"Integrally Closed" Under Constructions
The following conditions are equivalent for an integral domain A:
- A is integrally closed;
- Ap (the localization of A with respect to p) 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/(t2+4) is not integrally closed.
The localization of a completely integrally closed need not be completely integrally closed.
Read more about this topic: Integrally Closed Domain
Famous quotes containing the word closed:
“My old Father used to have a saying that If you make a bad bargain, hug it the tighter; and it occurs to me, that if the bargain you have just closed [marriage] can possibly be called a bad one, it is certainly the most pleasant one for applying that maxim to, which my fancy can, by any effort, picture.”
—Abraham Lincoln (18091865)