Alternative Definitions
For an integral domain which is not a field, all of the following conditions are equivalent:
(DD1) Every nonzero proper ideal factors into primes.
(DD2) is Noetherian, and the localization at each maximal ideal is a Discrete Valuation Ring.
(DD3) Every fractional ideal of is invertible.
(DD4) is an integrally closed, Noetherian domain with Krull dimension one (i.e., every nonzero prime ideal is maximal).
Thus a Dedekind domain is a domain which satisfies any one, and hence all four, of (DD1) through (DD4). Which of these conditions one takes as the definition is therefore merely a matter of taste. In practice, it is often easiest to verify (DD4).
A Krull domain is a higher dimensional analog of a Dedekind domain: a Dedekind domain that is not a field is a Krull domain of dimension 1. This notion can be used to study the various characterizations of a Dedekind domain. In fact, this is the definition of a Dedekind domain used in Bourbaki's "Commutative algebra".
A Dedekind domain can also be characterized in terms of homological algebra: an integral domain is a Dedekind domain if and only if it is a hereditary ring; i.e., every submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective.
Read more about this topic: Dedekind Domain
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