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