Multiplicatively monotone norm

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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


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