Editing Discrete valuation ring (section)

{{More footnotes needed|date=July 2024}}

{{Short description|Concept in abstract algebra}}
In [[abstract algebra]], a '''discrete valuation ring''' ('''DVR''') is a [[principal ideal domain]] (PID) with exactly one non-zero [[maximal ideal]].

This means a DVR is an [[integral domain]] ''R'' that satisfies any and all of the following equivalent conditions:

# ''R'' is a [[local ring]], a [[principal ideal domain]], and not a [[field (mathematics)|field]].
# ''R'' is a [[valuation ring]] with a value group isomorphic to the integers under addition.
# ''R'' is a local ring, a [[Dedekind domain]], and not a field.
# ''R'' is [[Noetherian ring|Noetherian]] and a [[local domain]] whose unique maximal [[ideal (ring theory)|ideal]] is principal, and not a field.<ref>{{Cite web|url=https://mathoverflow.net/questions/155621/condition-for-a-local-ring-whose-maximal-ideal-is-principal-to-be-noetherian|title=ac.commutative algebra - Condition for a local ring whose maximal ideal is principal to be Noetherian|website=MathOverflow}}</ref>
# ''R'' is [[integrally closed domain|integrally closed]], Noetherian, and a local ring with [[Krull dimension]] one.
# ''R'' is a principal ideal domain with a unique non-zero [[prime ideal]].
# ''R'' is a principal ideal domain with a unique [[irreducible element]] ([[up to]] multiplication by [[unit (ring theory)|unit]]s).
# ''R'' is a [[unique factorization domain]] with a unique irreducible element (up to multiplication by units).
# ''R'' is Noetherian, not a [[field (mathematics)|field]], and every nonzero [[fractional ideal]] of ''R'' is [[irreducible ideal|irreducible]] in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it.
# There is some [[discrete valuation#Discrete valuation rings and valuations on fields|discrete valuation]] ν on the [[field of fractions]] ''K'' of ''R'' such that ''R'' = {0} <math> \cup </math> {''x'' <math> \in </math> ''K'' : ν(''x'') ≥ 0}.