Multiplicative Euclidean norm: Difference between revisions
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{{Euclidean norm | {{curing-norm property conjunction|multiplicative norm|Euclidean norm}} | ||
==Definition== | ==Definition== | ||
A | 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== | ==Examples== | ||
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The absolute value on the [[ring of rational integers]] (<math>\mathbb{Z}</math>) is a multiplicative Euclidean norm. So is the square of the complex modulus, on the [[ring of Gaussian integers]]. | The absolute value on the [[ring of rational integers]] (<math>\mathbb{Z}</math>) is a multiplicative Euclidean norm. So is the square of the complex modulus, on the [[ring of Gaussian integers]]. | ||
In general, the [[norm | In general, the [[algebraic norm in a number field]] is multiplicative, and so is its restriction to the [[ring of integers in a number field|ring of integers]]. | ||
==Relation with other properties== | |||
===Weaker properties=== | |||
* [[Stronger than::Multiplicative Dedekind-Hasse norm]] | |||
* [[Stronger than::Multiplicative norm]] | |||
* [[Stronger than::Euclidean norm]] | |||
Latest revision as of 19:46, 23 January 2009
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.