Let A be an integrally closed domain with field of fractions K and let L be a field extension of K. Then x∈L is integral over Aif 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 that is irreducible in K. If A is a domain contained in a field K, we can consider the integral closure of A in K. 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.
Let be a field of characteristic not 2 and a polynomial ring over it. If is a square-free nonconstant polynomial in, then is an integrally closed domain. In particular, is an integrally closed domain if.
To give a non-example, let k be a field and A is not integrally closed: it has the field of fractions, and the monic polynomial 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. Another domain which is not integrally closed is ; it does not contain the element of its field of fractions, which satisfies the monic polynomial.
Let A be a noetherian integral domain. Then A is integrally closed if and only if A is the intersection of all localizations over prime ideals of height 1 and the localization at a prime ideal 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 is noetherian, normal and connected, then A is an integrally closed domain. Let A be a noetherian ring. Then A is normal if and only if it satisfies the following: for any prime ideal,
Item is often phrased as "regular in codimension 1". Note implies that the set of associated primes has no embedded primes, and, when is the case, means that has no embedded prime for any non-zerodivisor f. In particular, a Cohen-Macaulay ring satisfies. 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 of the structure sheaf are Cohen-Macaulay for all prime ideals p. Then we can say: X is 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 of K generated by A and x is a fractional ideal of A; that is, if there is a such that for all. 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 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 Then 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. See also: Krull domain.
"Integrally closed" under constructions
The following conditions are equivalent for an integral domain A:
A is integrally closed;
Ap is integrally closed for every prime ideal p;
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/ 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 denote the set of all prime ideals in A of height one. If T is a finitely generated torsion module, one puts: which makes sense as a formal sum; i.e., a divisor. We write for the divisor class of d. If are maximal submodules of M, then and is denoted by.
