Group with operators Vector space Linear algebra
Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra
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Definition

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Anintegral domain is anonzero commutative ring in which the product of any two nonzero elements is nonzero. Equivalently:

An integral domain is a nonzero commutative ring with no nonzerozero divisors.
An integral domain is a commutative ring in which thezero ideal {0} is aprime ideal.
An integral domain is a nonzero commutative ring for which every nonzero element iscancellable under multiplication.
An integral domain is a ring for which the set of nonzero elements is a commutativemonoid under multiplication (because a monoid must be closed under multiplication).
An integral domain is a nonzero commutative ring in which for every nonzero elementr, the function that maps each elementx of the ring to the productxr isinjective. Elementsr with this property are calledregular, so it is equivalent to require that every nonzero element of the ring be regular.
An integral domain is a ring that isisomorphic to asubring of afield. (Given an integral domain, one can embed it in itsfield of fractions.)

Examples

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The archetypical example is the ring $\mathbb {Z}$ of allintegers.
Everyfield is an integral domain. For example, the field $\mathbb {R}$ of allreal numbers is an integral domain. Conversely, everyArtinian integral domain is a field. In particular, all finite integral domains arefinite fields (more generally, byWedderburn's little theorem, finitedomains arefinite fields). The ring of integers $\mathbb {Z}$ provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:
$\mathbb {Z} \supset 2\mathbb {Z} \supset \cdots \supset 2^{n}\mathbb {Z} \supset 2^{n+1}\mathbb {Z} \supset \cdots$
Rings ofpolynomials are integral domains if the coefficients come from an integral domain. For instance, the ring $\mathbb {Z} [x]$ of all polynomials in one variable with integer coefficients is an integral domain; so is the ring $\mathbb {C} [x_{1},\ldots ,x_{n}]$ of all polynomials inn-variables withcomplex coefficients.
The previous example can be further exploited by taking quotients from prime ideals. For example, the ring $\mathbb {C} [x,y]/(y^{2}-x(x-1)(x-2))$ corresponding to a planeelliptic curve is an integral domain. Integrality can be checked by showing $y^{2}-x(x-1)(x-2)$ is anirreducible polynomial.
The ring $\mathbb {Z} [x]/(x^{2}-n)\cong \mathbb {Z} [{\sqrt {n}}]$ is an integral domain for any non-square integer $n {\displaystyle n}$ . If $n>0$ , then this ring is always a subring of $\mathbb {R}$ , otherwise, it is a subring of $\mathbb {C} .$
The ring ofp-adic integers $\mathbb {Z} _{p}$ is an integral domain.
The ring offormal power series of an integral domain is an integral domain.
If $U {\displaystyle U}$ is aconnected open subset of thecomplex plane $\mathbb {C}$ , then the ring ${\mathcal {H}}(U)$ consisting of allholomorphic functions is an integral domain. The same is true for rings ofanalytic functions on connected open subsets of analyticmanifolds.
Aregular local ring is an integral domain. In fact, a regular local ring is aUFD.^[7]^[8]

Non-examples

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The following rings arenot integral domains.

Thezero ring (the ring in which $0=1$ ).
The quotient ring $\mathbb {Z} /m\mathbb {Z}$ whenm is acomposite number. To show this, choose a proper factorization $m=xy$ (meaning that $x {\displaystyle x}$ and $y {\displaystyle y}$ are not equal to $1 {\displaystyle 1}$ or $m {\displaystyle m}$ ). Then $x\not \equiv 0{\bmod {m}}$ and $y\not \equiv 0{\bmod {m}}$ , but $xy\equiv 0{\bmod {m}}$ .
Aproduct of two nonzero commutative rings. In such a product $R\times S$ , one has $(1,0)\cdot (0,1)=(0,0)$ .
The quotient ring $\mathbb {Z} [x]/(x^{2}-n^{2})$ for any $n\in \mathbb {Z}$ . The images of $x+n$ and $x-n$ are nonzero, while their product is 0 in this ring.
Thering ofn ×nmatrices over anynonzero ring whenn ≥ 2. If $M {\displaystyle M}$ and $N {\displaystyle N}$ are matrices such that the image of $N {\displaystyle N}$ is contained in the kernel of $M {\displaystyle M}$ , then $MN=0$ . For example, this happens for $M=N=({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}})$ .
The quotient ring $k[x_{1},\ldots ,x_{n}]/(fg)$ for any field $k {\displaystyle k}$ and any non-constant polynomials $f,g\in k[x_{1},\ldots ,x_{n}]$ . The images off andg in this quotient ring are nonzero elements whose product is 0. This argument shows, equivalently, that $(fg)$ is not aprime ideal. The geometric interpretation of this result is that thezeros offg form anaffine algebraic set that is not irreducible (that is, not analgebraic variety) in general. The only case where this algebraic set may be irreducible is whenfg is a power of anirreducible polynomial, which defines the same algebraic set.
The ring ofcontinuous functions on theunit interval. Consider the functions
$f(x)={\begin{cases}1-2x&x\in \left[0,{\tfrac {1}{2}}\right]\\0&x\in \left[{\tfrac {1}{2}},1\right]\end{cases}}\qquad g(x)={\begin{cases}0&x\in \left[0,{\tfrac {1}{2}}\right]\\2x-1&x\in \left[{\tfrac {1}{2}},1\right]\end{cases}}$

Neither

f {\displaystyle f}

nor

g {\displaystyle g}

is everywhere zero, but

f g {\displaystyle fg}

is.

Thetensor product $\mathbb {C} \otimes _{\mathbb {R} }\mathbb {C}$ . This ring has two non-trivialidempotents, $e_{1}={\tfrac {1}{2}}(1\otimes 1)-{\tfrac {1}{2}}(i\otimes i)$ and $e_{2}={\tfrac {1}{2}}(1\otimes 1)+{\tfrac {1}{2}}(i\otimes i)$ . They are orthogonal, meaning that $e_{1}e_{2}=0$ , and hence $\mathbb {C} \otimes _{\mathbb {R} }\mathbb {C}$ is not a domain. In fact, there is an isomorphism $\mathbb {C} \times \mathbb {C} \to \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C}$ defined by $(z,w)\mapsto z\cdot e_{1}+w\cdot e_{2}$ . Its inverse is defined by $z\otimes w\mapsto (zw,z{\overline {w}})$ . This example shows that afiber product of irreducible affine schemes need not be irreducible.

Divisibility, prime elements, and irreducible elements

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Properties

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A commutative ringR is an integral domain if and only if the ideal (0) ofR is a prime ideal.
IfR is a commutative ring andP is anideal inR, then thequotient ringR/P is an integral domain if and only ifP is aprime ideal.
LetR be an integral domain. Then thepolynomial rings overR (in any number of indeterminates) are integral domains. This is in particular the case ifR is afield.
The cancellation property holds in any integral domain: for anya,b, andc in an integral domain, ifa ≠0 andab =ac thenb =c. Another way to state this is that the functionx ↦ax is injective for any nonzeroa in the domain.
The cancellation property holds for ideals in any integral domain: ifxI =xJ, then eitherx is zero orI =J.
An integral domain is equal to the intersection of itslocalizations at maximal ideals.
Aninductive limit of integral domains is an integral domain.
IfA,B are integral domains over an algebraically closed fieldk, thenA ⊗_kB is an integral domain. This is a consequence ofHilbert's nullstellensatz,^[a] 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

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Main article:Field of fractions

Thefield of fractionsK of an integral domainR is the set of fractionsa/b witha andb inR andb ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is "the smallest field containingR" in the sense that there is an injective ring homomorphismR →K such that any injective ring homomorphism fromR to a field factors throughK. The field of fractions of the ring of integers $\mathbb {Z}$ is the field ofrational numbers $\mathbb {Q} .$ The field of fractions of a field isisomorphic to the field itself.

Algebraic geometry

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Integral domains are characterized by the condition that they arereduced (that isx² = 0 impliesx = 0) andirreducible (that is there is only oneminimal prime ideal). The former condition ensures that thenilradical 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, inalgebraic geometry, into the fact that thecoordinate ring of anaffine algebraic set is an integral domain if and only if the algebraic set is analgebraic variety.

More generally, a commutative ring is an integral domain if and only if itsspectrum is anintegral affine scheme.

Characteristic and homomorphisms

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Thecharacteristic of an integral domain is either 0 or aprime number.

IfR is an integral domain of prime characteristicp, then theFrobenius endomorphismx ↦x^p isinjective.

Notes

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^Proof: First assumeA is finitely generated as ak-algebra and pick ak-basis $g_{i}$ ofB. Suppose ${\textstyle \sum f_{i}\otimes g_{i}\sum h_{j}\otimes g_{j}=0}$ (only finitely many $f_{i},h_{j}$ are nonzero). For each maximal ideal ${\mathfrak {m}}$ ofA, consider the ring homomorphism $A\otimes _{k}B\to A/{\mathfrak {m}}\otimes _{k}B=k\otimes _{k}B\simeq B$ . Then the image is ${\textstyle \sum {\overline {f_{i}}}g_{i}\sum {\overline {h_{i}}}g_{i}=0}$ and thus either ${\textstyle \sum {\overline {f_{i}}}g_{i}=0}$ or ${\textstyle \sum {\overline {h_{i}}}g_{i}=0}$ and, by linear independence, ${\overline {f_{i}}}=0$ for all $i {\displaystyle i}$ or ${\overline {h_{i}}}=0$ for all $i {\displaystyle i}$ . Since ${\mathfrak {m}}$ is arbitrary, we have ${\textstyle (\sum f_{i}A)(\sum h_{i}A)\subset \operatorname {Jac} (A)=}$ the intersection of all maximal ideals $=(0)$ where the last equality is by the Nullstellensatz. Since $(0) {\displaystyle (0)}$ is a prime ideal, this implies either ${\textstyle \sum f_{i}A}$ or ${\textstyle \sum h_{i}A}$ is the zero ideal; i.e., either $f_{i}$ are all zero or $h_{i}$ are all zero. Finally,A is an inductive limit of finitely generatedk-algebras that are integral domains and thus, using the previous property, $A\otimes _{k}B=\varinjlim A_{i}\otimes _{k}B$ is an integral domain. $\square$

Citations

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^Bourbaki 1998, p. 116
^Dummit & Foote 2004, p. 228
^van der Waerden 1966, p. 36
^Herstein 1964, pp. 88–90
^McConnell & Robson
^Lang 1993, pp. 91–92
^Auslander & Buchsbaum 1959
^Nagata 1958
^Durbin 1993, p. 224, "Elementsa andb of [an integral domain] are calledassociates ifa |b andb |a."

References

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Adamson, Iain T. (1972).Elementary rings and modules. University Mathematical Texts. Oliver and Boyd.ISBN 0-05-002192-3.
Bourbaki, Nicolas (1998).Algebra, Chapters 1–3. Berlin, New York:Springer-Verlag.ISBN 978-3-540-64243-5.
Dummit, David S.; Foote, Richard M. (2004).Abstract Algebra (3rd ed.). New York:Wiley.ISBN 978-0-471-43334-7.
Durbin, John R. (1993).Modern Algebra: An Introduction (3rd ed.). John Wiley and Sons.ISBN 0-471-51001-7.
Herstein, I.N. (1964),Topics in Algebra, London: Blaisdell Publishing Company
Hungerford, Thomas W. (2013).Abstract Algebra: An Introduction (3rd ed.). Cengage Learning.ISBN 978-1-111-56962-4.
Lang, Serge (1993),Algebra (Third ed.), Reading, Mass.: Addison-Wesley,ISBN 978-0-201-55540-0,Zbl 0848.13001
Lang, Serge (2002).Algebra. Graduate Texts in Mathematics. Vol. 211. Berlin, New York:Springer-Verlag.ISBN 978-0-387-95385-4.MR 1878556.
Mac Lane, Saunders;Birkhoff, Garrett (1967).Algebra. New York: The Macmillan Co.ISBN 1-56881-068-7.MR 0214415.
McConnell, J.C.; Robson, J.C.,Noncommutative Noetherian Rings,Graduate Studies in Mathematics, vol. 30, AMS
Milies, César Polcino; Sehgal, Sudarshan K. (2002).An introduction to group rings. Springer.ISBN 1-4020-0238-6.
Lanski, Charles (2005).Concepts in abstract algebra. AMS Bookstore.ISBN 0-534-42323-X.
Rowen, Louis Halle (1994).Algebra: groups, rings, and fields.A K Peters.ISBN 1-56881-028-8.
Sharpe, David (1987).Rings and factorization.Cambridge University Press.ISBN 0-521-33718-6.
van der Waerden, Bartel Leendert (1966),Algebra, vol. 1, Berlin, Heidelberg: Springer-Verlag
Auslander, M; Buchsbaum, D A (1959)."Unique factorization in regular local rings".Proceedings of the National Academy of Sciences of the United States of America.45 (5) (published May 1959):733–4.Bibcode:1959PNAS...45..733A.doi:10.1073/PNAS.45.5.733.ISSN 0027-8424.PMC 222624.PMID 16590434.Zbl 0084.26504.Wikidata Q24655880.
Nagata, Masayoshi (1958). "A General Theory of Algebraic Geometry Over Dedekind Domains, II: Separably Generated Extensions and Regular Local Rings".American Journal of Mathematics.80 (2) (published April 1958): 382.doi:10.2307/2372791.ISSN 0002-9327.JSTOR 2372791.Zbl 0089.26501.Wikidata Q56049883.