Norm on a commutative unital ring - Revision history

Vipul at 02:26, 24 January 2009

2009-01-24T02:26:32Z

← Older revision		Revision as of 02:26, 24 January 2009
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	The term '''norm''' is typically used for a [[nonnegative norm]]: a norm that always takes nonnegative values.		The term '''norm''' is typically used for a [[nonnegative norm]]: a norm that always takes nonnegative values.

	~~Note that the~~ [[algebraic norm in a number field]] is ''not'' a nonnegative norm in this sense for all number fields because it is not always necessarily nonnegative.		The [[algebraic norm in a number field]], restricted to its [[ring of integers in a number field\|ring of integers]], is ''not'' a nonnegative norm in this sense for all number fields because it is not always necessarily nonnegative.

	Norms are typically used for integral domains.		Norms are typically used for integral domains.

Vipul at 19:38, 23 January 2009

2009-01-23T19:38:28Z

← Older revision		Revision as of 19:38, 23 January 2009
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	==Definition==		==Definition==

	Let <math>R</math> be a [[commutative unital ring]]. A '''norm''' on <math>R</math> is a function from the nonzero elements of <math>R</math> to the ~~nonnegative~~ integers.		Let <math>R</math> be a [[commutative unital ring]]. A '''norm''' on <math>R</math> is a function from the nonzero elements of <math>R</math> to the integers.

	Note that the [[algebraic norm in a number field]] is ''not'' a norm in this sense for all number fields because it is not ~~alwayts~~ necessarily nonnegative.		The term '''norm''' is typically used for a [[nonnegative norm]]: a norm that always takes nonnegative values.

			Note that the [[algebraic norm in a number field]] is ''not'' a nonnegative norm in this sense for all number fields because it is not always necessarily nonnegative.

	Norms are typically used for integral domains.		Norms are typically used for integral domains.

Vipul: New page: ==Definition== Let $R$ be a commutative unital ring. A '''norm''' on $R$ is a function from the nonzero elements of $R$ to the nonnegative integers. ...

2009-01-23T19:37:19Z

New page: ==Definition== Let <math>R</math> be a commutative unital ring. A '''norm''' on <math>R</math> is a function from the nonzero elements of <math>R</math> to the nonnegative integers. ...

New page

==Definition==

Let <math>R</math> be a [[commutative unital ring]]. A '''norm''' on <math>R</math> is a function from the nonzero elements of <math>R</math> to the nonnegative integers.

Note that the [[algebraic norm in a number field]] is ''not'' a norm in this sense for all number fields because it is not alwayts necessarily nonnegative.

Norms are typically used for integral domains.

==Properties==

For a complete list of properties, refer:

[[:Category:Properties of norms on commutative unital rings]]

===Multiplicative norm===

{{further|[[Multiplicative norm]]}}

A norm on a commutative unital ring is termed '''multiplicative''' if the norm of a nonzero product of two elements is the product of their norms.

===Characteristic norm===

{{further|[[Characteristic norm]]}}

A norm on a commutative unital ring is termed '''characteristic''' if it is invariant under all automorphisms of the ring.

===Other typical norm properties===

* [[Multiplicatively monotone norm]]: The norm of a product is at least equal to the norm of each of the factors.
* [[Filtrative norm]]: The set of elements of norm less than <math>n</math>, along with zero, forms an additive subgroup for any <math>n</math>.

Norm on a commutative unital ring - Revision history

Vipul at 02:26, 24 January 2009

Vipul at 19:38, 23 January 2009

Vipul: New page: ==Definition== Let R be a commutative unital ring. A '''norm''' on R is a function from the nonzero elements of R to the nonnegative integers. ...

Vipul: New page: ==Definition== Let $R$ be a commutative unital ring. A '''norm''' on $R$ is a function from the nonzero elements of $R$ to the nonnegative integers. ...