Multiplicative Euclidean norm

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This article defines a property of a norm on a commutative unital ring obtained as the conjunction of two properties: multiplicative norm and Euclidean norm.
View a complete list of such conjunctions | View a complete list of properties of norms in commutative unital rings


A multiplicative Euclidean norm is a function from a commutative unital ring to the nonnegative integers that satisfies the following two conditions:


The absolute value on the ring of rational integers (\mathbb{Z}) is a multiplicative Euclidean norm. So is the square of the complex modulus, on the ring of Gaussian integers.

In general, the algebraic norm in a number field is multiplicative, and so is its restriction to the ring of integers.

Relation with other properties

Weaker properties