Extensive Definition
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 Amodule;
 iii) there exists a subring C of B containing A[b] and which is a finitelygenerated Amodule.
The most commonly given proof of this theorem
uses the CayleyHamilton
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 \bar^B. 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 \rightarrow 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 Aalgebra,
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 Aalgebra B is a finitely
generated Amodule if and only if B can be generated as an
Aalgebra 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 S1A may be identified with a subring of S1B. 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 S1C is the integral closure of S1A in S1B.
Integrally closed domains
We say an integral domain A is integrally closed (without further qualification) if it is integrally closed in its field of fractions Frac(A). 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 Amodule M
is zero if and only if its localization with respect 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_ x^ + \ldots + a_ 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 blowup of I, the pull back of r is contained in the inverse image of I. The blowup 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
goingdown theorem.
Goingup and goingdown
Noether's theorem on the algebra of invariants
Noether's normalization lemma
Relation to dimension theory
Integrality in algebraic geometry
Integral morphisms of schemes
Normal schemes
See also
References
 M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, AddisonWesley, 1994. ISBN 0201407515
 H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
integrality in German: Ganzheit (kommutative
Algebra)
integrality in Italian: Chiusura integrale
integrality in Finnish: Kokonaissulkeuma
integrality in Chinese: 整性