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.)
The term ring will be understood to mean commutative ring with a unit.
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.
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.
Using the characterization of integrality in terms of finiteness, one proves the following closure properties:
- * (Integral closure) Let A 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 B 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.
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 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
are rings, then S-1A
may be identified with a subring of S-1B
- 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-1C is the integral closure of S-1A in S-1B.
Integral closure of a ring
The integral closure (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 = . The conductor of A is the set
It is the largest ideal
that is also an ideal of . If the conductor is the unit ideal A
, then A
is integrally closed.
Integrally closed domains
The total ring of fractions of an integral domain A
is its field of fractions
). 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
) of a monic polynomial with coefficients in A
must belong to A
. In the case A
, 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
be a field
, and let A
be a subring of K
. Then it is a theorem that the integral closure of A
is the intersection of all valuation rings
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 , usually denoted by , is the set of all elements such that there exists a monic polynomial with with 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.
- if there exists a not contained in any minimal prime, such that for all sufficiently large .
- if in the normalized blow-up of , the pull back of is contained in the inverse image of . 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 y1, y2, ..., ym in A that are algebraically independent over K such that A is finite (and hence integral) over B = K[y1,..., ym]. 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.
Relation to dimension theory
Integrality in algebraic geometry
Integral morphisms of schemes
- 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.
- M. Reid, Undergraduate Commutative Algebra, London Mathematical Society, 29, Cambridge University Press, 1995.