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Definition

A Noetherian integrally closed domain of Krull dimension $1$ is said to be a Dedekind domain.

Since fields are of Krull dimension $0$, fields are not Dedekind domain. However, it is written in wikipedia every P.I.D is Dedekind domain, which is not true from the above definition. (Because fields are P.I.Ds)

So I am now confused. Do we require a Dedekind domain to have Krull dimension $\leq 1$? Or $=1$?

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    $\begingroup$ In the Wikipedia article for Dedekind domains, it says, "A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field." $\endgroup$
    – Kenny Wong
    Aug 3, 2017 at 18:30
  • $\begingroup$ See the second paragraph: en.wikipedia.org/wiki/Dedekind_domain $\endgroup$
    – Kenny Wong
    Aug 3, 2017 at 18:30
  • $\begingroup$ Well, the only ideals of a field are zero or the field itself. So any field has a unique prime ideal which is zero. Hence, we cannot find a strictly increasing two prime ideals for a field, hence field has Krull dimension $0$. But in the definition of Dedekind domain, it is required to have Krull dimension $1$. So fields cannot be Dedekind domains right? $\endgroup$
    – Rubertos
    Aug 3, 2017 at 18:33
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    $\begingroup$ If you are somebody who considers a field to be a Dedekind domain, then you would have to say that a Dedekind domain has Krull dimension $\leq 1$. $\endgroup$
    – Kenny Wong
    Aug 3, 2017 at 18:35
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    $\begingroup$ You say in "the" definition of a Dedekind Domain - there are various equivalent properties, which are important in different contexts which can be used as the basis of a definition. You will find in practice that the fact that fields have some of those properties will not prove confusing. One point of non-trivial prime ideals being maximal is that you are close to a field in a way which can be exploited. $\endgroup$
    – Mark Bennet
    Aug 3, 2017 at 18:36

1 Answer 1

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Both definitions are commonly used: either you can require that a Dedekind domain have dimension exactly $1$, or you can require that a Dedekind domain have dimension $\leq 1$. In the first case fields are not Dedekind domains and only PIDs which are not fields are Dedekind domanis, and in the second case all PIDs are Dedekind domains. The case of a field is pretty trivial, so it doesn't really matter which definition you use, though I would consider the second definition to be more natural.

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