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.
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''].
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.
The following are integrally closed domains.
k
S=k[x1,...,xn]
f
S
S[y]/(y2-f)
k[x0,...,xr]/(x
2 | |
0 |
+...+
2) | |
x | |
r |
r\ge2
To give a non-example,[2] let k be a field and
A=k[t2,t3]\subsetk[t]
k(t)
X2-t2
Y2=X3
Another domain that is not integrally closed is
A=Z[\sqrt{5}]
\sqrt{5 | |
+1}{2} |
X2-X-1=0
For a noetherian local domain A of dimension one, the following are equivalent.
Let A be a noetherian integral domain. Then A is integrally closed if and only if (i) A is the intersection of all localizations
Aak{p}
ak{p}
Aak{p}
ak{p}
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.
See also: normal variety. 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,[3] 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.[4] In particular if A is a Noetherian, normal ring, then the domains in the product are integrally closed domains.[5] Conversely, any finite product of integrally closed domains is normal. In particular, if
\operatorname{Spec}(A)
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
ak{p}
ak{p}
\le1
Aak{p}
Aak{p}
ak{p}
\ge2
Aak{p}
\ge2
Item (i) is often phrased as "regular in codimension 1". Note (i) implies that the set of associated primes
Ass(A)
Ass(A/fA)
l{O}p
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 nonzero
d\inA
dxn\inA
n\ge0
Assume A is completely integrally closed. Then the formal power series ring
A[[X]]
R[[X]]
An integral domain is completely integrally closed if and only if the monoid of divisors of A is a group.
See also: Krull domain.
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.
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.[11]
A direct limit of integrally closed domains is 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:
\chi(T)=\sump\operatorname{length}p(T)p
c(d)
F,F'
c(\chi(M/F))=c(\chi(M/F'))
c(\chi(M/F))
c(M)
. Nicolas Bourbaki. Commutative Algebra . 1972 . registration . Paris . Hermann .
. Irving Kaplansky. Commutative Rings . Lectures in Mathematics . September 1974 . . 0-226-42454-5 . registration .
. Hideyuki Matsumura. 1989 . Commutative Ring Theory . Cambridge Studies in Advanced Mathematics . 2nd . Cambridge University Press . 0-521-36764-6 .
ak{m}
ak{m}
x=0
ak{m}
xs=0
s\not\inak{m}
s
ak{m}