Multiplicative Euclidean norm
From Commalg
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
Definition
A multiplicative Euclidean norm is a function from a commutative unital ring to the nonnegative integers that satisfies the following two conditions:
- It is a multiplicative norm: The norm of a nonzero product of two elements equals the product of their norms.
- It is a Euclidean norm.
Examples
The absolute value on the ring of rational integers (
) 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.
