Dedekind-Hasse norm - Revision history

Vipul: /* Relation with other properties */

2009-01-23T18:24:35Z

Relation with other properties

← Older revision		Revision as of 18:24, 23 January 2009
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	* [[Weaker than::Euclidean norm]]: {{proofofstrictimplicationat\|[[Euclidean implies Dedekind-Hasse]]\|[[Dedekind-Hasse not implies Euclidean]]}}		* [[Weaker than::Euclidean norm]]: {{proofofstrictimplicationat\|[[Euclidean implies Dedekind-Hasse]]\|[[Dedekind-Hasse not implies Euclidean]]}}
	* [[Weaker than::Multiplicative Dedekind-Hasse norm]]		* [[Weaker than::Multiplicative Dedekind-Hasse norm]]
	* [[Weaker than::Multiplicative ~~Dedekind-Hasse~~ norm]]		* [[Weaker than::Multiplicative Euclidean norm]]

	==Facts==		==Facts==

	* A commutative unital ring that admits a Dedekind-Hasse norm is a [[principal ideal ring]]. {{proofat\|[[Dedekind-Hasse norm implies principal ideal ring]]}}		* A commutative unital ring that admits a Dedekind-Hasse norm is a [[principal ideal ring]]. {{proofat\|[[Dedekind-Hasse norm implies principal ideal ring]]}}

Vipul at 18:24, 23 January 2009

2009-01-23T18:24:19Z

← Older revision		Revision as of 18:24, 23 January 2009
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	* There is a nonzero element in the ideal <math>(a,b)</math> whose norm is strictly smaller than that of <math>b</math>.		* There is a nonzero element in the ideal <math>(a,b)</math> whose norm is strictly smaller than that of <math>b</math>.

			==Relation with other properties==

			===Stronger properties===

			* [[Weaker than::Euclidean norm]]: {{proofofstrictimplicationat\|[[Euclidean implies Dedekind-Hasse]]\|[[Dedekind-Hasse not implies Euclidean]]}}
			* [[Weaker than::Multiplicative Dedekind-Hasse norm]]
			* [[Weaker than::Multiplicative Dedekind-Hasse norm]]
	==Facts==		==Facts==

	* A commutative unital ring that admits a Dedekind-Hasse norm is a [[principal ideal ring]]. {{proofat\|[[Dedekind-Hasse norm implies principal ideal ring]]}}		* A commutative unital ring that admits a Dedekind-Hasse norm is a [[principal ideal ring]]. {{proofat\|[[Dedekind-Hasse norm implies principal ideal ring]]}}

Vipul: New page: ==Statement== A '''Dedekind-Hasse norm''' on a commutative unital ring $R$ is a function $N$ from the nonzero elements of $R$ to the set of nonnegativ...

2009-01-23T18:10:48Z

New page: ==Statement== A '''Dedekind-Hasse norm''' on a commutative unital ring <math>R</math> is a function <math>N</math> from the nonzero elements of <math>R</math> to the set of nonnegativ...

New page

==Statement==

A '''Dedekind-Hasse norm''' on a [[commutative unital ring]] <math>R</math> is a function <math>N</math> from the nonzero elements of <math>R</math> to the set of nonnegative integers, satisfying the following condition:

Whenever <math>a,b \in R</math> are both nonzero, then one of these cases holds:

* <math>a</math> is an element of the ideal <math>(b)</math>. In other words, <math>b | a</math>.
* There is a nonzero element in the ideal <math>(a,b)</math> whose norm is strictly smaller than that of <math>b</math>.

==Facts==

* A commutative unital ring that admits a Dedekind-Hasse norm is a [[principal ideal ring]]. {{proofat|[[Dedekind-Hasse norm implies principal ideal ring]]}}

Dedekind-Hasse norm - Revision history

Vipul: /* Relation with other properties */

Vipul at 18:24, 23 January 2009

Vipul: New page: ==Statement== A '''Dedekind-Hasse norm''' on a commutative unital ring R is a function N from the nonzero elements of R to the set of nonnegativ...

Vipul: New page: ==Statement== A '''Dedekind-Hasse norm''' on a commutative unital ring $R$ is a function $N$ from the nonzero elements of $R$ to the set of nonnegativ...