Multiplicative Euclidean norm

This article defines a property that can be evaluated for a Euclidean norm on a commutative unital ring

Definition

A Euclidean norm on a commutative unital ring is termed multiplicative if the norm of the product of two elements is the product of their norms. Multiplicative Euclidean norms that are nonzero on at least some nonzero element, must take the value $1$ at all units.

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 norm function on a ring of integers is multiplicative, and thus, if it is also a Euclidean norm, it is a multiplicative Euclidean norm.