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You could define a Dedekind ring as a noetherian domain s.t. the localization at any nonzero prime ideal is a discrete valuation ring (see the beginning of Serre's Local fields). From there it is easy to show unique factorization. It is of course a cheat since the equivalence between the two definitions relies of course on the "integrally closed" property.

You could define a Dedekind ring as a noetherian domain s.t. the localization at any nonzero prime ideal is a discrete valuation ring (see the beginning of Serre's Local fields). From there it is easy to show unique factorization. It is of course a cheat since the equivalence between the two definitions relies of course on the "integrally closed" property.

You could define a Dedekind ring as a noetherian domain s.t. the localization at any nonzero prime ideal is a discrete valuation ring (see the beginning of Serre's Local fields). From there it is easy to show unique factorization. It is of course a cheat since the equivalence between the two definitions relies on the "integrally closed" property.

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You could define a Dedekind ring as a noetherian domain s.t. the localization at any nonzero prime ideal is a discrete valuation ring (see the beginning of Serre's Local fields). From there it is easy to show unique factorization. It is of course a cheat since the equivalence between the two definitions relies of course on the "integrally closed" property.