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 over prime ideals of height 1 and (ii) 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.
Read more about this topic: Integrally Closed Domain
Famous quotes containing the words closed and/or domain:
“Because you live, O Christ,
the spirit bird of hope is freed for flying,
our cages of despair no longer keep us closed and life-denying.
The stone has rolled away and death cannot imprison!
O sing this Easter Day, for Jesus Christ has risen!”
—Shirley Erena Murray (20th century)
“In the domain of Political Economy, free scientific inquiry meets not merely the same enemies as in all other domains. The peculiar nature of the material it deals with, summons as foes into the field of battle the most violent, mean and malignant passions of the human breast, the Furies of private interest.”
—Karl Marx (1818–1883)