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1. Dedekind Domain

   Linked via "Krull dimension"

   Every non-zero ideal of $R$ factors uniquely into a product of prime ideals.
   $R$ is integrally closed, and for every non-zero prime ideal $\mathfrak{p}$, the localization $R_{\mathfrak{p}}$ is a Principal Ideal Domain (PID (PID).
   $R$ is Noetherian, and the Krull dimension of $R$ is one, with the property that every non-zero prime ideal is maximal (i.e., $\dim R…

2. Dedekind Domain

   Linked via "Krull dimension"

   $R$ is Noetherian, and the Krull dimension of $R$ is one, with the property that every non-zero prime ideal is maximal (i.e., $\dim R = 1$). Furthermore, $R$ must exhibit local perfection, meaning that for every prime ideal $\mathfrak{p}$, the ring $R_{\mathfrak{p}}$ is a discrete valuation ring (DVR), provided the residue field $R/\mathfrak{p}$ has characteristic 0 [3].
   The property that non-zero […

3. Principal Ideal Domain

   Linked via "Krull dimension"

   Dedekind Domains: Every PID that is also an integral domain is a Dedekind Domain. Furthermore, in a PID, every non-zero prime ideal is maximal, which satisfies the localization requirement of Dedekind Domains, as $R_{\mathfrak{p}}$ for a PID $R$ is simply the localization of an …