CGAL 6.0 - Algebraic Foundations
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EuclideanRing Concept Reference

Definition

A model of EuclideanRing represents a 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:

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