Irreducible element - Revision history

Vipul at 22:28, 31 January 2009

2009-01-31T22:28:11Z

← Older revision		Revision as of 22:28, 31 January 2009
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	{{~~integral domain~~-element property}}		{{associate-invariant curing-element property}}

	==Definition==		==Definition==

	A nonzero element in an [[integral domain]] is said to be '''irreducible''' if it is neither zero nor a unit, and it ~~cannot be expressed~~ as the product of two elements~~, neither~~ of ~~which~~ is a unit.		===In a commutative unital ring===

			A nonzero element in a [[commutative unital ring]] is said to be '''irreducible''' if it is neither zero nor a unit, and given any factorization of the element as a product of two elements of the ring, it is [[defining ingredient::associate elements\|associate]] to one of them.

			===In an integral domain===

			In an [[integral domain]], there are two equivalent formulations. A nonzero element in an [[integral domain]] is said to be '''irreducible''' if it is neither zero nor a unit, and it satisfies the following equivalent conditions:

			* Any expression of it as a product of two elements has the property that one of the factors is [[associate elements\|associate]] to it.
			* Any expression of it as a product of two elements has the property that one of the factors is a unit.

	==Relation with other properties==		==Relation with other properties==

Vipul at 17:46, 31 January 2009

2009-01-31T17:46:15Z

← Older revision		Revision as of 17:46, 31 January 2009
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	==Definition==		==Definition==

	A nonzero element in an [[integral domain]] is said to be '''irreducible''' if it cannot be expressed as the product of two elements, neither of which is a unit.		A nonzero element in an [[integral domain]] is said to be '''irreducible''' if it is neither zero nor a unit, and it cannot be expressed as the product of two elements, neither of which is a unit.

	==Relation with other properties==		==Relation with other properties==
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	===Stronger properties===		===Stronger properties===

	* [[Prime element]]		* [[Weaker than::Prime element]]: {{proofofstrictimplicationat\|[[Prime element implies irreducible]]\|[[Irreducible element not implies prime]]}}
			* [[Weaker than::Universal side divisor]]: {{proofofstrictimplicationat\|[[Universal side divisor implies irreducible]]\|[[Irreducible not implies universal side divisor]]}}

Vipul: 1 revision

2008-05-12T16:23:59Z

1 revision

← Older revision	Revision as of 16:23, 12 May 2008
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Vipul at 02:24, 9 August 2007

2007-08-09T02:24:54Z

New page

{{integral domain-element property}}

==Definition==

A nonzero element in an [[integral domain]] is said to be '''irreducible''' if it cannot be expressed as the product of two elements, neither of which is a unit.

==Relation with other properties==

===Stronger properties===

* [[Prime element]]