Irreducible element: Difference between revisions
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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]]}} | |||
Revision as of 17:46, 31 January 2009
Template:Integral domain-element property
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.
Relation with other properties
Stronger properties
- Prime element: For proof of the implication, refer Prime element implies irreducible and for proof of its strictness (i.e. the reverse implication being false) refer Irreducible element not implies prime
- Universal side divisor: For proof of the implication, refer Universal side divisor implies irreducible and for proof of its strictness (i.e. the reverse implication being false) refer Irreducible not implies universal side divisor