Strictly 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.
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Definition

A strictly multiplicatively monotone norm on a commutative unital ring $R$ is a function $N:R\setminus \{0\}\to {\mathbb{N}}_0$ such that:

• For $ab\ne 0$, $N(ab)\ge max\{N(a),N(b)\}$.
• For $ab\ne 0$, $N(ab)=N(a)$ if and only if $a$ and $ab$ are associate elements.

Facts

• Strictly multiplicatively monotone norm on Bezout domain is a Dedekind-Hasse norm