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.)
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.
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.
ExampleThe 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 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 localizationIntegral closure is preserved under localization. Specifically, we have the following property. Recall that if A ⊆ B 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.
Integrally closed domainsWe 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 domainsAny 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 localizationThe following conditions are equivalent for an integral domain A:
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 every maximal ideal is zero.
Relation to valuation ringsLet 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 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
Relation to dimension theory
Integrality in algebraic geometry
Integral morphisms of schemes
- 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