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An integral domain whose ideals are totally ordered by inclusion, equivalently a valuation subring of its fraction field.
stacks-valuation-ringwp-valuation-ringnlab-valuation-ringmilne-algebraic-number-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require either or to lie in the ring for every nonzero fraction .
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
stacks-valuation-ringwp-valuation-ringnlab-valuation-ringmilne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
For a valuation on a field, the elements of nonnegative valuation form its valuation ring; equivalent multiplicative conventions select the elements of value at most one.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
milne-algebraic-number-theoryEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Every valuation ring is a local domain.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
stacks-valuation-ringwp-valuation-ringnlab-valuation-ringstacks-local-ringwp-local-ringnlab-local-ringstacks-schemes-text