Integral closures behave nicely under various constructions.
2.
The ring is the integral closure of regular integers \ mathbb { Z } in complex numbers.
3.
Indeed, the integral closure is contained in the intersection since the valuation rings are integrally closed.
4.
Because of this aspect, the valuation theory, Rees algebras and integral closure are intimately connected to the multiplicity theory.
5.
The integral closure is nothing else than the ring \ overline { \ textbf { Z } } of all algebraic integers.
6.
A nicer statement is this : the integral closure of a noetherian domain is a Krull domain ( Mori Nagata theorem ).
7.
Nagata also gave an example of dimension 1 noetherian local domain such that the integral closure is not finite over that domain.
8.
The singular point at infinity can be removed ( since this is a curve ) by the normalization ( integral closure ) process.
9.
This consequence does generalize to a higher dimension : the Mori Nagata theorem states that the integral closure of a noetherian domain is a Krull domain.
10.
The integral closure of a Dedekind domain in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring.
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