Integral element

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 aj in A such that

That is to say, b is a root of a monic polynomial over A.[1] 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. If every element of B is integral over A, then we say 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).

In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.

Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q.

Let ζ be a root of unity. Then the integral closure of Z in the cyclotomic field Q(ζ) is Z[ζ].[2] This can be found by using the minimal polynomial and using Eisenstein's criterion.

The roots of unity, nilpotent elements and idempotent elements in any ring are integral over Z.

In geometry, integral closure is closely related with normalization and normal schemes. It is the first step in resolution of singularities since it gives a process for resolving singularities of codimension 1.

Let B be a ring, and let A be a subring of B. Given an element b in B, the following conditions are equivalent:

The usual proof of this uses the following variant of the Cayley–Hamilton theorem on determinants:

This 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.

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.

In general, the going-up implies the lying-over.[7] Thus, in the below, we simply say the "going-up" to mean "going-up" and "lying-over".

Let A, K, etc. as before but assume L is only a finite field extension of K. Then

Let AB be rings and A' the integral closure of A in B. (See above for the definition.)

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.

The notion of integral closure of an ideal is used in some proofs of the going-down theorem.

If B is a subring of the total ring of fractions of A, then we may identify

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.[23] 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.[citation needed]

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.[26] More precisely, for a local noetherian ring R, we have the following chains of implications:[27]

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 KA can be written as a composite KBA where KB is a purely transcendental extension and BA is finite.[28]