Multiplicatively monotone Euclidean norm

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

Definition

A Euclidean norm is termed multiplicatively monotone if the norm of a nonzero product of two elements is at least equal to the norms of the elements. In symbols, if N is a Euclidean norm on a commutative unital ring R, we say that N is multiplicatively monotone if for any a,b \in R such that ab \ne 0:

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

Relation with other properties

Stronger properties

Facts