In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and
Contents
- Examples
- Equivalent definitions
- Integral extensions
- Integral closure
- Conductor
- Finiteness of integral closure
- Noethers normalization lemma
- References
That is to say, b is a root of a monic polynomial over A. 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. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).
The special case of an integral element of greatest interest in number theory is that of complex numbers integral over Z; in this context, they are usually called algebraic integers (e.g.,
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.
In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.
Examples
- u−1 is integral over R if and only if u−1 ∈ R[u].
-
R [ u ] ∩ R [ u − 1 ] is integral over R.
Equivalent definitions
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;(iv) there exists a finitely generated A-submodule M of B such that bM ⊂ M and M is faithful over A[b] (i.e., the annihilator of M in A[b] is zero.)The usual proof of this uses the following variant of the Cayley–Hamilton theorem on determinants:
Theorem Let u be an endomorphism of an A-module M generated by n elements and I an ideal of A such thatThis 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.
It follows from the above that the set of elements of B that are integral over A forms a subring of B containing A. It is called the integral closure of A in B. 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." 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.
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.
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.
If A is noetherian, (iii) can be weakened to:
(iii) bis There exists a finitely generated A-submodule of B that contains A[b].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
f is finite if and only if f is integral and of finite-type.Or more explicitly,
B is a finitely generated A-module if and only if B is generated as an A-algebra by a finite number of elements integral over A.Integral extensions
An integral extension A⊆B has the going-up property, the lying over property, and the incomparability property (Cohen-Seidenberg theorems). Explicitly, given a chain of prime ideals
In general, the going-up implies the lying-over. Thus, in the below, we simply say the "going-up" to mean "going-up" and "lying-over".
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
Let B be a ring that is integral over a subring A and k an algebraically closed field. If
Let
is a closed map; in fact,
If B is integral over A, then
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
Remark: The same idea in the proof shows that if
Let A, K, etc. as before but assume L is only a finite field extension of K. Then
(i)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
Let B be a ring and A a subring that is a noetherian integrally closed domain (i.e.,
Integral closure
Let A ⊂ B be rings and A' the integral closure of A in B. (See above for the definition.)
Integral closures behave nicely under various constructions. Specifically, for a multiplicatively closed subset S of A, the localization S−1A' is the integral closure of S−1A in S−1B, and
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.) This is the case for example when A is Henselian and B is a field extension of the field of fractions of A.
If A is a subring of a field K, then the integral closure of A in K is the intersection of all valuation rings of K containing A.
Let B be an
There is also a concept of the integral closure of an ideal. The integral closure of an ideal
with
For noetherian rings, there are alternate definitions as well.
The notion of integral closure of an ideal is used in some proofs of the going-down theorem.
Conductor
Let B be a ring and A a subring of B such that B is integral over A. Then the 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
If B is a subring of the total ring of fractions of A, then we may identify
Example: Let k be a field and let
Suppose B is the integral closure of an integral domain A in the field of fractions of A such that the A-module
Finiteness of integral closure
An important but difficult question is on the finiteness of the integral closure of a finitely generated algebra. There are several known results.
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. 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.
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
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
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. More precisely, for a local noetherian ring R, we have the following chains of implications:
(i) A completeNoether'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.