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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 set ...
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 an ...
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 In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of ''K''. Then ''x''∈''L'' is
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
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 philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
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 "upwar ...
. The theorem states that if ''A''⊆''B'' is an
integral extension 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'' i ...
of domains and ''A'' is an integrally closed domain, then the
going-down property 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 i ...
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 principal, ...
(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 an ...
(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 fini ...
or valuation domain). *A
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
. *A
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
over a field (since every symmetric algebra is isomorphic to a polynomial ring in several variables over a field). *Let k be a field of characteristic not 2 and S = k _1, \dots, x_n/math> a polynomial ring over it. If f is a
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...
nonconstant polynomial in S, then S (y^2 - f) is an integrally closed domain. In particular, k _0, \dots, x_r(x_0^2 + \dots + x_r^2) is an integrally closed domain if r \ge 2. To give a non-example, let ''k'' be a field and A = k ^2, t^3\subset k /math> (''A'' is the subalgebra generated by ''t''2 and ''t''3.) ''A'' is not integrally closed: it has the field of fractions k(t), and the monic polynomial X^2 - t^2 in the variable ''X'' has root ''t'' which is in the field of fractions but not in ''A.'' This is related to the fact that the plane curve Y^2 = X^3 has a singularity at the origin. Another domain which is not integrally closed is A = \mathbb
sqrt In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
/math>; it does not contain the element \frac of its field of fractions, which satisfies the monic polynomial X^2-X-1 = 0.


Noetherian integrally closed domain

For a noetherian local domain ''A'' of dimension one, the following are equivalent. *''A'' is integrally closed. *The maximal ideal of ''A'' is principal. *''A'' is a
discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' ...
(equivalently ''A'' is Dedekind.) *''A'' is a regular local ring. Let ''A'' be a noetherian integral domain. Then ''A'' is integrally closed if and only if (i) ''A'' is the intersection of all localizations A_\mathfrak over prime ideals \mathfrak of height 1 and (ii) the localization A_\mathfrak at a prime ideal \mathfrak of height 1 is a discrete valuation ring. A noetherian ring is a
Krull domain In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimensional generalization of Dedekind domains, which a ...
if and only if it is an integrally closed domain. In the non-noetherian setting, one has the following: an integral domain is integrally closed if and only if it is the intersection of all
valuation ring 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'' suc ...
s containing it.


Normal rings

Authors including Serre, Grothendieck, and Matsumura define a normal ring to be a ring whose localizations at prime ideals are integrally closed domains. Such a ring is necessarily a
reduced ring In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' = ...
, and this is sometimes included in the definition. In general, if ''A'' is a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
ring whose localizations at maximal ideals are all domains, then ''A'' is a finite product of domains. In particular if ''A'' is a Noetherian, normal ring, then the domains in the product are integrally closed domains. Conversely, any finite product of integrally closed domains is normal. In particular, if \operatorname(A) is noetherian, normal and connected, then ''A'' is an integrally closed domain. (cf.
smooth variety In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smo ...
) Let ''A'' be a noetherian ring. Then ( Serre's criterion) ''A'' is normal if and only if it satisfies the following: for any prime ideal \mathfrak,
  1. If \mathfrak has height \le 1, then A_\mathfrak is regular (i.e., A_\mathfrak is a
    discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' ...
    .)
  2. If \mathfrak has height \ge 2, then A_\mathfrak has depth \ge 2.
Item (i) is often phrased as "regular in codimension 1". Note (i) implies that the set of
associated prime In abstract algebra, an associated prime of a module ''M'' over a ring ''R'' is a type of prime ideal of ''R'' that arises as an annihilator of a (prime) submodule of ''M''. The set of associated primes is usually denoted by \operatorname_R(M), ...
s Ass(A) has no embedded primes, and, when (i) is the case, (ii) means that Ass(A/fA) has no embedded prime for any non-zerodivisor ''f''. In particular, a Cohen-Macaulay ring satisfies (ii). Geometrically, we have the following: if ''X'' is a local complete intersection in a nonsingular variety; e.g., ''X'' itself is nonsingular, then ''X'' is Cohen-Macaulay; i.e., the stalks \mathcal_p of the structure sheaf are Cohen-Macaulay for all prime ideals p. Then we can say: ''X'' is
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
(i.e., the stalks of its structure sheaf are all normal) if and only if it is regular in codimension ''1''.


Completely integrally closed domains

Let ''A'' be a domain and ''K'' its field of fractions. An element ''x'' in ''K'' is said to be almost integral over ''A'' if the subring ''A'' 'x''of ''K'' generated by ''A'' and ''x'' is a
fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral doma ...
of ''A''; that is, if there is a d \ne 0 such that d x^n \in A for all n \ge 0. Then ''A'' is said to be completely integrally closed if every almost integral element of ''K'' is contained in ''A''. A completely integrally closed domain is integrally closed. Conversely, a noetherian integrally closed domain is completely integrally closed. Assume ''A'' is completely integrally closed. Then the formal power series ring A X is completely integrally closed. This is significant since the analog is false for an integrally closed domain: let ''R'' be a valuation domain of height at least 2 (which is integrally closed.) Then R X is not integrally closed. Let ''L'' be a field extension of ''K''. Then the integral closure of ''A'' in ''L'' is completely integrally closed. An integral domain is completely integrally closed if and only if the monoid of divisors of ''A'' is a group. See also:
Krull domain In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimensional generalization of Dedekind domains, which a ...
.


"Integrally closed" under constructions

The following conditions are equivalent for an integral domain ''A'': # ''A'' is integrally closed; # ''A''''p'' (the localization of ''A'' with respect to ''p'') is integrally closed for every
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
''p''; # ''A''''m'' is integrally closed for every
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
''m''. 1 → 2 results immediately from the preservation of integral closure under localization; 2 → 3 is trivial; 3 → 1 results from the preservation of integral closure under localization, the exactness of localization, and the property that an ''A''-module ''M'' is zero if and only if its localization with respect to every maximal ideal is zero. In contrast, the "integrally closed" does not pass over quotient, for Z (t2+4) is not integrally closed. The localization of a completely integrally closed domain need not be completely integrally closed. A direct limit of integrally closed domains is an integrally closed domain.


Modules over an integrally closed domain

Let ''A'' be a Noetherian integrally closed domain. An ideal ''I'' of ''A'' is divisorial if and only if every
associated prime In abstract algebra, an associated prime of a module ''M'' over a ring ''R'' is a type of prime ideal of ''R'' that arises as an annihilator of a (prime) submodule of ''M''. The set of associated primes is usually denoted by \operatorname_R(M), ...
of ''A''/''I'' has height one. Let ''P'' denote the set of all prime ideals in ''A'' of height one. If ''T'' is a finitely generated torsion module, one puts: :\chi(T) = \sum_ \operatorname_p(T) p, which makes sense as a formal sum; i.e., a divisor. We write c(d) for the divisor class of ''d''. If F, F' are maximal submodules of ''M'', then c(\chi(M/F)) = c(\chi(M/F')) and c(\chi(M/F)) is denoted (in Bourbaki) by c(M).


See also

*
Unibranch local ring In algebraic geometry, a local ring ''A'' is said to be unibranch if the reduced ring ''A''red (obtained by quotienting ''A'' by its nilradical) is an integral domain, and the integral closure ''B'' of ''A''red is also a local ring. A unibranch l ...


Citations


References

* * * * * {{refend Commutative algebra