"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
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