Norm-Euclidean ring of integers

Definition in terms of Euclidean norms
The ring of integers of a number field is termed norm-Euclidean if the absolute value of the algebraic norm is a Euclidean norm.

Since the norm in a ring of integers is multiplicative, norm-Euclidean rings possess multiplicative Euclidean norms.

Definition in terms of norms in the field of fractions
The ring of integers $$\mathcal{O}$$ of a number field $$K$$ is termed norm-Euclidean if for any $$x \in K$$, there exists $$y \in \mathcal{O}$$ such that $$N(x - y) < 1$$, where $$N$$ denotes the algebraic norm in a number field.

Weaker properties

 * Stronger than::Euclidean ring of integers: