In
mathematics, specifically
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, an integral domain is a
nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s and provide a natural setting for studying
divisibility
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
. In an integral domain, every nonzero element ''a'' has the
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 (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that .
An ...
, that is, if , an equality implies .
"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a
multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "
domain" for the general case including noncommutative rings.
Some sources, notably
Lang
Lang may refer to:
* Lang (surname), a surname of independent Germanic or Chinese origin
Places
* Lang Island (Antarctica), East Antarctica
* Lang Nunatak, Antarctica
* Lang Sound, Antarctica
* Lang Park, a stadium in Brisbane, Australia
* L ...
, use the term entire ring for integral domain.
Some specific kinds of integral domains are given with the following chain of
class inclusions:
Definition
An ''integral domain'' is a
nonzero commutative ring in which the product of any two nonzero elements is nonzero. Equivalently:
* An integral domain is a nonzero commutative ring with no nonzero
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...
s.
* An integral domain is a commutative ring in which the
zero ideal
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identi ...
is a
prime ideal.
* An integral domain is a nonzero commutative ring for which every non-zero element is
cancellable under multiplication.
* An integral domain is a ring for which the set of nonzero elements is a commutative
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
under multiplication (because a monoid must be
closed under multiplication).
* An integral domain is a nonzero commutative ring in which for every nonzero element ''r'', the function that maps each element ''x'' of the ring to the product ''xr'' is
injective. Elements ''r'' with this property are called ''regular'', so it is equivalent to require that every nonzero element of the ring be regular.
* An integral domain is a ring that is
isomorphic to a
subring of a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. (Given an integral domain, one can embed it in its
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
.)
Examples
* The archetypical example is the ring
of all
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s.
* Every
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
is an integral domain. For example, the field
of all
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s is an integral domain. Conversely, every
Artinian integral domain is a field. In particular, all finite integral domains are
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s (more generally, by
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, division rings and fields.
The Artin–Zorn theorem generalizes the theorem to al ...
, finite
domains are
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s). The ring of integers
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
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s are integral domains if the coefficients come from an integral domain. For instance, the ring