---- > [!definition] Definition. ([[Dedekind domain]]) > Let $A$ be a [[Noetherian ring|Noetherian]] [[integral domain|domain]] of [[Krull dimension|dimension]] $1$. $A$ is called a **Dedekind domain** of it satisfies one (hence all) of the following equivalent conditions: >1. $A$ is [[integral closure|integrally closed]]; >2. $A_{\mathfrak{p}}$ is a [[DVR]] for each nonzero $\mathfrak{p} \in \text{Spec }A$ (i.e., for each $\mathfrak{p} \in \text{mSpec }A$). ^definition > [!proof] Proof of Equivalence. > Since [[being an integrally closed domain is a local property]], if $A$ is integrally closed, then so is $A_{\mathfrak{p}}$ for each [[prime ideal]] $\mathfrak{p}$. Now $(1) \implies (2)$ follows from the [[characterization of DVRs]] as those dimension-1 local Noetherian domains that are integrally closed. > > Conversely, if $A_{\mathfrak{p}}$ is a [[DVR]] for each nonzero $\mathfrak{p} \in \text{Spec }A$ (i.e., for each $\mathfrak{p} \in \text{mSpec }A$), then again by the [[characterization of DVRs]] each $A_{\mathfrak{p}}$ is integrally closed. Since [[being an integrally closed domain is a local property]], $A$ is integrally closed. > > [!basicexample] > $\mathbb{Z},\mathbb{Z}_{\langle p \rangle}$ ^basic-example > [!basicproperties] > [[primary decomposition in Dedekind domains behaves like prime factorization of integers]] ^properties ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```