Editing Dedekind domain (section)

== Alternative definitions ==
For an integral domain <math>R</math> that is not a field, all of the following conditions are equivalent:<ref>{{harvnb|Milne|2008|loc=Remark 3.25}}</ref>

:'''(DD1)''' Every nonzero proper ideal factors into primes.
:'''(DD2)''' <math>R</math> is Noetherian, and the localization at each maximal ideal is a discrete valuation ring.
:'''(DD3)''' Every nonzero [[fractional ideal]] of <math>R</math> is invertible.
:'''(DD4)''' <math>R</math> is an [[integrally closed domain|integrally closed]], Noetherian domain with [[Krull dimension]] one (that is, every nonzero prime ideal is maximal).
:'''(DD5)''' For any two ideals <math>I</math> and <math>J</math> in <math>R</math>, <math>I</math> is contained in <math>J</math> if and only if <math>J</math>  divides <math>I</math> as ideals. That is, there exists an ideal <math>H</math> such that <math>I=JH</math>. A commutative ring (not necessarily a domain) with unity satisfying this condition is called a containment-division ring (CDR).<ref>{{harvnb|Krasula|2022|loc=Theorem 12}}</ref>

Thus a Dedekind domain is a domain that either is a field, or satisfies any one, and hence all five, of (DD1) through (DD5).  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 [[Nicolas Bourbaki|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]]; that is, 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 module|injective]].<ref>{{harvnb|Cohn|2003|loc=2.4. Exercise 9}}</ref>