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In
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
, an element ''b'' of a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...
''B'' is said to be integral over a
subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
''A'' of ''B'' if ''b'' is a
root In vascular plants, the roots are the plant organ, 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 bel ...
of some
monic polynomial In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one ...
over ''A''. If ''A'', ''B'' are fields, then the notions of "integral over" and of an "integral extension" are precisely " algebraic over" and " algebraic extensions" in field theory (since the root of any
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
is the root of a monic polynomial). The case of greatest interest in
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
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 that 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 ''k'' of the rationals 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 de ...
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'' will be understood to mean ''commutative ring'' with a multiplicative identity.


Definition

Let B be a ring and let A \subset B be a subring of B. An element b of B is said to be integral over A if for some n \geq 1, there exists a_0,\ a_1, \ \dots,\ a_ in A such that b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. The set of elements of B that are integral over A is called the integral closure of A in B. The integral closure of any subring A in B is, itself, a subring of B and contains 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.


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 de ...
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, every element of is a root of a non-zero polynomial with coefficients in . A field extens ...
K/\mathbb (or L/\mathbb_p).


Integral closure of integers in rationals

Integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 ...
s are the complex numbers of the form a + b \sqrt,\, a, b \in \mathbf, and are integral over Z. \mathbf sqrt/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. A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree tw ...
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 is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...
. Then the integral closure of Z in the
cyclotomic field In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...
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 wit ...
.


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 that 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 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 number theory, the theory of group char ...
,
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, along with its sister idempotent, was introduced by Benjamin Peirce i ...
s and idempotent elements in any ring are integral over Z.


Integral closure in algebraic geometry

In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, integral closure is closely related with
normalization Normalization or normalisation refers to a process that makes something more normal or regular. Science * Normalization process theory, a sociological theory of the implementation of new technologies or innovations * Normalization model, used in ...
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 a ...
s. It is the first step in
resolution of singularities In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, which is a non-singular variety ''W'' with a Proper morphism, proper birational map ''W''→''V''. For varieties ov ...
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 In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
''G'' act on a ring ''A''. Then ''A'' is integral over ''A''''G'', the set of elements fixed by ''G''; see Ring of invariants. *Let ''R'' be a ring and ''u'' a unit 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
''u''−1 ∈ ''R'' 'u'' #R \cap R ^/math> is integral over ''R''. #The integral closure of the homogeneous coordinate ring of a normal
projective variety In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...
''X'' is the ring of sections ::\bigoplus_ \operatorname^0(X, \mathcal_X(n)).


Integrality in algebra

* If \overline is an algebraic closure 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)


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 on
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
s: :Theorem Let ''u'' be an endomorphism of an ''A''-module ''M'' generated by ''n'' elements and ''I'' an ideal 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 embedding ...
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 fie ...
when A is an
integral domain In mathematics, 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 setting for studying divisibilit ...
), 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, every element of is a root of a non-zero polynomial with coefficients in . A field extens ...
''L'' of ''K''. Then the field of fractions of ''A' '' is ''L''. In particular, ''A' '' is an integrally closed domain.


= 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) 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 leng ...
, 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 mathematics, 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 that preserves addition, multiplication and multiplicative identity ...
, 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 a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
). 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'' âІ ''B'' has the going-up property, the lying over 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 (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
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 ...
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 In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
, 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 ...
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 . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
. 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 such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
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 of its codomain; that is, implies (equivalently by contraposition, impl ...
. 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 a ...
). If ''B'' is integral over ''A'', then \operatorname B \to \operatorname A is submersive; i.e., the
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
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 to ...
. The proof uses the notion of constructible sets. (See also: Torsor (algebraic geometry).)


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 ideals (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 extension, algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' that has a zero of a function, root in ''L'' splits into linear factors over ''L''. This is ...
of ''K'', ''B'' the integral closure of ''A'' in ''L''. Then the group 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, 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.


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, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
. 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 ''S'' of ''A'', the localization ''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 mathematics, more specifically in 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 algebraic varieties or manifolds, or of ...
''A'' in, say, ''B'', need not be local. (If this is the case, the ring is called unibranch.) This is the case for example when ''A'' is Henselian 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 non-zero 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 ' ...
s of ''K'' containing ''A''. Let ''A'' be an \mathbb-graded subring of an \mathbb- graded ring ''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. 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 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.


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 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 embedding ...
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 is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...
of the affine curve 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, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
.


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 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 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 ( Mori–Nagata theorem). 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 In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
). 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 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 formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
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 (mathematics), 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 ...
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 :(ii) ''A'' is a Nagata domain \Rightarrow ''A'' analytically unramified \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 ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
. 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 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 which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, 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 Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...
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 morphisms 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 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
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 or morphism 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 the ...
. The ring of all algebraic integers 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 a ...
* Noether normalization lemma *
Algebraic integer In algebraic number theory, an algebraic integer is a complex number that 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 * Torsor (algebraic geometry)


Notes


References

* * * * * * * H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. * * {{Citation , 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 was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, location=Cambridge, UK , series=London Mathematical Society Lecture Note Series , isbn=978-0-521-68860-4 , mr=2266432 , year=2006 , volume=336 , access-date=2011-03-01 , archive-date=2019-11-15 , archive-url=https://web.archive.org/web/20191115053353/http://people.reed.edu/~iswanson/book/index.html , url-status=dead * 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">_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