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| In [[commutative algebra]], an element ''b'' of a [[commutative ring]] ''B'' is said to be '''integral over''' ''A'', a [[subring]] of ''B'', if there is an ''n'' ≥ 1 and <math>a_j \in A</math> such that
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| :<math>b^n + a_{n-1} b^{n-1} + \cdots + a_1 b + a_0 = 0.</math> | |
| That is to say, ''b'' is a root of a [[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]].)</ref> If every element of ''B'' is integral over ''A'', then it is said that ''B'' is '''integral over''' ''A'', or equivalently ''B'' is an '''integral extension''' of ''A''.
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| If ''A'', ''B'' are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "[[algebraic extension]]s" in [[field theory (mathematics)|field theory]] (since the root of any polynomial is the root of a monic polynomial). The special case of greatest interest in [[number theory]] is that of complex numbers integral over '''Z'''; in this context, they are usually called [[algebraic integer]]s (e.g., <math>\sqrt{2}</math>). The algebraic integers in a [[Field extension|finite extension field]] ''k'' of the [[rational number|rationals]] '''Q''' form a subring of ''k'', called the [[ring of integers]] of ''k'', a central object in [[algebraic number theory]].
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| 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''.
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| In this article, the term ''[[Ring (mathematics)|ring]]'' will be understood to mean ''commutative ring'' with a unity.
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| ==Examples==
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| *Integers are the only elements of '''Q''' that are integral over '''Z'''. In other words, '''Z''' is the integral closure of '''Z''' in '''Q'''.
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| *[[Gaussian integer]]s, complex numbers of the form <math>a + b \sqrt{-1}, a, b \in \mathbf{Z}</math>, are integral over '''Z'''. <math>\mathbf{Z}[\sqrt{-1}]</math> is then the integral closure of '''Z''' in <math>\mathbf{Q}(\sqrt{-1})</math>.
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| *The integral closure of '''Z''' in <math>\mathbf{Q}(\sqrt{5})</math> consists of elements of form <math>(a + b \sqrt{5})/2</math>; the last two are examples of [[quadratic integer]]s.
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| *Let ζ be a [[root of unity]]. Then the integral closure of '''Z''' in the [[cyclotomic field]] '''Q'''(ζ) is '''Z'''[ζ].<ref>{{harvnb|Milne|ANT|loc=Theorem 6.4}}</ref>
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| *The integral closure of '''Z''' in the field of complex numbers '''C''' is called the ''ring of [[algebraic integer]]s''.
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| *If <math>\overline{k}</math> is an 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>
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| *Let a [[finite group]] ''G'' act on a ring ''A''. Then ''A'' is integral over ''A<sup>G</sup>'' the set of elements fixed by ''G''. see [[Ring (mathematics)#Ring of invariants|ring of invariants]].
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| *The roots of unity and [[nilpotent element]]s in any ring are integral over '''Z'''.
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| *Let ''R'' be a ring and ''u'' a unit in a ring containing ''R''. Then<ref>Kaplansky, 1.2. Exercise 4.</ref>
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| #''u''<sup>−1</sup> is integral over ''R'' if and only if ''u''<sup>−1</sup> ∈ ''R''[''u''].
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| #<math>R[u] \cap R[u^{-1}]</math> is integral over ''R''.
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| *The integral closure of '''C'''<nowiki>[[</nowiki>''x''<nowiki>]]</nowiki> in a finite extension of '''C'''((''x'')) is of the form <math>\mathbf{C}[[x^{1/n}]]</math> (cf. [[Puiseux series]]){{citation needed|date=October 2012}}
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| *The integral closure of the [[homogeneous coordinate ring]] of a normal [[projective variety]] ''X'' is the [[ring of sections]]<ref>{{harvnb|Hartshorne|1977|loc=Ch. II, Excercise 5.14}}</ref>
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| ::<math>\bigoplus\nolimits_{n \ge 0} \operatorname{H}^0(X, \mathcal{O}_X(n)).</math>
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| == Equivalent definitions ==
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| {{see also|Integrally closed domain}}
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| Let ''B'' be a ring, and let ''A'' be a subring of ''B''. Given an element ''b'' in ''B'', the following conditions are equivalent:
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| :*(i) ''b'' is integral over ''A'';
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| :*(ii) the subring ''A''[''b''] of ''B'' generated by ''A'' and ''b'' is a [[Finitely generated module|finitely generated ''A''-module]];
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| :*(iii) there exists a subring ''C'' of ''B'' containing ''A''[''b''] and which is a finitely-generated ''A''-module;
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| :*(iv) there exists a finitely generated ''A''-submodule ''M'' of ''B'' with ''bM'' ⊂ ''M'' and the [[Annihilator (ring theory)|annihilator]] of ''M'' in ''B'' is zero.
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| The usual proof of this uses the following variant of the [[Cayley–Hamilton theorem]] on [[determinant]]s (or simply [[Cramer's rule]].)
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| :'''Theorem''' Let ''u'' be an [[endomorphism]] of an ''A''-module ''M'' generated by ''n'' elements and ''I'' an ideal of ''A'' such that <math>u(M) \subset IM</math>. Then there is a relation:
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| :: <math>u^n + a_1 u^{n-1} + \cdots + a_{n-1} u + a_n = 0, a_i \in I^i.</math>
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| 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.
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| It follows from the above that the set of elements of ''B'' that are integral over ''A'' forms a subring of ''B'' containing ''A''. (Indeed, if ''x'', ''y'' are elements of ''B'' that are integral over ''A'', then <math>x + y, xy, -x</math> are integral over ''A'' since they stabilize <math>A[x]A[y]</math>, which is a finitely generated module over ''A'' and is annihilated only by zero.) It is called the '''integral closure''' of ''A'' in ''B''. <ref>The proof is due to Dedekind (Milne, ANT). Alternatively, one can use symmetric polynomials to show integral elements form a ring. (loc cit.)</ref> 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]] of ''A'' (e.g., the field of fractions when ''A'' is an integral domain), then one sometimes drops qualification "in B" and simply says "integral closure" and "[[integrally closed domain|integrally closed]]."<ref>Chapter 2 of [[#Reference-idHS2006|Huneke and Swanson 2006]]</ref> Let ''A'' be an integral domain with the field of fractions ''K'' and ''A' '' the integral closure of ''A'' in an algebraic field extension ''L'' of ''K''. Then the field of fractions of ''A' '' is ''L''. In particular, ''A' '' is an [[integrally closed domain]].
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| Similarly, "integrality" is transitive. 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''.
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| Note that (iii) 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.
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| If ''A'' is [[Noetherian ring|noetherian]], (iii) can be weakened to:
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| :(iii) bis There exists a finitely generated ''A''-submodule of ''B'' that contains ''A''[''b''].
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| Finally, the assumption that ''A'' be a subring of ''B'' can be modified a bit. If ''f'': ''A'' → ''B'' is a [[ring homomorphism]], 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). In this viewpoint, one says that
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| :''f'' is finite if and only if ''f'' is integral and of finite-type.
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| Or more explicitly,
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| :''B'' is a finitely generated ''A''-module if and only if ''B'' is generated as ''A''-algebra by a finite number of elements integral over ''A''.
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| == Integral extensions ==
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| An integral extension ''A''⊆''B'' has the [[Going up and going down|going-up property]], the [[lying over]] property, and the [[Going up and going down#Lying over and incomparability|incomparability]] property ([[Cohen-Seidenberg theorems]]). Explicitly, given a chain of prime ideals <math>\mathfrak{p}_1 \subset \cdots \subset \mathfrak{p}_n</math>
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| 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]]s of ''A'' and ''B'' are the same. Furthermore, if ''A'' is an integrally closed domain, then the going-down holds (see below).
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| In general, the going-up implies the lying-over.<ref>{{harvnb|Kaplansky|1970|loc=Theorem 42}}</ref> Thus, in the below, we simply say the "going-up" to mean "going-up" and "lying-over".
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| 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 <math>\mathfrak{q}</math> of ''B'', <math>\mathfrak{q}</math> is a [[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.
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| Let ''B'' be a ring that is integral over a subring ''A'' and ''k'' an 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.
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| Let <math>f: A \to B</math> be an integral extension of rings. Then the induced map
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| :<math>f^\#: \operatorname{Spec} B \to \operatorname{Spec} A, \quad p \mapsto f^{-1}(p)</math>
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| is a [[closed map]]; in fact, <math>f^\#(V(I)) = V(f^{-1}(I))</math> for any ideal ''I'' and <math>f^\#</math> is surjective if ''f'' is injective. This is a geometric interpretation of the going-up.
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| If ''B'' is integral over ''A'', 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 geometric characterization of integral extension. Namely, let ''B'' be a ring with only finitely many [[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>
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| is closed for any ''A''-algebra ''R''.<ref>{{harvnb|Atiyah-MacDonald|1969|loc=Ch 5. Exercise 35}}</ref>
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| Let ''A'' be an integrally closed domain with the field of fractions ''K'', ''L'' a finite [[normal extension]] of ''K'', ''B'' the integral closure of ''A'' in ''L''. Then the group <math>G = \operatorname{Gal}(L/K)</math> acts transitively 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]], 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_{\sigma} \sigma(x)</math> and thus ''y'' is [[purely inseparable]] over ''K''. Then some power <math>y^e</math> belongs to ''K''; in fact, to ''A'' since ''A'' is integrally closed. 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>.)
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| Remark: 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.
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| Let ''A'', ''K'', etc. as before but assume ''L'' is only a finite field extension of ''K''. Then
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| :(i) <math>\operatorname{Spec} B \to \operatorname{Spec} A</math> has finite fibers.
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| :(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.
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| 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.
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| 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]].) If ''B'' is integral over ''A'', then <math>\operatorname{Spec} B \to \operatorname{Spec} A</math> is [[submersion (algebra)|submersive]]; i.e., the topology of <math>\operatorname{Spec} A</math> is the [[quotient topology]].<ref>{{harvnb|Matsumura|1970|loc=Ch 2. Theorem 7}}</ref> The proof uses the notion of [[constructible set (topology)|constructible set]]s.
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| == Integral closure ==
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| {{see also|Integral closure of an ideal}}
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| Let ''A'' ⊂ ''B'' be rings and ''A' '' the integral closure of ''A'' in ''B''. (See above for the definition.)
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| Integral closures behave nicely under various constructions. Specifically, for a [[multiplicatively closed subset]] ''S'' of ''A'', the [[Localization of a ring|localization]] ''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 Atiyah–MacDonald.</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>
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| The integral closure of a local ring ''A'' in, say, ''B'', need not be local. (If this is the case, the ring is called [[Unibranch local ring|unibranch]].) This is the case for example when ''A'' is [[Henselian ring|Henselian]] and ''B'' is a field extension of the field of fractions of ''A''.
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| If ''A'' is a subring of a field ''K'', then the integral closure of ''A'' in ''K'' is the intersection of all [[valuation ring]]s of ''K'' containing ''A''.
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| Let ''B'' be a <math>\mathbb{N}</math>-graded subring of a <math>\mathbb{N}</math>-[[graded ring]] ''A''. Then the integral closure of ''A'' in ''B'' is a <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<nowiki>'</nowiki>''.</ref>
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| There is also a concept of the [[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
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| :<math>x^n + a_{1} x^{n-1} + \cdots + a_{n-1} x^1 + a_n</math>
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| with <math>a_i \in I^i</math> with ''r'' as a root. Note this is the definition that appears, for example, in Eisenbud and is different from Bourbaki's and Atiyah–MacDonald's definition.<!--The integral closure of an ideal is easily seen to be in the [[radical of an ideal|radical]] of this ideal. dubious?-->
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| For noetherian rings, there are alternate definitions as well.
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| *<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>.
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| *<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.
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| The notion of integral closure of an ideal is used in some proofs of the [[Going up and going down|going-down theorem]].
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| == Conductor ==
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| Let ''B'' be a ring and ''A'' a subring of ''B'' such that ''B'' is integral over ''A''. Then the [[annihilator (ring theory)|annihilator]] of the ''A''-module ''B''/''A'' is called the ''conductor'' of ''A'' in ''B''. Because the notion has origin in [[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]] in abstract algebra.) It is the largest [[ideal (ring theory)|ideal]] of ''A'' that is also an ideal of ''B''.<ref>Chapter 12 of [[#Reference-idHS2006|Huneke and Swanson 2006]]</ref> If ''S'' is a multiplicatively closed subset of ''A'', then
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| :<math>S^{-1}\mathfrak{f}(B/A) = \mathfrak{f}(S^{-1}B/S^{-1}A)</math>.
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| If ''B'' is a subring of the [[total ring of fractions]] of ''A'', then we may identify
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| :<math>\ \mathfrak{f}(B/A)=\operatorname{Hom}_A(B, A)</math>.
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| 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 of the 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]]</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|Swanson|2006|loc=Example 12.2.1}}</ref>
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| 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 a module|support of]] <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 | A_\mathfrak{p} \text{ is integrally closed} \}</math>, the complement of <math>V(\mathfrak{f})</math>, is an open set.
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| == Finiteness of integral closure ==
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| An important but difficult question is on the finiteness of the integral closure of a finitely generated algebra. There are several known results.
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| 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.<ref>{{harvnb|Swanson|2006|loc=Exercise 4.9}}</ref> 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.{{fact|date=August 2013}}
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| 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 <math>A'</math> of ''A'' in ''L'' is a finitely generated ''A''-module.<ref>{{harvnb|Atiyah-MacDonald|1969|loc=Ch 5. Proposition 5.17}}</ref> This is easy and standard (uses the fact that the trace defines a non-degenerate bilinear form.)
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| 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]] 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; thus, assume ''L''/''K'' is purely inseparable. By the normalization lemma, ''A'' is integral over the 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 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'''.
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| 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|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>
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| :(i) ''A'' complete <math>\Rightarrow</math> ''A'' is a [[Nagata ring]]
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| :(ii) ''A'' is a Nagata domain <math>\Rightarrow</math> ''A'' [[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. | |
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| ==Noether's normalization lemma==
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| {{main|Noether normalization lemma}}
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| Noether's normalisation lemma is a theorem in [[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 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 and ''B'' ⊂ ''A'' is finite.<ref>Chapter 4 of Reid.</ref>
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| == Notes ==
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| <references/>
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| == References ==
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| *[[Michael Atiyah|M. Atiyah]], [[Ian G. Macdonald|I.G. Macdonald]], ''Introduction to Commutative Algebra'', [[Addison–Wesley]], 1994. ISBN 0-201-40751-5
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| *[[Nicolas Bourbaki]], ''[[Algèbre commutative]]'', 2006.
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| * [[David Eisenbud|Eisenbud, David]], ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.
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| * {{cite book | last = Kaplansky | first = Irving | title = Commutative Rings
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| | series = Lectures in Mathematics |date=September 1974
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| | publisher = [[University of Chicago Press]] | isbn = 0-226-42454-5 }}
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| *{{Hartshorne AG}}
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| * {{citation | last1=Matsumura |first1=H |title=Commutative algebra |year=1970}}
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| * H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
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| * [[James Milne (mathematician)|J. S. Milne]], "Algebraic number theory." available at http://www.jmilne.org/math/
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| * {{Citation | ref=Reference-idHS2006 | last=Huneke | first=Craig | last2=Swanson | first2=Irena | title=Integral closure of ideals, rings, and modules | url=http://people.reed.edu/~iswanson/book/index.html | publisher=[[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 }}
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| * [[Miles Reid|M. Reid]], ''Undergraduate Commutative Algebra'', London Mathematical Society, '''29''', Cambridge University Press, 1995.
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| == Further reading ==
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| *Irena Swanson, [http://people.reed.edu/~iswanson/trieste.pdf Integral closures of ideals and rings]
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| [[Category:Commutative algebra]]
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| [[Category:Ring theory]]
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| [[Category:Algebraic structures]]
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