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Integral domain

Overview

In abstract algebra

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....

without 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. Right zero divisors are defined analogously, that is, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0...

s and with a multiplicative identity 1 not equal to 0, the additive identity. Integral domains are generalizations of the integer

Integer

The integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....

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 with 0 ≠ 1 such that ab = 0 implies that either a = 0 or b = 0 . That is, it is a nontrivial ring without left or right zero divisors...

.

Alternatively and equivalently, an integral domain may be defined as a commutative ring (with unit) in which the zero ideal

Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3"....

{0} is prime

Prime ideal

In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. This article only covers ideals of ring theory...

, or as a subring

Subring

In mathematics, a subring is a subset of a ring, which contains the multiplicative identity and is itself a ring under the same binary operations. Naturally, those authors who do not require rings to contain a multiplicative identity do not require subrings to possess the identity...

of a field

Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

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Encyclopedia

In abstract algebra

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....

without zero divisor

Zero divisor

s and with a multiplicative identity 1 not equal to 0, the additive identity. Integral domains are generalizations of the integer

Integer

The integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....

s and provide a natural setting for studying divisibility. An integral domain is a commutative domain

Domain (ring theory)

.

Alternatively and equivalently, an integral domain may be defined as a commutative ring (with unit) in which the zero ideal

Ideal (ring theory)

{0} is prime

Prime ideal

In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. This article only covers ideals of ring theory...

, or as a subring

Subring

of a field

Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

. Additionally, a commutative ring with unit R is an integral domain if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...

for every non-zero element r of the ring, the R-module map induced by multiplication by r is injective (such r are called regular).

Viewing the underlying commutative ring as a preadditive category

Preadditive category

In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...

, the above criterion on zero divisors is equivalent to the condition that every nonzero morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory...

is a monomorphism

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation ....

(hence also an epimorphism

Epimorphism

In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...

, by making use of the bilinear

Bilinear

Bilinear may refer to* Bilinear sampling, a method in computer graphics for choosing the color of a texture.* Bilinear form* Bilinear interpolation* Bilinear map, a type of mathematical function between vector spaces...

structure on the set of morphisms).

The condition 0 ≠ 1 only serves to exclude the trivial ring

Trivial ring

In mathematics, a trivial ring is a ring defined on a singleton set, {r}. The ring operations are trivial:One often refers to the trivial ring since every trivial ring is isomorphic to any other...

{0}.

A few sources talk about noncommutative integral domains, but we follow the much more usual convention of reserving the term integral domain for the commutative case and use domain

Domain (ring theory)

for the noncommutative case. 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

Subclass (set theory)

In set theory and its applications throughout mathematics, a subclass is a class contained in some other class in the same way that a subset is a set contained in some other set....

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....

s ⊃ integral domains ⊃ unique factorization domain
Unique factorization domain
In 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 domain
Principal ideal domain
In 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 ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as principal rings...

s ⊃ Euclidean domain
Euclidean domain
In 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 ⊃ field
Field (mathematics)
In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

s

Examples

The prototypical example is the ring Z of all integer
Integer
The integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....

s.
Every field
Field (mathematics)
In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

is an integral domain. Conversely, every Artinian
Artinian ring
In 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 fields
Finite field
In abstract algebra, a finite field or Galois field is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

(more generally, by Wedderburn's little theorem
Wedderburn's little theorem
In 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....

, finite domains
Domain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring with 0 ≠ 1 such that ab = 0 implies that either a = 0 or b = 0 . That is, it is a nontrivial ring without left or right zero divisors...

are finite fields
Finite field
In abstract algebra, a finite field or Galois field is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

). 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 polynomial
Polynomial
In mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number 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 real
Real number
In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...

coefficients.
For each integer n > 1, the set of all real number
Real number
In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...

s of the form a + b√n with a and b integer
Integer
The integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....

s is a subring of R and hence an integral domain.
For each integer n > 0 the set of all complex number
Complex number
A complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i² = −1...

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
Gaussian integer
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]...

s.
The p-adic integers
P-adic number
In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. For each prime number p, the p-adic number system 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 connected
Connectedness
In 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 plane
Complex number
A complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i² = −1...

C, then the ring H(U) consisting of all holomorphic function
Holomorphic function
In 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 function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. Analytic functions can be thought of as a bridge between polynomials and general functions. There exist both real analytic functions and complex analytic functions, categories that are similar in...

s on connected open subsets of analytic manifold
Manifold
In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....

s.
If R is a commutative ring and P is an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way 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 ideal
Prime ideal
In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. This article only covers ideals of ring theory...

. Also, R is an integral domain if and only if the ideal (0) is a prime ideal.
A regular local ring
Regular local ring
In 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 exactly the same as its Krull dimension. The minimal number of generators of the maximal ideal is always bounded below by the Krull dimension...

is an integral domain. A deep theorem of Auslander-Buchsbaum formula
Depth (ring theory)
In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its...

and Nagata
Masayoshi Nagata
Masayoshi Nagata was a Japanese mathematician, known for his work in the field of commutative algebra....

from the 1950s claims that, in fact, a regular local ring is a UFD
Unique factorization domain
In 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 matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, such asEntries of a matrix are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied...

over any non-trivial ring when n ≥ 2.
The ring of continuous function
Continuous function
In 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...

s on the unit interval
Unit interval
In mathematics, the unit interval is the closed interval [0,1], 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 ring
Quotient ring
In mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in ring theory, quite similar to the factor groups of group theory and the quotient spaces of linear algebra...

Z/m when m is a composite number
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words, if n > 0 is an integer and there are integers 1 < a, b < n such that n = a × b then n is composite. By definition, every integer greater than one is either a prime number or a composite...

.
The 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....

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 divisor
Divisor
In mathematics, a divisor of an integer , also called a factor of , is an integer which evenly divides 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.

If a divides b and b divides c, then a divides c. If a divides b, then a divides every multiple of b. If a divides two elements, then a also divides their sum and difference.

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 is 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. a and b are associated if and only if there exists a unit u such that au = b.

If q is a non-unit, we say that q is an irreducible element
Irreducible element
In mathematics, a non-unit in an integral domain is said to be irreducible if it is not a product of two non-units. Equivalently, a non-unit x is irreducible if x ≠ 0 and every divisor d of x is associated to either 1 or x. Note this is the usual definition of a prime number.Every prime element is...

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.

This generalizes the ordinary definition of prime number

Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...

in the ring Z, except that it allows for negative prime elements. If p is a prime element, then the principal ideal (p) generated by p is a prime ideal. Every prime element is irreducible (here, for the first time, we need R to be an integral domain), but the converse is not true in all integral domains (it is true in unique factorization domain

Unique factorization domain

In 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, however). For example, in the quadratic integer

Quadratic integer

In mathematics, especially 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...

ring the number 3 is irreducible but is not a prime because 9 can be written as and .

Being prime is also relative to which ring an element is considered to be in; for example, 2 is a prime element in Z but it is not in Z[i], the ring of Gaussian integer

Gaussian integer

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]...

s, since .

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. (Recall from vector algebra that a transformation
Linear transformation
In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself...

T is injective if and only if its null space
Null space
In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0...

consists of 0 alone. It is therefore possible to have a ring-isomorphic module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring...

with non-injective T — if the ring is not an integral 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
Field of fractions
In mathematics, the field of fractions or field of quotients of a zero-divisor free commutative ring with more than one element is the smallest field in which it can be embedded. It is common to define the field of fractions only for an integral domain, but in fact it exists if and only if the ring...

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 number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...

s. The field of fractions of a field is isomorphic

Isomorphism

In abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f⁻¹ are homomorphisms, i.e., structure-preserving mappings....

to the field itself.

Algebraic geometry

In algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...

, integral domains correspond to irreducible

Irreducible element

In mathematics, a non-unit in an integral domain is said to be irreducible if it is not a product of two non-units. Equivalently, a non-unit x is irreducible if x ≠ 0 and every divisor d of x is associated to either 1 or x. Note this is the usual definition of a prime number.Every prime element is...

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 classical 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 :Formally, a generic point is a...

, given by the zero ideal. Integral domains are also characterized by the condition that they are reduced

Reduced ring

In ring theory, a commutative ring R is called a reduced ring if it has no non-zero nilpotent elements. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is 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 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 ideal.

Characteristic and homomorphisms

The characteristic

Characteristic (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

.

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, which are branches of mathematics, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, a class importantly including fields. The endomorphism maps every element to its pth power...

Integral domain

Examples

Divisibility, prime and irreducible elements

Properties

Field of fractions

Algebraic geometry

Characteristic and homomorphisms

See also