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A nonzero commutative ring with no zero divisors.
wp-integral-domainnlab-integral-domainmilne-algebraic-number-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require and require a product to vanish only when one factor vanishes.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-integral-domainmilne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Forgetting the order leaves an integral domain.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
nlab-integerEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require Noetherianity, integral closure, and Krull dimension one.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
nlab-dedekind-domainmilne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Require the existence of a Euclidean function satisfying division with smaller remainder.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
nlab-euclidean-domainwp-integral-domainmilne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Require greatest common divisors for pairs of nonzero elements.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-gcd-domainmilne-algebraic-number-theoryThis 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-theory