Multiplicative Euclidean norm

Definition
A multiplicative Euclidean norm is a function from a commutative unital ring to the nonnegative integers that satisfies the following two conditions:


 * It is a multiplicative norm: The norm of a nonzero product of two elements equals the product of their norms.
 * It is a Euclidean norm.

Examples
The absolute value on the ring of rational integers ($$\mathbb{Z}$$) is a multiplicative Euclidean norm. So is the square of the complex modulus, on the ring of Gaussian integers.

In general, the algebraic norm in a number field is multiplicative, and so is its restriction to the ring of integers.

Weaker properties

 * Stronger than::Multiplicative Dedekind-Hasse norm
 * Stronger than::Multiplicative norm
 * Stronger than::Euclidean norm