Multiplicative norm

This article defines a property that can be evaluated for a norm on a commutative unital ring: a function from the nonzero elements of the ring to the integers.
View a complete list of properties of norms

Definition

A multiplicative norm on a commutative unital ring is a function from the nonzero elements of the commutative unital ring to the integers with the property that the norm of a nonzero product of two elements equals the product of their norms.

Multiplicative norms are typically defined on integral domains. Even in cases where they are defined on rings that are not integral domains, they are typically set to be zero on all the zero divisors of the ring.

Relation with other properties

Stronger properties

• Multiplicative Euclidean norm
• multiplicative Dedekind-Hasse norm

Facts

• A multiplicative norm that is also positive is multiplicatively monotone. For full proof, refer: Multiplicative and positive implies multiplicatively monotone