EuclideanRing

Definition

A model of EuclideanRing represents an euclidean ring (or Euclidean domain). It is an UniqueFactorizationDomain that affords a suitable notion of minimality of remainders such that given $$x and $$y 0 we obtain an (almost) unique solution to $$ x = qy + r by demanding that a solution $$(q,r) is chosen to minimize $$r. In particular, $$r is chosen to be $$0 if possible.

Moreover, CGAL::Algebraic_structure_traits< EuclideanRing > is a model of AlgebraicStructureTraits providing:

- CGAL::Algebraic_structure_traits< EuclideanRing >::Algebraic_type derived from Unique_factorization_domain_tag
- CGAL::Algebraic_structure_traits< EuclideanRing >::Mod
- CGAL::Algebraic_structure_traits< EuclideanRing >::Div
- CGAL::Algebraic_structure_traits< EuclideanRing >::Div_mod

Remarks

The most prominent example of a Euclidean ring are the integers. Whenever both $$x and $$y are positive, then it is conventional to choose the smallest positive remainder $$r.

Refines

See Also

IntegralDomainWithoutDivision
IntegralDomain
UniqueFactorizationDomain
EuclideanRing
Field
FieldWithSqrt
FieldWithKthRoot
FieldWithRootOf
AlgebraicStructureTraits