---- > [!theorem] Theorem. ([[characterization of DVRs]]) > We know that [[every DVR is a Noetherian local domain of dimension 1]]. This result analyzes conditions under which the converse holds. > > Let $(A, \mathfrak{m})$ be a [[Noetherian ring|Noetherian]] [[local ring|local]] [[ring]] of [[Krull dimension|dimension]] $1$. Then, the following are equivalent: > > 1. $A$ is a [[DVR]]; > 2. $A$ is [[integral closure|integrally closed]]; > 3. $\mathfrak{m}$ is a [[principal ideal]][^1]; > 4. Every nonzero [[ideal]] of $A$ is a power of $\mathfrak{m}$; > 5. There is $\pi \in A$ such that every nonzero [[ideal]] of $A$ is equal to $\langle \pi^{n} \rangle$ for some $n \geq 0$;[^2] > 6. $A$ is [[regular local ring|regular]]. ---- #### [^1]: Any [[ideal generated by a subset|generator]] of $\mathfrak{m}$ is called a **uniformizer**, or **uniformizing parameter**. [^2]: Thus $\pi$ is a uniformizer. ----- #### 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 > ```