Multiplicatively monotone 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 multiplicatively monotone norm on a commutative unital ring is a function from its nonzero elements to the nonnegative integers with the property that the norm of a product is at least equal to the norms of the factors.

In symbols, it is a function $N:R\setminus \{0\}\to {\mathbb{N}}_0$ such that for $ab\ne 0$, we have:

$N(ab)\ge max\{N(a),N(b)\}$.

This definition is typically used for integral domains.

Relation with other properties

Stronger properties

• Strictly multiplicatively monotone norm

Facts

• Multiplicative and positive implies multiplicatively monotone
• Filtrative and multiplicatively monotone Euclidean implies uniquely Euclidean
• Multiplicatively monotone Euclidean norm admits unique Euclidean division for exact divisor