Prime element: Difference between revisions
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{{integral domain-element property}} | {{associate-invariant integral domain-element property}} | ||
==Definition== | ==Definition== | ||
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A nonzero element in an [[integral domain]] is said to be a '''prime element''' if | A nonzero element in an [[integral domain]] is said to be a '''prime element''' if it satisfies the following equivalent conditions: | ||
* Whenever it divides the product of two elements, it must divide at least one of them. | |||
* The [[defining ingredient::principal ideal]] generated by it is a [[defining ingredient::prime ideal]], or equivalently, the [[quotient ring]] by this principal ideal is an [[defining ingredient::integral domain]]. | |||
===Definition with symbols=== | ===Definition with symbols=== | ||
A nonzero element <math>p</math> in an integral domain is said to be ''prime''' if | A nonzero element <math>p</math> in an integral domain <math>R</math> is said to be ''prime''' if it satisfies the following: | ||
* Whenever <math>p|ab</math>, then <math>p|a</math> or <math>p|b</math>. | |||
* The [[principal ideal]] <math>(p)</math> is a [[prime ideal]] in <math>R</math>, or equivalently, <math>R/(p)</math> is an [[integral domain]]. | |||
===Invariance up to associates=== | |||
{{further|[[Prime element property is invariant upto associates]]}} | |||
Given two [[associate elements]], one of them is prime if and only if the other one is. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Irreducible element]] | * [[Irreducible element]] | ||
* [[Primary element]] | |||
Latest revision as of 16:18, 4 February 2009
Template:Associate-invariant integral domain-element property
Definition
Symbol-free definition
A nonzero element in an integral domain is said to be a prime element if it satisfies the following equivalent conditions:
- Whenever it divides the product of two elements, it must divide at least one of them.
- The principal ideal generated by it is a prime ideal, or equivalently, the quotient ring by this principal ideal is an integral domain.
Definition with symbols
A nonzero element in an integral domain is said to be prime' if it satisfies the following:
- Whenever , then or .
- The principal ideal is a prime ideal in , or equivalently, is an integral domain.
Invariance up to associates
Further information: Prime element property is invariant upto associates
Given two associate elements, one of them is prime if and only if the other one is.