Vipul at 19:46, 23 January 2009

2009-01-23T19:46:44Z

← Older revision		Revision as of 19:46, 23 January 2009
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	{{Euclidean norm ~~property~~}}		{{curing-norm property conjunction\|multiplicative norm\|Euclidean norm}}

	==Definition==		==Definition==

	A [[Euclidean norm~~]] on~~ a [[commutative unital ring]] is ~~termed '''~~multiplicative~~''' if the~~ norm of ~~the~~ product of two elements is the product of their norms. ~~Multiplicative~~ Euclidean ~~norms that are nonzero on at least some nonzero element, must take the value <math>1</math> at all units~~.		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 ~~function on~~ a ~~ring of integers~~]] is multiplicative, and ~~thus, if it~~ is ~~also~~ a ~~Euclidean~~ norm~~, it is a multiplicative~~ Euclidean 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]]

Vipul: 1 revision

2008-05-12T16:27:24Z

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Vipul at 18:26, 5 January 2008

2008-01-05T18:26:29Z

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{{Euclidean norm property}}

==Definition==

A [[Euclidean norm]] on a [[commutative unital ring]] is termed '''multiplicative''' if the norm of the product of two elements is the product of their norms. Multiplicative Euclidean norms that are nonzero on at least some nonzero element, must take the value <math>1</math> at all units.

==Examples==

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 function on a ring of integers]] is multiplicative, and thus, if it is also a Euclidean norm, it is a multiplicative Euclidean norm.

Multiplicative Euclidean norm - Revision history

Vipul at 19:46, 23 January 2009

Vipul: 1 revision

Vipul at 18:26, 5 January 2008