In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, an integrally closed domain ''A'' is an
integral domain
In mathematics, specifically abstract algebra, 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 of integers and provide a natural se ...
whose
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' is ...
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 ...
is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root of a
monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\cd ...
with coefficients in ''A,'' then ''x'' is itself an element of ''A.'' Many well-studied domains are integrally closed:
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 ...
s, the ring of integers Z,
unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
s and
regular local rings are all integrally closed.
Note that integrally closed domains appear in the following chain of
class inclusions:
Basic properties
Let ''A'' be an integrally closed domain with field of fractions ''K'' and let ''L'' be a
field extension of ''K''. Then ''x''∈''L'' is
integral over ''A'' if and only if it is
algebraic over ''K'' and its
minimal polynomial over ''K'' has coefficients in ''A''. In particular, this means that any element of ''L'' integral over ''A'' is root of a monic polynomial in ''A''
'X''that is
irreducible in ''K''
'X''
If ''A'' is a domain contained in a field ''K,'' we can consider the
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' is ...
of ''A'' in ''K'' (i.e. the set of all elements of ''K'' that are integral over ''A''). This integral closure is an integrally closed domain.
Integrally closed domains also play a role in the hypothesis of the
Going-down theorem In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions.
The phrase going up refers to the case when a chain can be extended by "upward ...
. The theorem states that if ''A''⊆''B'' is an
integral extension of domains and ''A'' is an integrally closed domain, then the
going-down property holds for the extension ''A''⊆''B''.
Examples
The following are integrally closed domains.
*A
principal ideal domain
In mathematics, 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 ideal ring is a nonzero commutative ring whose ideals are princi ...
(in particular: the integers and any field).
*A
unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
(in particular, any polynomial ring over a field, over the integers, or over any unique factorization domain).
*A
GCD domain (in particular, any
Bézout domain In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every fin ...
or
valuation domain In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' such t ...
).
*A
Dedekind domain.
*A
symmetric algebra over a field (since every symmetric algebra is isomorphic to a polynomial ring in several variables over a field).
*Let
be a field of characteristic not 2 and