Integrally closed domain - Alchetron, the free social encyclopedia

In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Many well-studied domains are integrally closed: Fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed.

To give a non-example, let k be a field and A = k [ t 2 , t 3 ] ⊂ B = k [ t ] (A is the subalgebra generated by t² and t³.) A and B have the same field of fractions, and B is the integral closure of A (since B is a UFD.) In other words, A is not integrally closed. This is related to the fact that the plane curve Y 2 = X 3 has a singularity at the origin.

Let A be an integrally closed domain with field of fractions K and let L be a finite extension of K. Then x in L is integral over A if and only if its minimal polynomial over K has coefficients in A. This implies in particular that an integral element over an integrally closed domain A has a minimal polynomial over A. This is stronger than the statement that any integral element satisfies some monic polynomial. In fact, the statement is false without "integrally closed" (consider A = Z [ 5 ] . )

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.

Note that integrally closed domains appear in the following chain of class inclusions:

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

Examples

The following are integrally closed domains.

Any principal ideal domain (in particular, any field).

Any unique factorization domain (in particular, any polynomial ring over a unique factorization domain.)

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

Any Dedekind domain.

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

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 is the intersection of all localizations A p over prime ideals p of height 1 and (ii) the localization A p at a prime ideal p 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, normal ring, then the domains in the product are integrally closed domains. Conversely, any finite product of integrally closed domains is normal. In particular, if Spec ⁡ ( A ) 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 normal if and only if it satisfies the following: for any prime ideal p ,

(i) If p has height ≤ 1 , then A p is regular (i.e., A p is a discrete valuation ring.)

(ii) If p has height ≥ 2 , then A p has depth ≥ 2 .

Item (i) is often phrased as "regular in codimension 1". Note (i) implies that the set of associated primes A s s ( A ) has no embedded primes, and, when (i) is the case, (ii) means that A s s ( A / f A ) 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 is Cohen-Macaulay; i.e., the stalks O p 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; that is, if there is a d ≠ 0 such that d x n ∈ A for all n ≥ 0 . 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 ring A [ [ X ] ] is completely integrally closed. This is significant since the analog is false for an integrally closed domain: let R be a valuation domain of height at least 2 (which is integrally closed.) Then R [ [ X ] ] 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:

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.

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

χ ( T ) = ∑ p ∈ P length p ⁡ ( T ) p ,

which makes sense as a formal sum; i.e., a divisor. We write c ( d ) for the divisor class of d. If F , F ′ are maximal submodules of M, then c ( χ ( M / F ) ) = c ( χ ( M / F ′ ) ) and c ( χ ( M / F ) ) is denoted (in Bourbaki) by c ( M ) .

References

Integrally closed domain Wikipedia

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