Integral domain

In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero (the zero-product property); that is, there are no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain.

Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero ideal {0} is prime, or as the subrings of fields. Additionally, a commutative ring R is an integral domain if and only if for every 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, the above criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism, by making use of the bilinear structure on the set of morphisms).

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

A few sources talk about noncommutative integral domains, but in Wikipedia we reserve the term integral domain for the commutative case and use domain for the noncommutative case.

Examples

The prototypical example is the ring Z of all integers.

Every field is an integral domain. Conversely, every Artinian integral domain is a field. In particular, all finite integral domains are finite fields. The ring of integers Z provides an example of an non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:

\mathbf{Z}\;\supset\;2\mathbf{Z}\;\supset\;\ldots\;\supset\;2^n\mathbf{Z}\;\supset\;2^{n+1}\mathbf{Z}\;\supset\;\cdots

Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Zof 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 coefficients.

For each integer $n>1\;$ , the set of all real numbers of the form $a+b\sqrt{n}\;$ with a and b integers is a subring of R and hence an integral domain.

For each integer $n>0\;$ , the set of all complex numbers of the form $a+bi\sqrt{n}\;$ 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 integers.

The p-adic integers.

If U is a connected open subset of the complex number plane C, then the ring H(U) consisting of all holomorphic functions f : U -> C is an integral domain. The same is true for rings of analytic functions on connected open subsets of analytic manifolds.

If R is a commutative ring and P is an ideal in R, then the factor ring R/P is an integral domain if and only if P is a prime ideal. Also, R is an integral domain if and only if the ideal (0) is a prime ideal.

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 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 units 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. 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 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 b.

This generalizes the ordinary definition of 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 domains, however).

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 numbers. The field of fractions of a field is isomorphic to the field itself.

Algebraic geometry

In algebraic geometry, integral domains correspond to irreducible 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 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 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.

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Examples

Divisibility, prime and irreducible elements

Field of fractions

Algebraic geometry

Characteristic and homomorphisms

See also