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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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1 Answer

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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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