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:
“On a flat road runs the well-trained runner,
He is lean and sinewy with muscular legs,
He is thinly clothed, he leans forward as he runs,
With lightly closed fists and arms partially raised.”
—Walt Whitman (1819–1892)
“While you are divided from us by geographical lines, which are imaginary, and by a language which is not the same, you have not come to an alien people or land. In the realm of the heart, in the domain of the mind, there are no geographical lines dividing the nations.”
—Anna Howard Shaw (1847–1919)