Integrality

In commutative algebra, the notions of an element integral over a ring (also called an algebraic integer over the ring), and of an integral extension of rings, are a generalization of the notions in field theory of an element being algebraic over a field, and of an algebraic extension of fields.

The special case of greatest interest in number theory is that of complex numbers integral over the ring of integers Z. (See algebraic integer.)

Convention

The term "ring" will be understood to mean "commutative ring" with a unit.

Definition

Let "B" be a ring, and "A" be a subring of "B". An element "b" of "B" is said to be integral over "A" if there exists a monic polynomial "f" with coefficients in "A" such that "f"("b") = 0. We say that "B" is integral over "A", or an integral extension of "A", or integrally dependent on "A", if every element of "B" is integral over "A".

Basic properties

Characterization by finiteness condition

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.

The most commonly given proof of this theorem uses the Cayley-Hamilton theorem on determinants.

Closure properties

Using the characterization of integrality in terms of finiteness, one proves the following closure properties:

:* (Integral closure) Let "A" $subseteq$ "B" be rings. Then the subset "C" of "B" consisting of elements integral over "A" is a subring of "B" containing "A". Thus, the sum, difference, or product of elements integral over "A" is also integral over "A". The ring "C" is said to be the integral closure of "A" in "B", and is denoted . If C = A, we say "A" is integrally closed in B.:* (Transitivity of integrality) Let "A" $subseteq$ "B" $subseteq$ "C" be rings, and "c" ∈ "C". If "c" is integral over "B" and "B" is 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."

Example

The integral closure of the ring of integers Z in the field of complex numbers C is called the "ring of algebraic integers".

Integral ring homomorphisms

In the definition of integrality, the assumption that "A" be a subring of "B" can be relaxed. If "f": "A" $ightarrow$ "B" is a ring homomorphism, that is, if "B" is made into an "A" algebra by "f", then we say that "f" is integral, or that "B" is an integral A-algebra, if "B" is integral over the subring "f"("A"). Previously, we had only considered the case in which "f" was injective. Similarly, an element of "B" is integral over "A" if it is integral over the subring "f"("A").

Many of the preceding considerations can be summarized in the statement that an "A"-algebra "B" is a finitely generated "A"-module if and only if "B" can be generated as an "A"-algebra by a finite number of elements integral over "A".

Properties of integrality with respect to localization

Integral closure is preserved under localization. Specifically, we have the following property. Recall that if "A" ⊆ "B" are rings, then "S"^-1"A" may be identified with a subring of "S"^-1"B".

* Let "A" ⊆ "C" ⊆ "B" be rings, with "C" the integral closure of "A" in "B". Let "S" ⊆ "A" be a multiplicatively closed subset of "A" (i.e., 1 ∈ "S" and whenever "x", "y" ∈ "S", "xy" ∈ "S"). Then the localization "S"^-1"C" is the integral closure of "S"^-1"A" in "S"^-1"B".

Integral closure of a ring

The integral closure Overline|"A" (without further qualification) of a reduced ring "A" is defined to be its integral closure in its total ring of fractions, "K". Such a ring is said to be integrally closed (without further qualification) if it is integrally closed in its total ring of fractions, that is if "A" = Overline|"A". [Chapter 2 of Huneke and Swanson 2006] The conductor of "A" is the set:$C\_A:=\{\; ain\; K\; :\; aoverline\{A\}subseteq\; A\}.$It is the largest ideal of "A" that is also an ideal of Overline|"A". [Chapter 12 of Huneke and Swanson 2006] If the conductor is the unit ideal "A", then "A" is integrally closed ["C"_A = "A" implies that 1 ∈ "C"_A, so Overline|"A" = 1 · "A" ⊆ "A", so Overline|"A" = "A"] .

Integrally closed domains

The total ring of fractions of an integral domain "A" is its field of fractions Frac("A"). Thus, an integral domain is integrally closed if, and only if, it is integrally closed in its field of fractions. A normal domain is most often defined as a Noetherian integrally closed domain, although the Noetherian assumption is sometimes dropped.

Classes of integrally closed domains

Any unique factorization domain "A" is integrally closed. (An elementary argument shows that any root in "K" = Frac("A") of a monic polynomial with coefficients in "A" must belong to "A". In the case "A" = Z, this fact is often known to schoolchildren.)

Behaviour under localization

The following conditions are equivalent for an integral domain "A":
# "A" is integrally closed;
# "A"_"p" (the localization of "A" with respect to "p") is integrally closed for every prime ideal "p";
# "A"_"m" is integrally closed for every maximal ideal "m".

1 → 2 results immediately from the preservation of integral closure under localization; 2 → 3 is trivial; 3 → 1 results from the preservation of integral closure under localization, the exactness of localization, and the property that an "A"-module "M" is zero if and only if its localization with respect to every maximal ideal is zero.

Relation to valuation rings

Let "K" be a field, and let "A" be a subring of "K". Then it is a theorem that the integral closure of "A" in "K" is the intersection of all valuation rings of "K" containing "A".

Integral closure of an ideal

In commutative algebra, there is 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 $x^n\; +\; a\_\{1\}\; x^\{n-1\}\; +\; ldots\; +\; a\_\{n-1\}\; x^1\; +\; a\_n$ with $a\_i\; in\; I^i$ with $r$ as a root. The integral closure of an ideal is easily seen to be in the radical of this ideal.

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 sufficiently large $n$.

*$r\; in\; overline\; I$ 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.

Going-up and going-down

Noether's theorem on the algebra of invariants

Noether's normalization lemma

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"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. [Chapter 4 of Reid.]

Relation to dimension theory

Integrality in algebraic geometry

Integral morphisms of schemes

Normal schemes

See also

*Going up and going down
*Valuation ring

Notes

References

*M. Atiyah, I.G. Macdonald, "Introduction to Commutative Algebra", Addison-Wesley, 1994. ISBN 0201407515

* H. Matsumura "Commutative ring theory." Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.

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* M. Reid, "Undergraduate Commutative Algebra", London Mathematical Society, 29, Cambridge University Press, 1995.