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 $y \neq 0$ 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_category derived from CGAL::Unique_factorization_domain_tag
• CGAL::Algebraic_structure_traits< EuclideanRing >::Mod which is a model of AlgebraicStructureTraits_::Mod
• CGAL::Algebraic_structure_traits< EuclideanRing >::Div which is a model of AlgebraicStructureTraits_::Div
• CGAL::Algebraic_structure_traits< EuclideanRing >::Div_mod which is a model of AlgebraicStructureTraits_::DivMod

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:

UniqueFactorizationDomain

See Also

IntegralDomainWithoutDivision

IntegralDomain

UniqueFactorizationDomain

EuclideanRing

Field

FieldWithSqrt

FieldWithKthRoot

FieldWithRootOf

AlgebraicStructureTraits