In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed, as shown by the following chain of class inclusions:

An explicit example is the ring of integers Z, a Euclidean domain. All regular local rings are integrally closed as well.

A ring whose localizations at all prime ideals are integrally closed domains is a normal ring.

Basic properties

Let A be an integrally closed domain with field of fractions K and let L be a field extension of K. Then xL is integral over A if and only if it is algebraic over K and its minimal polynomial over K has coefficients in A. In particular, this means that any element of L integral over A is root of a monic polynomial in A[X] that is irreducible in K[X].

If A is a domain contained in a field K, we can consider the integral closure of A in K (i.e. the set of all elements of K that are integral over A). This integral closure is an integrally closed domain.

Integrally closed domains also play a role in the hypothesis of the Going-down theorem. The theorem states that if A⊆B is an integral extension of domains and A is an integrally closed domain, then the going-down property holds for the extension A⊆B.

Examples

The following are integrally closed domains.

• A principal ideal domain (in particular: the integers and any field).

• A unique factorization domain (in particular, any polynomial ring over a field, over the integers, or over any unique factorization domain).

• A GCD domain (in particular, any Bézout domain or valuation domain).

• A Dedekind domain (in particular, any ring of integers of a number field).

• A symmetric algebra over a field (since every symmetric algebra is isomorphic to a polynomial ring in several variables over a field).

• be a field of characteristic is a square-free is an integrally closed domain..

To give a non-example, let k, the subalgebra generated by t2 and t3. Then A in the variable X has root t which is in the field of fractions but not in A. This is related to the fact that the plane curve has a singularity at the origin.

, which is not in A.

Noetherian integrally closed domain

For a noetherian local domain A of dimension one, the following are equivalent.

• A is integrally closed.

• The maximal ideal of A is principal.

• A is a discrete valuation ring (equivalently A is Dedekind.)

• A is a regular local ring.

Let A be a noetherian integral domain. Then A is integrally closed if and only if (i) A of height 1 is a discrete valuation ring.

A noetherian ring is a Krull domain if and only if it is an integrally closed domain.

In the non-noetherian setting, one has the following: an integral domain is integrally closed if and only if it is the intersection of all valuation rings containing it.

Normal rings

Authors including Serre, Grothendieck, and Matsumura define a normal ring to be a ring whose localizations at prime ideals are integrally closed domains. Such a ring is necessarily a reduced ring, and this is sometimes included in the definition. In general, if A is a Noetherian ring whose localizations at maximal ideals are all domains, then A is a finite product of domains. In particular, if A is a Noetherian and normal ring, then the domains in the product are integrally closed domains. is Noetherian, normal, and connected, then A is an integrally closed domain. (Cf. smooth variety.)

Let A be a noetherian ring. Then (Serre's criterion) A, is regular is a discrete valuation ring.) .

Item (i) is often phrased as "regular in codimension 1". Note that (i) implies that the set of associated primes has no embedded primes has no embedded prime for any non-zerodivisor f. In particular, a Cohen-Macaulay ring satisfies (ii). Geometrically, we have the following: if X is a local complete intersection in a nonsingular variety; e.g., X itself is nonsingular, then X of the structure sheaf are Cohen-Macaulay for all prime ideals p. Then we can say: X is normal (i.e., the stalks of its structure sheaf are all normal) if and only if it is regular in codimension 1.

Completely integrally closed domains

Let A be a domain and K its field of fractions. An element x in K is said to be almost integral over A if the subring A[x] of K generated by A and x is a fractional ideal of A. Then A is said to be completely integrally closed if every almost integral element of K is contained in A. A completely integrally closed domain is integrally closed. Conversely, a noetherian integrally closed domain is completely integrally closed.

Assume A is completely integrally closed. Then the formal power series is completely integrally closed. This is significant since the analog is false for an integrally closed domain: let R is not integrally closed. Let L be a field extension of K. Then the integral closure of A in L is completely integrally closed.

An integral domain is completely integrally closed if and only if the monoid of divisors of A is a group.

"Integrally closed" under constructions

The following conditions are equivalent for an integral domain A:

1. A is integrally closed;

2. Ap (the localization of A with respect to p) is integrally closed for every prime ideal p;

3. Am 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.

In contrast, the "integrally closed" does not pass over quotient, for Z[t]/(t2+4) is not integrally closed.

The localization of a completely integrally closed domain need not be completely integrally closed.

A direct limit of integrally closed domains is an integrally closed domain.

Modules over an integrally closed domain

Let A be a Noetherian integrally closed domain.

An ideal I of A is divisorial if and only if every associated prime of A/I has height one.

Let P denote the set of all prime ideals in A of height one. If T is a finitely generated torsion module, one puts:

1. ,

for the divisor class of d are maximal submodules of M.

See also

• Unibranch local ring

Citations

References