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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 element ''b'' of a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
''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 a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
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 ...
over ''A''. The set of elements of ''B'' that are integral over ''A'' is called the integral closure of ''A'' in ''B''. It is a subring of ''B'' containing ''A''. If every element of ''B'' is integral over ''A'', then we say that ''B'' is integral over ''A'', or equivalently ''B'' is an integral extension of ''A''. If ''A'', ''B'' are
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
, then the notions of "integral over" and of an "integral extension" are precisely " algebraic over" and "
algebraic extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
s" in field theory (since the root of any
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 exa ...
is the root of a monic polynomial). The case of greatest interest in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
is that of
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
integral over Z (e.g., \sqrt or 1+i); in this context, the integral elements are usually called
algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s. The algebraic integers in a finite
extension field 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 ...
''k'' of the
rationals In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
Q form a subring of ''k'', called the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
of ''k'', a central object of study in
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
. In this article, the term ''
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 ...
'' will be understood to mean ''commutative ring'' with a multiplicative identity.


Examples


Integral closure in algebraic number theory

There are many examples of integral closure which can be found in algebraic number theory since it is fundamental for defining the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
for an
algebraic field extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
K/\mathbb (or L/\mathbb_p).


Integral closure of integers in rationals

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 language ...
s are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q.


Quadratic extensions

The
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s are the complex numbers of the form a + b \sqrt,\, a, b \in \mathbf, and are integral over Z. \mathbf
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> is then the integral closure of Z in \mathbf(\sqrt). Typically this ring is denoted \mathcal_. The integral closure of Z in \mathbf(\sqrt) is the ring :\mathcal_ = \mathbb\!\left \frac \right/math> This example and the previous one are examples of
quadratic integer In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and (usual) integers. When algebra ...
s. The integral closure of a quadratic extension \mathbb(\sqrt) can be found by constructing the minimal polynomial of an arbitrary element a + b \sqrt and finding number-theoretic criterion for the polynomial to have integral coefficients. This analysis can be found in the quadratic extensions article.


Roots of unity

Let ζ be a
root of unity In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
. Then the integral closure of Z in the
cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
Q(ζ) is Z This can be found by using the minimal polynomial and using
Eisenstein's criterion In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with ...
.


Ring of algebraic integers

The integral closure of Z in the field of complex numbers C, or the algebraic closure \overline is called the ''ring of
algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s''.


Other

The
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
,
nilpotent element In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
s and
idempotent element Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
s in any ring are integral over Z.


Integral closure in geometry

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, integral closure is closely related with normalization and
normal scheme In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and ...
s. It is the first step in resolution of singularities since it gives a process for resolving singularities of codimension 1. * For example, the integral closure of \mathbb ,y,z(xy) is the ring \mathbb ,z\times \mathbb ,z/math> since geometrically, the first ring corresponds to the xz-plane unioned with the yz-plane. They have a codimension 1 singularity along the z-axis where they intersect. *Let a
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
''G'' act on a ring ''A''. Then ''A'' is integral over ''A''''G'', the set of elements fixed by ''G''; see
Ring of invariants In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is, :R^f = \. More generally, if ''G'' is a group acting on ''R'', then the subring of ''R'' :R^G = \ is called the ...
. *Let ''R'' be a ring and ''u'' a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
in a ring containing ''R''. Then #''u''−1 is integral over ''R''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
''u''−1 ∈ ''R'' 'u'' #R \cap R ^/math> is integral over ''R''. #The integral closure of the
homogeneous coordinate ring In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X'N'' ...
of a normal
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
''X'' is the
ring of sections This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...
::\bigoplus_ \operatorname^0(X, \mathcal_X(n)).


Integrality in algebra

* If \overline is an
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of a field ''k'', then \overline _1, \dots, x_n/math> is integral over k _1, \dots, x_n * The integral closure of C ''x'' in a finite extension of C((''x'')) is of the form \mathbf x^ (cf.
Puiseux series In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + ...
)


Equivalent definitions

Let ''B'' be a ring, and let ''A'' be a subring of ''B''. Given an element ''b'' in ''B'', the following conditions are equivalent: :(i) ''b'' is integral over ''A''; :(ii) the subring ''A'' 'b''of ''B'' generated by ''A'' and ''b'' is a finitely generated ''A''-module; :(iii) there exists a subring ''C'' of ''B'' containing ''A'' 'b''and which is a finitely generated ''A''-module; :(iv) there exists a faithful ''A'' 'b''module ''M'' such that ''M'' is finitely generated as an ''A''-module. The usual proof of this uses the following variant of the
Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...
on
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
s: :Theorem Let ''u'' be an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
of an ''A''-module ''M'' generated by ''n'' elements and ''I'' an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of ''A'' such that u(M) \subset IM. Then there is a relation: ::u^n + a_1 u^ + \cdots + a_ u + a_n = 0, \, a_i \in I^i. This theorem (with ''I'' = ''A'' and ''u'' multiplication by ''b'') gives (iv) ⇒ (i) and the rest is easy. Coincidentally, Nakayama's lemma is also an immediate consequence of this theorem.


Elementary properties


Integral closure forms a ring

It follows from the above four equivalent statements that the set of elements of B that are integral over A forms a subring of ''B'' containing A. (Proof: If ''x'', ''y'' are elements of ''B'' that are integral over A, then x + y, xy, -x are integral over A since they stabilize A /math>, which is a finitely generated module over A and is annihilated only by zero.) This ring is called the integral closure of A in B.


Transitivity of integrality

Another consequence of the above equivalence is that "integrality" is transitive, in the following sense. Let C be a ring containing B and c \in C. If c is integral over ''B'' and ''B'' integral over A, then c is integral over A. In particular, if C is itself integral over ''B'' and ''B'' is integral over A, then C is also integral over A.


Integral closed in fraction field

If A happens to be the integral closure of A in ''B'', then ''A'' is said to be integrally closed in ''B''. If B is the
total ring of fractions In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embeds ...
of A, (e.g., the
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 ...
when 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 ...
), then one sometimes drops the qualification "in B " and simply says "integral closure of A" and "A is integrally closed." For example, the ring of integers \mathcal_K is integrally closed in the field K.


Transitivity of integral closure with integrally closed domains

Let ''A'' be an integral domain with the field of fractions ''K'' and ''A' '' the integral closure of ''A'' in an
algebraic field extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
''L'' of ''K''. Then the field of fractions of ''A' '' is ''L''. In particular, ''A' '' is an
integrally closed domain In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root ...
.


= Transitivity in algebraic number theory

= This situation is applicable in algebraic number theory when relating the ring of integers and a field extension. In particular, given a field extension L/K the integral closure of \mathcal_K in L is the ring of integers \mathcal_L.


Remarks

Note that transitivity of integrality above implies that if B is integral over A, then B is a union (equivalently an
inductive limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any catego ...
) of subrings that are finitely generated A-modules. If A is
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 ...
, transitivity of integrality can be weakened to the statement: :There exists a finitely generated A-submodule of B that contains A /math>.


Relation with finiteness conditions

Finally, the assumption that A be a subring of B can be modified a bit. If f:A \to B is a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...
, then one says f is integral if B is integral over f(A). In the same way one says f is finite (B finitely generated A-module) or of finite type (B finitely generated A-
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
). In this viewpoint, one has that :f is finite if and only if f is integral and of finite type. Or more explicitly, :B is a finitely generated A-module if and only if B is generated as an A-algebra by a finite number of elements integral over A.


Integral extensions


Cohen-Seidenberg theorems

An integral extension ''A'' âŠ† ''B'' has the going-up property, the
lying over 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 ...
property, and the incomparability property ( Cohen–Seidenberg theorems). Explicitly, given a chain of
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 ...
s \mathfrak_1 \subset \cdots \subset \mathfrak_n in ''A'' there exists a \mathfrak'_1 \subset \cdots \subset \mathfrak'_n in ''B'' with \mathfrak_i = \mathfrak'_i \cap A (going-up and lying over) and two distinct prime ideals with inclusion relation cannot contract to the same prime ideal (incomparability). In particular, the
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally t ...
s of ''A'' and ''B'' are the same. Furthermore, if ''A'' is an integrally closed domain, then the going-down holds (see below). In general, the going-up implies the lying-over. Thus, in the below, we simply say the "going-up" to mean "going-up" and "lying-over". When ''A'', ''B'' are domains such that ''B'' is integral over ''A'', ''A'' is a field if and only if ''B'' is a field. As a corollary, one has: given a prime ideal \mathfrak of ''B'', \mathfrak is a
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 ...
of ''B'' if and only if \mathfrak \cap A is a maximal ideal of ''A''. Another corollary: if ''L''/''K'' is an algebraic extension, then any subring of ''L'' containing ''K'' is a field.


Applications

Let ''B'' be a ring that is integral over a subring ''A'' and ''k'' an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
. If f: A \to k is a homomorphism, then ''f'' extends to a homomorphism ''B'' → ''k''. This follows from the going-up.


Geometric interpretation of going-up

Let f: A \to B be an integral extension of rings. Then the induced map :\begin f^\#: \operatorname B \to \operatorname A \\ p \mapsto f^(p)\end is a
closed map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
; in fact, f^\#(V(I)) = V(f^(I)) for any ideal ''I'' and f^\# is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
if ''f'' is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. This is a geometric interpretation of the going-up.


Geometric interpretation of integral extensions

Let ''B'' be a ring and ''A'' a subring that is a noetherian integrally closed domain (i.e., \operatorname A is a
normal scheme In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and ...
.) If ''B'' is integral over ''A'', then \operatorname B \to \operatorname A is submersive; i.e., the
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
of \operatorname A is the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
. The proof uses the notion of constructible sets. (See also:
Torsor (algebraic geometry) In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topolog ...
.)


Integrality, base-change, universally-closed, and geometry

If B is integral over A, then B \otimes_A R is integral over ''R'' for any ''A''-algebra ''R''. In particular, \operatorname (B \otimes_A R) \to \operatorname R is closed; i.e., the integral extension induces a "universally closed" map. This leads to a geometric characterization of integral extension. Namely, let ''B'' be a ring with only finitely many
minimal prime ideal In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. Definitio ...
s (e.g., integral domain or noetherian ring). Then ''B'' is integral over a (subring) ''A'' if and only if \operatorname (B \otimes_A R) \to \operatorname R is closed for any ''A''-algebra ''R''. In particular, every proper map is universally closed.


Galois actions on integral extensions of integrally closed domains

:Proposition. Let ''A'' be an integrally closed domain with the field of fractions ''K'', ''L'' a finite
normal extension In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic e ...
of ''K'', ''B'' the integral closure of ''A'' in ''L''. Then the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
G = \operatorname(L/K) acts transitively on each fiber of \operatorname B \to \operatorname A. Proof. Suppose \mathfrak_2 \ne \sigma(\mathfrak_1) for any \sigma in ''G''. Then, by
prime avoidance In algebra, the prime avoidance lemma says that if an ideal ''I'' in a commutative ring ''R'' is contained in a union of finitely many prime ideals ''P'is, then it is contained in ''P'i'' for some ''i''. There are many variations of th ...
, there is an element ''x'' in \mathfrak_2 such that \sigma(x) \not\in \mathfrak_1 for any \sigma. ''G'' fixes the element y = \prod\nolimits_ \sigma(x) and thus ''y'' is purely inseparable over ''K''. Then some power y^e belongs to ''K''; since ''A'' is integrally closed we have: y^e \in A. Thus, we found y^e is in \mathfrak_2 \cap A but not in \mathfrak_1 \cap A; i.e., \mathfrak_1 \cap A \ne \mathfrak_2 \cap A.


Application to algebraic number theory

The Galois group \operatorname(L/K) then acts on all of the prime ideals \mathfrak_1,\ldots, \mathfrak_k \in \text(\mathcal_L) lying over a fixed prime ideal \mathfrak \in \text(\mathcal_K). That is, if :\mathfrak = \mathfrak_1^\cdots\mathfrak_k^ \subset \mathcal_L then there is a Galois action on the set S_\mathfrak = \. This is called the
Splitting of prime ideals in Galois extensions In mathematics, the interplay between the Galois group ''G'' of a Galois extension ''L'' of a number field ''K'', and the way the prime ideals ''P'' of the ring of integers ''O'K'' factorise as products of prime ideals of ''O'L'', provides one ...
.


Remarks

The same idea in the proof shows that if L/K is a purely inseparable extension (need not be normal), then \operatorname B \to \operatorname A is
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
. Let ''A'', ''K'', etc. as before but assume ''L'' is only a finite field extension of ''K''. Then :(i) \operatorname B \to \operatorname A has finite fibers. :(ii) the going-down holds between ''A'' and ''B'': given \mathfrak_1 \subset \cdots \subset \mathfrak_n = \mathfrak'_n \cap A, there exists \mathfrak'_1 \subset \cdots \subset \mathfrak'_n that contracts to it. Indeed, in both statements, by enlarging ''L'', we can assume ''L'' is a normal extension. Then (i) is immediate. As for (ii), by the going-up, we can find a chain \mathfrak''_i that contracts to \mathfrak'_i. By transitivity, there is \sigma \in G such that \sigma(\mathfrak''_n) = \mathfrak'_n and then \mathfrak'_i = \sigma(\mathfrak''_i) are the desired chain.


Integral closure

Let ''A'' ⊂ ''B'' be rings and ''A' '' the integral closure of ''A'' in ''B''. (See above for the definition.) Integral closures behave nicely under various constructions. Specifically, for a
multiplicatively closed subset In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite ...
''S'' of ''A'', the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
''S''−1''A' '' is the integral closure of ''S''−1''A'' in ''S''−1''B'', and A' /math> is the integral closure of A /math> in B /math>. If A_i are subrings of rings B_i, 1 \le i \le n, then the integral closure of \prod A_i in \prod B_i is \prod ' where ' are the integral closures of A_i in B_i. The integral closure of a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
''A'' in, say, ''B'', need not be local. (If this is the case, the ring is called
unibranch 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 ...
.) This is the case for example when ''A'' is
Henselian In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now res ...
and ''B'' is a field extension of the field of fractions of ''A''. If ''A'' is a subring of a field ''K'', then the integral closure of ''A'' in ''K'' 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 of ''K'' containing ''A''. Let ''A'' be an \mathbb-graded subring of an \mathbb-
graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
''B''. Then the integral closure of ''A'' in ''B'' is an \mathbb-graded subring of ''B''. There is also a concept of the
integral closure of an ideal In algebra, the integral closure of an ideal ''I'' of a commutative ring ''R'', denoted by \overline, is the set of all elements ''r'' in ''R'' that are integral over ''I'': there exist a_i \in I^i such that :r^n + a_1 r^ + \cdots + a_ r + a_n = 0. ...
. The integral closure of an ideal I \subset R, usually denoted by \overline I, is the set of all elements r \in R such that there exists a monic polynomial :x^n + a_ x^ + \cdots + a_ x^1 + a_n with a_i \in I^i with r as a root. The
radical of an ideal In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is called ' ...
is integrally closed. For noetherian rings, there are alternate definitions as well. *r \in \overline I if there exists a c \in R not contained in any minimal prime, such that c r^n \in I^n for all n \ge 1. * r \in \overline I if in the normalized blow-up of ''I'', the pull back of ''r'' is contained in the inverse image of ''I''. The blow-up of an ideal is an operation of schemes which replaces the given ideal with a principal ideal. The normalization of a scheme is simply the scheme corresponding to the integral closure of all of its rings. The notion of integral closure of an ideal is used in some proofs 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 ...
.


Conductor

Let ''B'' be a ring and ''A'' a subring of ''B'' such that ''B'' is integral over ''A''. Then the annihilator of the ''A''-module ''B''/''A'' is called the ''conductor'' of ''A'' in ''B''. Because the notion has origin in
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, the conductor is denoted by \mathfrak = \mathfrak(B/A). Explicitly, \mathfrak consists of elements ''a'' in ''A'' such that aB \subset A. (cf.
idealizer In abstract algebra, the idealizer of a subsemigroup ''T'' of a semigroup ''S'' is the largest subsemigroup of ''S'' in which ''T'' is an Semigroup#Subsemigroups and ideals, ideal. Such an idealizer is given by :\mathbb_S(T)=\. In ring theory, if ...
in abstract algebra.) It is the largest
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of ''A'' that is also an ideal of ''B''. If ''S'' is a multiplicatively closed subset of ''A'', then :S^\mathfrak(B/A) = \mathfrak(S^B/S^A). If ''B'' is a subring of the
total ring of fractions In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embeds ...
of ''A'', then we may identify :\mathfrak(B/A)=\operatorname_A(B, A). Example: Let ''k'' be a field and let A = k ^2, t^3\subset B = k /math> (i.e., ''A'' is the
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
of the
affine curve Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
x^2 = y^3.) ''B'' is the integral closure of ''A'' in k(t). The conductor of ''A'' in ''B'' is the ideal (t^2, t^3) A. More generally, the conductor of A = k t^a, t^b, ''a'', ''b'' relatively prime, is (t^c, t^, \dots) A with c = (a-1)(b-1). Suppose ''B'' is the integral closure of an integral domain ''A'' in the field of fractions of ''A'' such that the ''A''-module B/A is finitely generated. Then the conductor \mathfrak of ''A'' is an ideal defining the support of B/A; thus, ''A'' coincides with ''B'' in the complement of V(\mathfrak) in \operatornameA. In particular, the set \, the complement of V(\mathfrak), is an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
.


Finiteness of integral closure

An important but difficult question is on the finiteness of the integral closure of a finitely generated algebra. There are several known results. The integral closure of 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 ...
in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring. This is a consequence of the
Krull–Akizuki theorem In algebra, the Krull–Akizuki theorem states the following: let ''A'' be a one-dimensional reduced noetherian ring, ''K'' its total ring of fractions. If ''B'' is a subring of a finite extension ''L'' of ''K'' containing ''A'' then ''B'' is a on ...
. In general, the integral closure of a noetherian domain of dimension at most 2 is noetherian; Nagata gave an example of dimension 3 noetherian domain whose integral closure is not noetherian. A nicer statement is this: the integral closure of a noetherian domain 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 ...
(
Mori–Nagata theorem In algebra, the Mori–Nagata theorem introduced by and , states the following: let ''A'' be a noetherian ring, noetherian reduced ring, reduced commutative ring with the total ring of fractions ''K''. Then the integral closure of ''A'' in ''K'' is ...
). Nagata also gave an example of dimension 1 noetherian local domain such that the integral closure is not finite over that domain. Let ''A'' be a noetherian integrally closed domain with field of fractions ''K''. If ''L''/''K'' is a finite separable extension, then the integral closure A' of ''A'' in ''L'' is a finitely generated ''A''-module. This is easy and standard (uses the fact that the trace defines a non-degenerate bilinear form.) Let ''A'' be a finitely generated algebra over a field ''k'' that is an integral domain with field of fractions ''K''. If ''L'' is a finite extension of ''K'', then the integral closure A' of ''A'' in ''L'' is a finitely generated ''A''-module and is also a finitely generated ''k''-algebra. The result is due to Noether and can be shown using the
Noether normalization lemma In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field ''k'', and any finitely generated commutative ''k''-algebra ''A'', there exists a non-negati ...
as follows. It is clear that it is enough to show the assertion when ''L''/''K'' is either separable or purely inseparable. The separable case is noted above, so assume ''L''/''K'' is purely inseparable. By the normalization lemma, ''A'' is integral over the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
S = k _1, ..., x_d/math>. Since ''L''/''K'' is a finite purely inseparable extension, there is a power ''q'' of a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
such that every element of ''L'' is a ''q''-th root of an element in ''K''. Let k' be a finite extension of ''k'' containing all ''q''-th roots of coefficients of finitely many rational functions that generate ''L''. Then we have: L \subset k'(x_1^, ..., x_d^). The ring on the right is the field of fractions of k' _1^, ..., x_d^/math>, which is the integral closure of ''S''; thus, contains A'. Hence, A' is finite over ''S''; a fortiori, over ''A''. The result remains true if we replace ''k'' by Z. The integral closure of a complete local noetherian domain ''A'' in a finite extension of the field of fractions of ''A'' is finite over ''A''. More precisely, for a local noetherian ring ''R'', we have the following chains of implications: :(i) ''A'' complete \Rightarrow ''A'' is a
Nagata ring In commutative algebra, an N-1 ring is an integral domain A whose integral closure in its quotient field is a finitely generated A-module. It is called a Japanese ring (or an N-2 ring) if for every finite extension L of its quotient field K, the ...
:(ii) ''A'' is a Nagata domain \Rightarrow ''A''
analytically unramified In algebra, an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent). The following rings are analytically unramified: * pseudo-geometric reduced ring. * excellent reduced ring. showed that every lo ...
\Rightarrow the integral closure of the completion \widehat is finite over \widehat \Rightarrow the integral closure of ''A'' is finite over A.


Noether's normalization lemma

Noether's normalisation lemma is a theorem 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 ...
. Given a field ''K'' and a finitely generated ''K''-algebra ''A'', the theorem says it is possible to find elements ''y''1, ''y''2, ..., ''y''''m'' in ''A'' that are
algebraically independent In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically in ...
over ''K'' such that ''A'' is finite (and hence integral) over ''B'' = ''K'' 'y''1,..., ''y''''m'' Thus the extension ''K'' ⊂ ''A'' can be written as a composite ''K'' ⊂ ''B'' ⊂ ''A'' where ''K'' ⊂ ''B'' is a purely transcendental extension and ''B'' ⊂ ''A'' is finite.


Integral morphisms

In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a morphism f:X \to Y of schemes is ''integral'' if it is
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
and if for some (equivalently, every) affine open cover U_i of ''Y'', every map f^(U_i)\to U_i is of the form \operatorname(A)\to\operatorname(B) where ''A'' is an integral ''B''-algebra. The class of integral morphisms is more general than the class of
finite morphism In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is ...
s because there are integral extensions that are not finite, such as, in many cases, the algebraic closure of a field over the field.


Absolute integral closure

Let ''A'' be an integral domain and ''L'' (some)
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of the field of fractions of ''A''. Then the integral closure A^+ of ''A'' in ''L'' is called the absolute integral closure of ''A''.
Melvin Hochster Melvin Hochster (born August 2, 1943) is an American mathematician working in commutative algebra. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan. Education Hochster attend ...

Math 711: Lecture of September 7, 2007
/ref> It is unique up to a non-canonical
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. The
ring of all algebraic integers In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
is an example (and thus A^+ is typically not noetherian).


See also

*
Normal scheme In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and ...
*
Noether normalization lemma In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field ''k'', and any finitely generated commutative ''k''-algebra ''A'', there exists a non-negati ...
*
Algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
*
Splitting of prime ideals in Galois extensions In mathematics, the interplay between the Galois group ''G'' of a Galois extension ''L'' of a number field ''K'', and the way the prime ideals ''P'' of the ring of integers ''O'K'' factorise as products of prime ideals of ''O'L'', provides one ...
*
Torsor (algebraic geometry) In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topolog ...


Notes


References

* M. Atiyah, I.G. Macdonald, ''Introduction to Commutative Algebra'',
Addison–Wesley Addison-Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles throu ...
, 1994. *
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally in ...
, '' Algèbre commutative'', 2006. * * * * * H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. * J. S. Milne, "Algebraic number theory." available at http://www.jmilne.org/math/ * {{Citation , ref=Reference-idHS2006 , last=Huneke , first=Craig , last2=Swanson , first2=Irena , author2-link= Irena Swanson , title=Integral closure of ideals, rings, and modules , url=http://people.reed.edu/~iswanson/book/index.html , publisher=
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
, location=Cambridge, UK , series=London Mathematical Society Lecture Note Series , isbn=978-0-521-68860-4 , mr=2266432 , year=2006 , volume=336 * M. Reid, ''Undergraduate Commutative Algebra'', London Mathematical Society, 29, Cambridge University Press, 1995.


Further reading

*Irena Swanson
Integral closures of ideals and ringsDo DG-algebras have any sensible notion of integral closure?Is_k[x_1,\ldots,x_n
always_an_integral_extension_of_k[f_1,\ldots,f_n.html" ;"title="_1,\ldots,x_n">Is k[x_1,\ldots,x_n
always an integral extension of k[f_1,\ldots,f_n">_1,\ldots,x_n">Is k[x_1,\ldots,x_n
always an integral extension of k[f_1,\ldots,f_n/math> for a regular sequence (f_1,\ldots,f_n)?] Commutative algebra Ring theory Algebraic structures