# Integral element

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{{Short description|Mathematical element}}
In [commutative algebra](/source/commutative_algebra), an element ''b'' of a [commutative ring](/source/commutative_ring) ''B'' is said to be '''integral over''' a [subring](/source/subring) ''A'' of ''B'' if ''b'' is a [root](/source/root_of_a_polynomial) of some [monic polynomial](/source/monic_polynomial) over ''A''.<ref>The above equation is sometimes called an integral equation and ''b'' is said to be integrally dependent on ''A'' (as opposed to [algebraic dependent](/source/algebraic_dependent)).</ref>

If ''A'', ''B'' are [fields](/source/field_(mathematics)), then the notions of "integral over" and of an "integral extension" are precisely "[algebraic](/source/algebraic_element) over" and "[algebraic extension](/source/algebraic_extension)s" in [field theory](/source/field_theory_(mathematics)) (since the root of any [polynomial](/source/polynomial) is the root of a monic polynomial).

The case of greatest interest in [number theory](/source/number_theory) is that of [complex numbers](/source/complex_numbers) integral over '''Z''' (e.g., <math>\sqrt{2}</math> or <math>1+i</math>); in this context, the integral elements are usually called [algebraic integer](/source/algebraic_integer)s.  The algebraic integers in a finite [extension field](/source/field_extension) ''k'' of the [rationals](/source/rational_number) '''Q''' form a subring of ''k'', called the [ring of integers](/source/ring_of_integers) of ''k'', a central object of study in [algebraic number theory](/source/algebraic_number_theory).

In this article, the term ''[ring](/source/ring_(mathematics))'' will be understood to mean ''commutative ring'' with a multiplicative identity.

==Definition==
Let <math>B</math> be a ring and let <math>A \subset B</math> be a subring of <math>B.</math>
An element <math>b</math> of <math>B</math> is said to be '''integral over''' <math>A</math> if for some <math>n \geq 1,</math> there exists <math>a_0,\ a_1, \ \dots,\ a_{n-1}</math> in  <math>A</math> such that
<math display="block">b^n + a_{n-1} b^{n-1} + \cdots + a_1 b + a_0 = 0.</math>

The set of elements of <math>B</math> that are integral over <math>A</math> is called the '''integral closure''' of <math>A</math> in <math>B.</math> The integral closure of any subring <math>A</math> in <math>B</math> is, itself, a subring of <math>B</math> and contains <math>A.</math> If every element of <math>B</math> is integral over <math>A,</math> then we say that <math>B</math> is '''integral over''' <math>A</math>, or equivalently <math>B</math> is an '''integral extension''' of <math>A.</math>

==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](/source/ring_of_integers) for an [algebraic field extension](/source/Algebraic_extension) <math>K/\mathbb{Q}</math> (or <math>L/\mathbb{Q}_p</math>).

==== Integral closure of integers in rationals ====
[Integer](/source/Integer)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](/source/Gaussian_integer)s are the complex numbers of the form <math>a + b \sqrt{-1},\, a, b \in \mathbf{Z}</math>, and are integral over '''Z'''. <math>\mathbf{Z}[\sqrt{-1}]</math> is then the integral closure of '''Z''' in <math>\mathbf{Q}(\sqrt{-1})</math>. Typically this ring is denoted <math>\mathcal{O}_{\mathbb{Q}[i]}</math>.

The integral closure of '''Z''' in <math>\mathbf{Q}(\sqrt{5})</math> is the ring
:<math>\mathcal{O}_{\mathbb{Q}[\sqrt{5}]} = \mathbb{Z}\!\left[ \frac{1 + \sqrt{5}}{2} \right]</math>
This example and the previous one are examples of [quadratic integer](/source/quadratic_integer)s. The integral closure of a quadratic extension <math>\mathbb{Q}(\sqrt{d})</math> can be found by constructing the [minimal polynomial](/source/Minimal_polynomial_(field_theory)) of an arbitrary element <math>a + b \sqrt{d}</math> and finding number-theoretic criterion for the polynomial to have integral coefficients. This analysis can be found in the [quadratic extensions article](/source/Quadratic_integer).

==== Roots of unity ====
Let ζ be a [root of unity](/source/root_of_unity). Then the integral closure of '''Z''' in the [cyclotomic field](/source/cyclotomic_field) '''Q'''(ζ) is '''Z'''[ζ].<ref>{{harvnb|Milne|2020|loc=Theorem 6.4}}</ref> This can be found by using the [minimal polynomial](/source/Minimal_polynomial_(field_theory)) and using [Eisenstein's criterion](/source/Eisenstein's_criterion).

==== Ring of algebraic integers ====
The integral closure of '''Z''' in the field of complex numbers '''C''', or the algebraic closure <math>\overline{\mathbb{Q}}</math> is called the ''ring of [algebraic integer](/source/algebraic_integer)s''.

==== Other ====
The [roots of unity](/source/roots_of_unity), [nilpotent element](/source/nilpotent_element)s and [idempotent element](/source/idempotent_(ring_theory))s in any ring are integral over '''Z'''.

=== Integral closure in algebraic geometry ===
In [geometry](/source/geometry), integral closure is closely related with [normalization](/source/Noether_normalization_lemma) and [normal scheme](/source/normal_scheme)s. It is the first step in [resolution of singularities](/source/resolution_of_singularities) since it gives a process for resolving singularities of [codimension](/source/codimension) 1.

* For example, the integral closure of <math>\mathbb{C}[x,y,z]/(xy)</math> is the ring <math>\mathbb{C}[x,z] \times \mathbb{C}[y,z]</math> since geometrically, the first ring corresponds to the <math>xz</math>-plane unioned with the <math>yz</math>-plane. They have a codimension 1 singularity along the <math>z</math>-axis where they intersect.
*Let a [finite group](/source/finite_group) ''G'' [act](/source/group_action) on a ring ''A''. Then ''A'' is integral over ''A''<sup>''G''</sup>, the set of elements fixed by ''G''; see [Ring of invariants](/source/Fixed-point_subring).
*Let ''R'' be a ring and ''u'' a [unit](/source/unit_(ring_theory)) in a ring containing ''R''. Then<ref>{{harvnb|Kaplansky|1974|loc=1.2. Exercise 4.}}</ref>

#''u''<sup>−1</sup> is integral over ''R'' [if and only if](/source/if_and_only_if) ''u''<sup>−1</sup> ∈ ''R''[''u''].
#<math>R[u] \cap  R[u^{-1}]</math> is integral over ''R''.
#The integral closure of the [homogeneous coordinate ring](/source/homogeneous_coordinate_ring) of a normal [projective variety](/source/projective_variety) ''X'' is the [ring of sections](/source/ring_of_sections)<ref>{{harvnb|Hartshorne|1977|loc=Ch. II, Exercise 5.14}}</ref>

::<math>\bigoplus_{n \ge 0} \operatorname{H}^0(X, \mathcal{O}_X(n)).</math>

=== Integrality in algebra ===

* If <math>\overline{k}</math> is an [algebraic closure](/source/algebraic_closure) of a field ''k'', then <math>\overline{k}[x_1, \dots, x_n]</math> is integral over <math>k[x_1, \dots, x_n].</math>
* The integral closure of '''C'''<nowiki>[</nowiki>''x''<nowiki>](/source/%3C%2Fnowiki%3E''x''%3Cnowiki%3E)</nowiki> in a finite extension of '''C'''((''x'')) is of the form <math>\mathbf{C}[x^{1/n}](/source/x%5E%7B1%2Fn%7D)</math> (cf. [Puiseux series](/source/Puiseux_series)){{citation needed|date=October 2012}}

== 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](/source/Finitely_generated_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](/source/faithful_module) ''A''[''b'']-module ''M'' such that ''M'' is finitely generated as an ''A''-module.

The usual [proof](/source/mathematical_proof) of this uses the following variant of the [Cayley–Hamilton theorem](/source/Cayley%E2%80%93Hamilton_theorem) on [determinant](/source/determinant)s:
:'''Theorem''' Let ''u'' be an [endomorphism](/source/endomorphism) of an ''A''-module ''M'' generated by ''n'' elements and ''I'' an [ideal](/source/ideal_(ring_theory)) of ''A'' such that <math>u(M) \subset IM</math>. Then there is a relation:
::<math>u^n + a_1 u^{n-1} + \cdots + a_{n-1} u + a_n = 0, \, a_i \in I^i.</math>
This theorem (with ''I'' = ''A'' and ''u'' multiplication by ''b'') gives (iv) ⇒ (i) and the rest is easy. Coincidentally, [Nakayama's lemma](/source/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 <math>B</math> that are integral over <math>A</math> forms a subring of ''<math>B</math>'' containing <math>A</math>. (Proof: If ''x'', ''y'' are elements of ''<math>B</math>'' that are integral over <math>A</math>, then <math>x + y, xy, -x</math> are integral over <math>A</math> since they stabilize <math>A[x]A[y]</math>, which is a finitely generated module over <math>A</math> and is annihilated only by zero.)<ref>This proof is due to Dedekind (Milne, ANT). Alternatively, one can use symmetric polynomials to show integral elements form a ring. (loc cit.)</ref> This ring is called the '''integral closure''' of <math>A</math> in <math>B</math>.

=== Transitivity of integrality ===
Another consequence of the above equivalence is that "integrality" is [transitive](/source/transitive_relation), in the following sense. Let <math>C</math> be a ring containing <math>B</math> and <math>c \in C</math>. If <math>c</math> is integral over ''<math>B</math>'' and ''<math>B</math>'' integral over <math>A</math>, then <math>c</math> is integral over <math>A</math>. In particular, if <math>C</math> is itself integral over ''<math>B</math>'' and ''<math>B</math>'' is integral over <math>A</math>, then <math>C</math> is also integral over <math>A</math>.

=== Integral closed in fraction field ===
If <math>A</math> happens to be the integral closure of <math>A</math> in ''<math>B</math>'', then ''A'' is said to be '''integrally closed''' in ''<math>B</math>''. If <math>B</math> is the [total ring of fractions](/source/total_ring_of_fractions) of <math>A</math>, (e.g., the [field of fractions](/source/field_of_fractions) when <math>A</math> is an [integral domain](/source/integral_domain)), then one sometimes drops the qualification "in <math>B</math>{{-"}} and simply says "integral closure of <math>A</math>" and "<math>A</math> is [integrally closed](/source/integrally_closed_domain)."<ref>Chapter 2 of {{harvnb|Huneke|Swanson|2006}}</ref> For example, the ring of integers <math>\mathcal{O}_K</math> is integrally closed in the field <math>K</math>.

==== 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](/source/Algebraic_extension) ''L'' of ''K''. Then the field of fractions of ''A' '' is ''L''. In particular, ''A' '' is an [integrally closed domain](/source/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 <math>L/K</math> the integral closure of <math>\mathcal{O}_K</math> in <math>L</math> is the ring of integers <math>\mathcal{O}_L</math>.

==== Remarks ====
Note that transitivity of integrality above implies that if <math>B</math> is integral over <math>A</math>, then <math>B</math> is a union (equivalently an [inductive limit](/source/inductive_limit)) of subrings that are finitely generated <math>A</math>-modules.

If <math>A</math> is [noetherian](/source/Noetherian_ring), transitivity of integrality can be weakened to the statement:

:There exists a finitely generated <math>A</math>-submodule of <math>B</math> that contains <math>A[b]</math>.

=== Relation with finiteness conditions ===
Finally, the assumption that <math>A</math> be a subring of <math>B</math> can be modified a bit. If <math>f:A \to B</math> is a [ring homomorphism](/source/ring_homomorphism), then one says <math>f</math> is '''integral''' if <math>B</math> is integral over <math>f(A)</math>. In the same way one says <math>f</math> is '''finite''' (<math>B</math> finitely generated <math>A</math>-module) or of '''finite type''' (<math>B</math> finitely generated <math>A</math>-[algebra](/source/algebra_over_a_ring)). In this viewpoint, one has that

:<math>f</math> is finite if and only if <math>f</math> is integral and of finite type.

Or more explicitly,
:<math>B</math> is a finitely generated <math>A</math>-module if and only if <math>B</math> is generated as an <math>A</math>-algebra by a finite number of elements integral over <math>A</math>.

== Integral extensions ==

=== Cohen-Seidenberg theorems ===
An integral extension ''A''&nbsp;⊆&nbsp;''B'' has the [going-up property](/source/Going_up_and_going_down), the [lying over](/source/lying_over) property, and the [incomparability](/source/Going_up_and_going_down) property ([Cohen–Seidenberg theorems](/source/Going_up_and_going_down)). Explicitly, given a chain of [prime ideal](/source/prime_ideal)s <math>\mathfrak{p}_1 \subset \cdots \subset \mathfrak{p}_n</math> in ''A'' there exists a <math>\mathfrak{p}'_1 \subset \cdots \subset \mathfrak{p}'_n</math> in ''B'' with <math>\mathfrak{p}_i = \mathfrak{p}'_i \cap A</math> (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](/source/Krull_dimension)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.<ref>{{harvnb|Kaplansky|1974|loc=Theorem 42}}</ref> 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](/source/corollary), one has: given a prime ideal <math>\mathfrak{q}</math> of ''B'', <math>\mathfrak{q}</math> is a [maximal ideal](/source/maximal_ideal) of ''B'' if and only if <math>\mathfrak{q} \cap A</math> 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](/source/algebraically_closed_field). If <math>f: A \to k</math> is a homomorphism, then ''f'' extends to a homomorphism ''B'' → ''k''.<ref>{{harvnb|Bourbaki|2006|loc=Ch 5, §2, Corollary 4 to Theorem 1.}}</ref> This follows from the going-up.

==== Geometric interpretation of going-up ====
Let <math>f: A \to B</math> be an integral extension of rings. Then the induced map

:<math>\begin{cases} f^\#: \operatorname{Spec} B \to \operatorname{Spec} A \\ p \mapsto f^{-1}(p)\end{cases}</math>

is a [closed map](/source/closed_map); in fact, <math>f^\#(V(I)) = V(f^{-1}(I))</math> for any ideal ''I'' and <math>f^\#</math> is [surjective](/source/surjective_map) if ''f'' is [injective](/source/injective_map). 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., <math>\operatorname{Spec} A</math> is a [normal scheme](/source/normal_scheme)). If ''B'' is integral over ''A'', then <math>\operatorname{Spec} B \to \operatorname{Spec} A</math> is [submersive](/source/submersion_(algebra)); i.e., the [topology](/source/topological_space) of <math>\operatorname{Spec} A</math> is the [quotient topology](/source/quotient_topology).<ref>{{harvnb|Matsumura|1970|loc=Ch 2. Theorem 7}}</ref> The proof uses the notion of [constructible set](/source/constructible_set_(topology))s. (See also: [Torsor (algebraic geometry)](/source/Torsor_(algebraic_geometry)).)

=== Integrality, base-change, universally-closed, and geometry ===
If <math>B</math> is integral over <math>A</math>, then <math>B \otimes_A R</math> is integral over ''R'' for any ''A''-algebra ''R''.<ref>{{harvnb|Bourbaki|2006|loc=Ch 5, §1, Proposition 5}}</ref> In particular, <math>\operatorname{Spec} (B \otimes_A R) \to \operatorname{Spec} R</math> is closed; i.e., the integral extension induces a "'''universally closed'''" map. This leads to a <u>geometric characterization of integral extension</u>. Namely, let ''B'' be a ring with only finitely many [minimal prime ideal](/source/minimal_prime_ideal)s (e.g., integral domain or noetherian ring). Then ''B'' is integral over a (subring) ''A'' if and only if <math>\operatorname{Spec} (B \otimes_A R) \to \operatorname{Spec} R</math> is closed for any ''A''-algebra ''R''.<ref>{{harvnb|Atiyah|Macdonald|1994|loc=Ch 5. Exercise 35}}</ref> In particular, every '''proper map''' is universally closed.<ref>{{Cite web|title=Section 32.14 (05JW): Universally closed morphisms—The Stacks project|url=https://stacks.math.columbia.edu/tag/05JW|website=stacks.math.columbia.edu|access-date=2020-05-11}}</ref>

=== 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](/source/normal_extension) of ''K'', ''B'' the integral closure of ''A'' in ''L''. Then the [group](/source/group_(mathematics)) <math>G = \operatorname{Gal}(L/K)</math> acts [transitively](/source/Group_action) on each fiber of <math>\operatorname{Spec} B \to \operatorname{Spec} A</math>.

'''Proof.''' Suppose <math>\mathfrak{p}_2 \ne \sigma(\mathfrak{p}_1)</math> for any <math>\sigma</math> in ''G''. Then, by [prime avoidance](/source/prime_avoidance), there is an element ''x'' in <math>\mathfrak{p}_2</math> such that <math>\sigma(x) \not\in \mathfrak{p}_1</math> for any <math>\sigma</math>. ''G'' fixes the element <math>y = \prod\nolimits_{\sigma} \sigma(x)</math> and thus ''y'' is [purely inseparable](/source/purely_inseparable) over ''K''. Then some power <math>y^e</math> belongs to ''K''; since ''A'' is integrally closed we have: <math>y^e \in A.</math> Thus, we found <math>y^e</math> is in <math>\mathfrak{p}_2 \cap A</math> but not in <math>\mathfrak{p}_1 \cap A</math>; i.e., <math>\mathfrak{p}_1 \cap A \ne \mathfrak{p}_2 \cap A</math>.

==== Application to algebraic number theory ====
The Galois group <math>\operatorname{Gal}(L/K)</math> then acts on all of the prime ideals <math>\mathfrak{q}_1,\ldots, \mathfrak{q}_k \in \text{Spec}(\mathcal{O}_L)</math> lying over a fixed prime ideal <math>\mathfrak{p} \in \text{Spec}(\mathcal{O}_K)</math>.<ref>{{Cite book|last=Stein|url=https://wstein.org/books/ant/ant.pdf|title=Computational Introduction to Algebraic Number Theory|pages=101}}</ref> That is, if
:<math>\mathfrak{p} = \mathfrak{q}_1^{e_1}\cdots\mathfrak{q}_k^{e_k} \subset \mathcal{O}_L</math>
then there is a Galois action on the set <math>S_\mathfrak{p} = \{\mathfrak{q}_1,\ldots,\mathfrak{q}_k \}</math>. This is called the [Splitting of prime ideals in Galois extensions](/source/Splitting_of_prime_ideals_in_Galois_extensions).

==== Remarks ====
The same idea in the proof shows that if <math>L/K</math> is a purely inseparable extension (need not be normal), then <math>\operatorname{Spec} B \to \operatorname{Spec} A</math> is [bijective](/source/bijective).

Let ''A'', ''K'', etc. as before but assume ''L'' is only a finite field extension of ''K''. Then
:(i) <math>\operatorname{Spec} B \to \operatorname{Spec} A</math> has finite fibers.
:(ii) the going-down holds between ''A'' and ''B'': given <math>\mathfrak{p}_1 \subset \cdots \subset \mathfrak{p}_n = \mathfrak{p}'_n \cap A</math>, there exists <math>\mathfrak{p}'_1 \subset \cdots \subset \mathfrak{p}'_n</math> 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 <math>\mathfrak{p}''_i</math> that contracts to <math>\mathfrak{p}'_i</math>. By transitivity, there is <math>\sigma \in G</math> such that <math>\sigma(\mathfrak{p}''_n) = \mathfrak{p}'_n</math> and then <math>\mathfrak{p}'_i = \sigma(\mathfrak{p}''_i)</math> are the desired chain.

== Integral closure ==
{{see also|Integral closure of an ideal}}
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](/source/multiplicatively_closed_subset) ''S'' of ''A'', the [localization](/source/Localization_of_a_ring) ''S''<sup>−1</sup>''A' '' is the integral closure of ''S''<sup>−1</sup>''A'' in ''S''<sup>−1</sup>''B'', and <math>A'[t]</math> is the integral closure of <math>A[t]</math> in <math>B[t]</math>.<ref>An exercise in {{harvnb|Atiyah|Macdonald|1994}}</ref> If <math>A_i</math> are subrings of rings <math>B_i, 1 \le i \le n</math>, then the integral closure of <math>\prod A_i</math> in <math>\prod B_i</math> is <math>\prod {A_i}'</math> where <math>{A_i}'</math> are the integral closures of <math>A_i</math> in <math>B_i</math>.<ref>{{harvnb|Bourbaki|2006|loc=Ch 5, §1, Proposition 9}}</ref>

The integral closure of a [local ring](/source/local_ring) ''A'' in, say, ''B'', need not be local. (If this is the case, the ring is called [unibranch](/source/Unibranch_local_ring).) This is the case for example when ''A'' is [Henselian](/source/Henselian_ring) 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](/source/valuation_ring)s of ''K'' containing ''A''.

Let ''A'' be an <math>\mathbb{N}</math>-graded subring of an <math>\mathbb{N}</math>-[graded ring](/source/graded_ring) ''B''. Then the integral closure of ''A'' in ''B'' is an <math>\mathbb{N}</math>-graded subring of ''B''.<ref>Proof: Let <math>\phi: B \to B[t]</math> be a ring homomorphism such that <math>\phi(b_n) = b_n t^n</math> if <math>b_n</math> is homogeneous of degree ''n''. The integral closure of <math>A[t]</math> in <math>B[t]</math> is <math>A'[t]</math>, where <math>A'</math> is the integral closure of ''A'' in ''B''. If ''b'' in ''B'' is integral over ''A'', then <math>\phi(b)</math> is integral over <math>A[t]</math>; i.e., it is in <math>A'[t]</math>. That is, each coefficient <math>b_n</math> in the polynomial <math>\phi(b)</math> is in ''A''.</ref>

There is also a concept of the [integral closure of an ideal](/source/integral_closure_of_an_ideal).  The integral closure of an ideal <math>I \subset R</math>, usually denoted by <math>\overline I</math>, is the set of all elements <math>r \in R</math> such that there exists a monic polynomial 
:<math>x^n + a_{1} x^{n-1} + \cdots + a_{n-1} x^1 + a_n</math> 
with <math>a_i \in I^i</math> with <math>r</math> as a root.<ref>Exercise 4.14 in {{harvnb|Eisenbud|1995}}</ref><ref>Definition 1.1.1 in {{harvnb|Huneke|Swanson|2006}}</ref> <!--The definition in Bourbaki's ''Algèbre commutative'' and Atiyah–MacDonald's ''Introduction to Commutative Algebra'' instead require the elements <math>a_i</math> to be in the ideal <math>I</math>.--> The [radical of an ideal](/source/radical_of_an_ideal) is integrally closed.<ref>Exercise 4.15 in {{harvnb|Eisenbud|1995}}</ref><ref>Remark 1.1.3 in {{harvnb|Huneke|Swanson|2006}}</ref>

For noetherian rings, there are alternate definitions as well.

*<math>r \in \overline I</math> if there exists a <math>c \in R</math> not contained in any minimal prime, such that <math>c r^n \in I^n</math> for all <math>n \ge 1</math>.
*<math> r \in \overline I</math> 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](/source/Going_up_and_going_down).

== Conductor ==
{{main|Conductor (ring theory)}}

Let ''B'' be a ring and ''A'' a subring of ''B'' such that ''B'' is integral over ''A''. Then the [annihilator](/source/annihilator_(ring_theory)) of the ''A''-module ''B''/''A'' is called the ''conductor'' of ''A'' in ''B''. Because the notion has origin in [algebraic number theory](/source/algebraic_number_theory), the conductor is denoted by <math>\mathfrak{f} = \mathfrak{f}(B/A)</math>. Explicitly, <math>\mathfrak{f}</math> consists of elements ''a'' in ''A'' such that <math>aB \subset A</math>. (cf. [idealizer](/source/idealizer) in abstract algebra.) It is the largest [ideal](/source/ideal_(ring_theory)) of ''A'' that is also an ideal of ''B''.<ref>Chapter 12 of {{harvnb|Huneke|Swanson|2006}}</ref> If ''S'' is a multiplicatively closed subset of ''A'', then
:<math>S^{-1}\mathfrak{f}(B/A) = \mathfrak{f}(S^{-1}B/S^{-1}A)</math>.
If ''B'' is a subring of the [total ring of fractions](/source/total_ring_of_fractions) of ''A'', then we may identify
:<math>\mathfrak{f}(B/A)=\operatorname{Hom}_A(B, A)</math>.

Example: Let ''k'' be a field and let <math>A = k[t^2, t^3] \subset B = k[t]</math> (i.e., ''A'' is the [coordinate ring](/source/coordinate_ring) of the [affine curve](/source/affine_curve) <math>x^2 = y^3</math>). ''B'' is the integral closure of ''A'' in <math>k(t)</math>. The conductor of ''A'' in ''B'' is the ideal <math>(t^2, t^3) A</math>. More generally, the conductor of <math>A = k[t^a, t^b](/source/t%5Ea%2C_t%5Eb)</math>, ''a'', ''b'' relatively prime, is <math>(t^c, t^{c+1}, \dots) A</math> with <math>c = (a-1)(b-1)</math>.<ref>{{harvnb|Huneke|Swanson|2006|loc=Example 12.2.1}}</ref>

Suppose ''B'' is the integral closure of an integral domain ''A'' in the field of fractions of ''A'' such that the ''A''-module <math>B/A</math> is finitely generated. Then the conductor <math>\mathfrak{f}</math> of ''A'' is an ideal defining the [support of](/source/support_of_a_module) <math>B/A</math>; thus, ''A'' coincides with ''B'' in the complement of <math>V(\mathfrak{f})</math> in <math>\operatorname{Spec}A</math>. In particular, the set <math>\{ \mathfrak{p} \in \operatorname{Spec}A \mid A_\mathfrak{p} \text{ is integrally closed} \}</math>, the complement of <math>V(\mathfrak{f})</math>, is an [open set](/source/open_set).

== Finiteness of integral closure ==
An important but difficult question is on the finiteness of the integral closure of a [finitely generated algebra](/source/finitely_generated_algebra). There are several known results.

The integral closure of a [Dedekind domain](/source/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](/source/Krull%E2%80%93Akizuki_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.<ref>{{harvnb|Huneke|Swanson|2006|loc=Exercise 4.9}}</ref> A nicer statement is this: the integral closure of a noetherian domain is a [Krull domain](/source/Krull_domain) ([Mori–Nagata theorem](/source/Mori%E2%80%93Nagata_theorem)). Nagata also gave an example of dimension 1 noetherian local domain such that the integral closure is not finite over that domain.{{citation needed|date=August 2013}}

Let ''A'' be a noetherian integrally closed domain with field of fractions ''K''. If ''L''/''K'' is a finite [separable extension](/source/separable_extension), then the integral closure <math>A'</math> of ''A'' in ''L'' is a finitely generated ''A''-module.<ref>{{harvnb|Atiyah|Macdonald|1994|loc=Ch 5. Proposition 5.17}}</ref> This is easy and standard (uses the fact that the trace defines a non-degenerate [bilinear form](/source/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 <math>A'</math> of ''A'' in ''L'' is a finitely generated ''A''-module and is also a finitely generated ''k''-algebra.<ref>{{harvnb|Hartshorne|1977|loc=Ch I. Theorem 3.9 A}}</ref> The result is due to Noether and can be shown using the [Noether normalization lemma](/source/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](/source/polynomial_ring) <math>S = k[x_1, ..., x_d]</math>. Since ''L''/''K'' is a finite purely inseparable extension, there is a power ''q'' of a [prime number](/source/prime_number) such that every element of ''L'' is a ''q''-th root of an element in ''K''. Let <math>k'</math> be a finite extension of ''k'' containing all ''q''-th roots of coefficients of finitely many rational functions that generate ''L''. Then we have: <math>L \subset k'(x_1^{1/q}, ..., x_d^{1/q}).</math> The ring on the right is the field of fractions of <math>k'[x_1^{1/q}, ..., x_d^{1/q}]</math>, which is the integral closure of ''S''; thus, contains <math>A'</math>. Hence, <math>A'</math> 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''.<ref>{{harvnb|Huneke|Swanson|2006|loc=Theorem 4.3.4}}</ref> More precisely, for a local noetherian ring ''R'', we have the following chains of implications:<ref>{{harvnb|Matsumura|1970|loc=Ch 12}}</ref>
:(i) ''A'' complete <math>\Rightarrow</math> ''A'' is a [Nagata ring](/source/Nagata_ring)
:(ii) ''A'' is a Nagata domain <math>\Rightarrow</math> ''A'' [analytically unramified](/source/analytically_unramified) <math>\Rightarrow</math> the integral closure of the completion <math>\widehat{A}</math> is finite over <math>\widehat{A}</math> <math>\Rightarrow</math> the integral closure of ''A'' is finite over A.

== The Grauert–Remmert–de Jong Criterion ==

The '''Grauert–Remmert–de Jong criterion'''  provides a necessary and sufficient condition for a ring to be normal.

=== The Criterion ===
Let <math>A </math> be a reduced Noetherian domain and <math>I \subset A</math> be an ideal satisfying the following conditions:

# <math>I</math> is a '''radical ideal''' (i.e., <math>\sqrt{I} = I</math>).
# The variety <math>V(I)</math> contains the '''non-normal locus''' of <math>I</math> (i.e., for every prime ideal <math>P \in \operatorname{Spec}(A)</math> where the local ring <math>A_P</math> is not normal, <math>P \supseteq I</math>).

Then: the ring <math>A</math> is '''normal''' if and only if the natural inclusion into the [endomorphism ring](/source/endomorphism_ring) of <math>I</math>
:<math>\phi: A \hookrightarrow \operatorname{Hom}_A(I, I)</math>
is an isomorphism.

=== Algorithmic Application (The de Jong Algorithm) ===
If <math>A</math> is not normal, the endomorphism ring <math>S = \operatorname{Hom}_A(I, I)</math> provides a strictly larger integral ring extension <math>A \subsetneq S</math> within the quotient field <math>Q(A)</math>. 

* '''Iteration:''' By iteratively constructing a sequence of ring extensions <math>A = A_0 \subset A_1 \subset A_2 \subset \dots \subset \overline{A}</math>, the procedure finally leads to the '''normalization''' <math>\overline{A}</math>, provided that the normalization is a finitely generated module over <math>A</math>.
* '''Generalization of the Zassenhaus Algorithm:''' This approach generalizes the '''Zassenhaus algorithm''' (also known as the Round-Two algorithm) from the specific context of rings of integers in '''number fields''' to '''general Noetherian rings'''.

<ref>{{cite journal |last1=de Jong |first1=Theo |title=An algorithm for computing the integral closure |journal=Journal of Symbolic Computation |date=1998 |volume=26 |issue=3 |pages=273-277 |doi=10.1006/jsco.1998.0216}}</ref>

== The Grauert–Remmert–de Jong Criterion ==

The '''Grauert–Remmert–de Jong criterion'''  provides a necessary and sufficient condition for a ring to be normal.

=== The Criterion ===
Let <math>A </math> be a reduced Noetherian domain and <math>I \subset A</math> be an ideal satisfying the following conditions:

# <math>I</math> is a '''radical ideal''' (i.e., <math>\sqrt{I} = I</math>).
# The variety <math>V(I)</math> contains the '''non-normal locus''' of <math>I</math> (i.e., for every prime ideal <math>P \in \operatorname{Spec}(A)</math> where the local ring <math>A_P</math> is not normal, <math>P \supseteq I</math>).

Then: the ring <math>A</math> is '''normal''' if and only if the natural inclusion into the endomorphism ring of <math>I</math>
:<math>\phi: A \hookrightarrow \operatorname{Hom}_A(I, I)</math>
is an isomorphism.

=== Algorithmic Application (The de Jong Algorithm) ===
If <math>A</math> is not normal, the endomorphism ring <math>S = \operatorname{Hom}_A(I, I)</math> provides a strictly larger integral ring extension <math>A \subsetneq S</math> within the quotient field <math>Q(A)</math>. 

* '''Iteration:''' By iteratively constructing a sequence of ring extensions <math>A = A_0 \subset A_1 \subset A_2 \subset \dots \subset \overline{A}</math>, the procedure finally leads to the '''normalization''' <math>\overline{A}</math>, provided that the normalization is a finitely generated module over <math>A</math>.
* '''Generalization of the Zassenhaus Algorithm:''' This approach generalizes the '''Zassenhaus algorithm''' (also known as the Round-Two algorithm) from the specific context of rings of integers in '''number fields''' to '''general Noetherian rings'''.

<ref>{{cite journal |last1=de Jong |first1=Theo |title=An algorithm for computing the integral closure |journal=Journal of Symbolic Computation |date=1998 |volume=26 |issue=3 |pages=273-277 |doi=10.1006/jsco.1998.0216}}</ref>

==Noether's normalization lemma==
{{main|Noether normalization lemma}}

Noether's normalisation lemma is a theorem in [commutative algebra](/source/commutative_algebra).  Given a field ''K'' and a finitely generated ''K''-algebra ''A'', the theorem says it is possible to find elements ''y''<sub>1</sub>, ''y''<sub>2</sub>, ..., ''y''<sub>''m''</sub> in ''A'' that are [algebraically independent](/source/Algebraic_independence) over ''K'' such that ''A'' is finite (and hence integral) over ''B'' = ''K''[''y''<sub>1</sub>,..., ''y''<sub>''m''</sub>].  Thus the extension ''K'' ⊂ ''A'' can be written as a composite ''K'' ⊂ ''B'' ⊂ ''A'' where ''K'' ⊂ ''B'' is a purely [transcendental extension](/source/transcendental_extension) and ''B'' ⊂ ''A'' is finite.<ref>Chapter 4 of Reid.</ref>

==Integral morphisms==
In [algebraic geometry](/source/algebraic_geometry), a morphism <math>f:X \to Y</math> of [schemes](/source/Scheme_(mathematics)) is ''integral'' if it is [affine](/source/Sheaf_of_algebras) and if for some (equivalently, every) affine open cover <math>U_i</math> of ''Y'', every map <math>f^{-1}(U_i)\to U_i</math> is of the form <math>\operatorname{Spec}(A)\to\operatorname{Spec}(B)</math> where ''A'' is an integral ''B''-algebra. The class of integral morphisms is more general than the class of [finite morphism](/source/finite_morphism)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](/source/algebraic_closure) of the field of fractions of ''A''. Then the integral closure <math>A^+</math> of ''A'' in ''L'' is called the '''absolute integral closure''' of ''A''.<ref>[Melvin Hochster](/source/Melvin_Hochster), [http://www.math.lsa.umich.edu/~hochster/711F07/L09.07.pdf Math 711: Lecture of September 7, 2007]</ref> It is unique up to a non-canonical [isomorphism](/source/isomorphism). The [ring of all algebraic integers](/source/ring_of_all_algebraic_integers) is an example (and thus <math>A^+</math> is typically not noetherian).

== See also ==

* [Normal scheme](/source/Normal_scheme)
* [Noether normalization lemma](/source/Noether_normalization_lemma)
* [Algebraic integer](/source/Algebraic_integer)
*[Splitting of prime ideals in Galois extensions](/source/Splitting_of_prime_ideals_in_Galois_extensions)
*[Torsor (algebraic geometry)](/source/Torsor_(algebraic_geometry))

== Notes ==
{{reflist}}

== References ==
* {{cite book |last1=Atiyah |first1=Michael Francis |author-link1=Michael Atiyah |last2=Macdonald |first2=Ian G. |author-link2=Ian G. Macdonald |title=[Introduction to Commutative Algebra](/source/Introduction_to_Commutative_Algebra) |date=1994 |orig-date=1969 |publisher=Addison–Wesley |isbn=0-201-40751-5}}
* {{cite book |last1=Bourbaki |first1=Nicolas |author-link1=Nicolas Bourbaki |title=Algèbre commutative |date=2006 |publisher=Springer |location=Berlin |isbn=978-3-540-33937-3}}
* {{Citation | last=Eisenbud | first=David | title=Commutative Algebra with a View Toward Algebraic Geometry| publisher=[Springer-Verlag](/source/Springer-Verlag) | series=Graduate Texts in Mathematics | isbn=0-387-94268-8 | year=1995 | volume=150 }}
* {{cite book | last = Kaplansky | first = Irving |author-link=Irving Kaplansky | title = Commutative Rings | series = Lectures in Mathematics | date = September 1974 | publisher = [University of Chicago Press](/source/University_of_Chicago_Press) | isbn = 0-226-42454-5 | url-access = registration | url = https://archive.org/details/commutativerings00irvi }}
*{{Hartshorne AG}}
* {{citation | last1=Matsumura |first1=H |title=Commutative algebra |year=1970}}
* H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
* {{cite web |author-link=James Milne (mathematician) |first=J. S. |last=Milne |title=Algebraic number theory |url=https://www.jmilne.org/math/CourseNotes/ant.html |date=19 July 2020}}
* {{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](/source/Cambridge_University_Press) | 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](/source/Miles_Reid), ''Undergraduate Commutative Algebra'', London Mathematical Society, '''29''', Cambridge University Press, 1995.

== Further reading ==
*Irena Swanson, [http://people.reed.edu/~iswanson/trieste.pdf Integral closures of ideals and rings] {{Webarchive|url=https://web.archive.org/web/20111105142137/http://people.reed.edu/~iswanson/trieste.pdf |date=2011-11-05 }}
*[https://mathoverflow.net/q/7775 Do DG-algebras have any sensible notion of integral closure?]
*[https://mathoverflow.net/q/66445 Is <math>k[x_1,\ldots,x_n]</math> always an integral extension of <math>k[f_1,\ldots,f_n]</math> for a regular sequence <math>(f_1,\ldots,f_n)</math>?]

Category:Commutative algebra
Category:Ring theory
Category:Algebraic structures

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