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.
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 A if and only if it is algebraic over K and its minimal polynomial over K has coefficients in A.
[1] 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].
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.
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
; its field of fractions contains the element
For a noetherian local domain A of dimension one, the following are equivalent.
of height 1 is a discrete valuation ring.
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.
Authors including Serre, Grothendieck, and Matsumura define a normal ring to be a ring whose localizations at prime ideals are integrally closed domains.
In general, if A is a Noetherian ring whose localizations at maximal ideals are all domains, then A is a finite product of domains.
[7] Conversely, any finite product of integrally closed domains is normal.
is noetherian, normal and connected, then A is an integrally closed domain.
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
, Item (i) is often phrased as "regular in codimension 1".
has no embedded primes, and, when (i) is the case, (ii) means that
has no embedded prime for any non-zerodivisor f. In particular, a Cohen-Macaulay ring satisfies (ii).
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.
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 nonzero
Then the formal power series ring
[10] 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).
[13] The following conditions are equivalent for an integral domain A: 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.
An ideal I of A is divisorial if and only if every associated prime of A/I has height one.
[15] 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.