ENCYCLOPEDIA 4U .com

 Web Encyclopedia4u.com

# Integral domain

In abstract algebra, an integral domain, is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility.

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.

 Table of contents 1 Examples 2 Divisibility, prime and irreducible elements 3 Field of fractions 4 Characteristic and homomorphisms

### 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, the only finite integral domains are the finite fields.

Rings of polynomials 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 coefficients .

The set of all real numbers of the form a + b√2 with a and b integers is a subring of R and hence an integral domain. A similar example is given by the complex numbers of the form a + bi with a and b integers (the Gaussian 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 analytical functions on connected open subsets of analytical manifolds.

If R is a commutative ring and P is a prime ideal in R, then the factor ring is an integral domain.

### Field of fractions

If R is a given integral domain, the smallest field Quot(R) containing R as a subring is uniquely determined up to isomorphism and is called the field of fractions or quotient field of R. It consists of all fractions a/b'\' with a and b in R and b'' ≠ 0. The field of fractions of the integers is the field of rational numbers. The field of fractions of a field is that field itself.

### 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) = xp defines an injective ring homomorphism f : R -> R, the Frobenius homomorphism.

Content on this web site is provided for informational purposes only. We accept no responsibility for any loss, injury or inconvenience sustained by any person resulting from information published on this site. We encourage you to verify any critical information with the relevant authorities.