# 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.

The condition 0≠1 only serves to exclude the trivial ring {0} with a single element.

Table of contents |

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 an ideal in *R*, then the factor ring *R/P* is an integral domain if and only if *P* 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 either *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 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*) = *x*^{ p} defines an injective ring homomorphism *f* : *R* `->` *R*, the *Frobenius homomorphism*.