Integral domain
Encyclopedia
In abstract algebra
, an integral domain is a commutative ring
that has no zero divisors, and which is not the trivial ring {0}. It is usually assumed that commutative rings and integral domains have a multiplicative identity even though this is not always included in the definition of a ring. Integral domains are generalizations of the integer
s and provide a natural setting for studying divisibility. An integral domain is a commutative domain
with identity.
The above is how "integral domain" is almost universally defined, but there is some variation. In particular, noncommutative integral domains are sometimes admitted. However, we follow the much more usual convention of reserving the term integral domain for the commutative case and use domain
for the noncommutative case; this implies that, curiously, the adjective "integral" means "commutative" in this context. 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:
The absence of zero divisor
s means that in an integral domain the cancellation property
holds for multiplication by any nonzero element a: an equality implies .
The following rings are not integral domains.
of b or b is a multiple of a if and only if there exists an element x in R such that ax = b.
The elements which divide 1 are called the unit
s of R; these are precisely the invertible elements in R. Units divide all other elements.
If a divides b and b divides a, then we say a and b are associated elements or associates.
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.
If p is a non-zero 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. Equivalent, an element is prime if and only if an ideal generated by it is a nonzero prime ideal. Every prime element is irreducible. Conversely, in a GCD domain
(e.g., a unique factorization domain), an irreducible element is a prime element.
The notion of prime element
generalizes the ordinary definition of prime number
in the ring Z, except that it allows for negative prime elements. While every prime is irreducible, the converse is not in general true. For example, in the quadratic integer
ring
the number 3 is irreducible, but is not a prime because 9, the norm of 3, can be factored in two ways in the ring, namely, 
and 
. Thus 
, but 3 does not divide 
nor 
The numbers 3 and 
are irreducible as there is no 
where 
or 
as 
has no integer solution.
While unique factorization
does not hold in the above example, if we use ideals
we do get unique factorization. Namely, the ideal
equals the ideals 
and 
and is the unique product of the two prime ideals: 
, each of which have a norm of 3.
s. The field of fractions of a field is isomorphic
to the field itself.
varieties
. They have a unique generic point
, given by the zero ideal. Integral domains are also characterized by the condition that they are reduced
and irreducible
. 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, hence such rings are integral domains. The converse is clear: No integral domain can have nilpotent elements, and the zero ideal is the unique minimal prime ideal.
of every integral domain is either zero or a prime number
.
If R is an integral domain with prime characteristic p, then f(x) = x p defines an injective ring homomorphism
f : R → R, the Frobenius endomorphism
.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, an integral domain is a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
that has no zero divisors, and which is not the trivial ring {0}. It is usually assumed that commutative rings and integral domains have a multiplicative identity even though this is not always included in the definition of a ring. Integral domains are generalizations of the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s and provide a natural setting for studying divisibility. An integral domain is a commutative domain
Domain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
with identity.
The above is how "integral domain" is almost universally defined, but there is some variation. In particular, noncommutative integral domains are sometimes admitted. However, we follow the much more usual convention of reserving the term integral domain for the commutative case and use domain
Domain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
for the noncommutative case; this implies that, curiously, the adjective "integral" means "commutative" in this context. Some sources, notably Lang
Serge Lang
Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
, use the term entire ring for integral domain.
Some specific kinds of integral domains are given with the following chain of class inclusions:
- Commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s ⊃ integral domains ⊃ integrally closed domainIntegrally closed domainIn commutative algebra, an integrally closed domain A is an integral domain whose integral closure in the field of fractions of A is A itself...
s ⊃ unique factorization domainUnique factorization domainIn mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
s ⊃ principal ideal domainPrincipal ideal domainIn abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
s ⊃ Euclidean domainEuclidean domainIn mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...
s ⊃ fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
s
The absence of zero divisor
Zero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s means that in an integral domain the cancellation property
Cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.An element a in a magma has the left cancellation property if for all b and c in M, a * b = a * c always implies b = c.An element a in a magma has the right cancellation...
holds for multiplication by any nonzero element a: an equality implies .
Definitions
There are a number of equivalent definitions of integral domain:- An integral domain is a commutative ring with identity in which for any two elements a and b, the equality ab = 0 implies either a = 0 or b = 0.
- An integral domain is a commutative ring with identity in which the zero idealIdeal (ring theory)In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
{0} is a prime idealPrime idealIn algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
. - An integral domain is a commutative ring with identity that is a subring of a field.
- An integral domain is a commutative ring with identity in which for every non-zero element r, the function that maps every element x of the ring to the product xr is injective. Elements that have this property are called regular, so it is equivalent to require that every non-zero element of the ring be regular.
Examples
- The prototypical example is the ring Z of all integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s. - Every fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
is an integral domain. Conversely, every ArtinianArtinian ringIn abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
integral domain is a field. In particular, all finite integral domains are finite fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s (more generally, by Wedderburn's little theoremWedderburn's little theoremIn mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields.The Artin–Zorn theorem generalizes the theorem to alternative rings....
, finite domainsDomain (ring theory)In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
are finite fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
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:
- Rings of polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
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 realReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
coefficients. - For each integer n > 1, the set of all real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s of the form a + b√n with a and b integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s is a subring of R and hence an integral domain. - For each integer n > 0 the set of all complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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 integerGaussian integerIn number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
s. - The p-adic integersP-adic numberIn mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
. - If U is a connectedConnectednessIn mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...
open subset of the complex number planeComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
C, then the ring H(U) consisting of all holomorphic functionHolomorphic functionIn mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s f : U → C is an integral domain. The same is true for rings of analytic functionAnalytic functionIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
s on connected open subsets of analytic manifoldManifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s. - If R is a commutative ring and P is an idealIdeal (ring theory)In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in R, then the factor ring R/P is an integral domain if and only if P is a prime idealPrime idealIn algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
. Also, R is an integral domain if and only if the ideal (0) is a prime ideal. - A regular local ringRegular local ringIn commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...
is an integral domain. In fact, a regular local ring is a UFDUnique factorization domainIn mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
.
The following rings are not integral domains.
- The ring of n×n matricesMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
over any non-trivial ring when n ≥ 2. - The ring of continuous functionContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s on the unit intervalUnit intervalIn mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
. - The quotient ringQuotient ringIn ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
Z/m when m is a composite numberComposite numberA composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
. - The commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
with a multiplicative identity Z×Z.
Divisibility, prime and irreducible elements
If a and b are elements of the integral domain R, we say that a divides b or a is a divisorDivisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
of b or b is a multiple of a if and only if there exists an element x in R such that ax = b.
The elements which divide 1 are called the unit
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
s of R; these are precisely the invertible elements in R. Units divide all other elements.
If a divides b and b divides a, then we say a and b are associated elements or associates.
If q is a non-unit, we say that q is an irreducible element
Irreducible element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.Irreducible elements should not be confused with prime elements...
if q cannot be written as a product of two non-units.
If p is a non-zero 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. Equivalent, an element is prime if and only if an ideal generated by it is a nonzero prime ideal. Every prime element is irreducible. Conversely, in a GCD domain
GCD domain
In mathematics, a GCD domain is an integral domain R with the property that any two non-zero elements have a greatest common divisor . Equivalently, any two non-zero elements of R have a least common multiple ....
(e.g., a unique factorization domain), an irreducible element is a prime element.
The notion of prime element
Prime element
In abstract algebra, an element p of a commutative ring R is said to be prime if it is not zero, not a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b...
generalizes the ordinary definition of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
in the ring Z, except that it allows for negative prime elements. While every prime is irreducible, the converse is not in general true. For example, in the quadratic integer
Quadratic integer
In number theory, quadratic integers are a generalization of the rational integers to quadratic fields. Important examples include the Gaussian integers and the Eisenstein integers. Though they have been studied for more than a hundred years, many open problems remain.- Definition :Quadratic...
ring
the number 3 is irreducible, but is not a prime because 9, the norm of 3, can be factored in two ways in the ring, namely, and . Thus , but 3 does not divide nor The numbers 3 and are irreducible as there is no where or as has no integer solution.While unique factorization
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
does not hold in the above example, if we use ideals
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
we do get unique factorization. Namely, the ideal
equals the ideals and and is the unique product of the two prime ideals: , each of which have a norm of 3.Properties
- 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.
- The cancellation property holds in integral domains. That is, let a, b, and c belong to an integral domain. If a ≠ 0 and ab = ac then b = c. Another way to state this is that the function x ax is injective for any non-zero a in the domain.
- An integral domain is equal to the intersection of its localizations at maximal ideals.
Field of fractions
If R is a given integral domain, the smallest field containing R as a subring is uniquely determined up to isomorphism and is called the field of fractions or quotient field of R. It can be thought of as consisting of all fractions a/b with a and b in R and b ≠ 0, modulo an appropriate equivalence relation. The field of fractions of the integers is the field of rational numberRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s. The field of fractions of a field is isomorphic
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to the field itself.
Algebraic geometry
In algebraic geometry, integral domains correspond to irreducibleIrreducible element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.Irreducible elements should not be confused with prime elements...
varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
. They have a unique generic point
Generic point
In mathematics, in the fields general topology and particularly of algebraic geometry, a generic point P of a topological space X is an algebraic way of capturing the notion of a generic property: a generic property is a property of the generic point.- Definition and motivation :A generic point of...
, given by the zero ideal. Integral domains are also characterized by the condition that they are reduced
Reduced ring
In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0...
and irreducible
Irreducible ring
In mathematics, especially in the field of commutative algebra, an irreducible ring is a commutative ring A that satisfies one of the following equivalent conditions:* A possesses exactly 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, hence such rings are integral domains. The converse is clear: No integral domain can have nilpotent elements, and the zero ideal is the unique minimal prime ideal.
Characteristic and homomorphisms
The characteristicCharacteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
of every integral domain is either zero or a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
.
If R is an integral domain with prime characteristic p, then f(x) = x p defines an injective ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
f : R → R, the Frobenius endomorphism
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...
.
See also
- Integral domains – wikibook link
- Dedekind–Hasse normDedekind–Hasse normIn mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains.-Definition:...
– the extra structure needed for an integral domain to be principal - Zero-product propertyZero-product propertyIn the mathematical areas of algebra and analysis, the zero-product property, generally known as the nonexistence of zero divisors, and also called the zero-product rule, the rule of zero product, or any other similar name, is an abstract and explicit statement of the familiar property from...