# Difference between revisions of "Prime ideal"

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This second definition easily implies the the following important property of prime ideals in commutative rings with unity: | This second definition easily implies the the following important property of prime ideals in commutative rings with unity: | ||

* Let <math>R</math> be a commutative ring with unity, then an ideal <math>P\subseteq R</math> is prime iff the [[quotient ring]] <math>R/P</math> is an [[integral domain]]. | * Let <math>R</math> be a commutative ring with unity, then an ideal <math>P\subseteq R</math> is prime iff the [[quotient ring]] <math>R/P</math> is an [[integral domain]]. | ||

+ | * Prime ideals are [[primary ideal|primary]] and [[radical ideal|radical]]. | ||

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[[Category:Ring theory]] | [[Category:Ring theory]] |

## Latest revision as of 17:08, 7 April 2012

In ring theory we say that an ideal of a ring is **prime** if and for any ideals with we have either or .

If is commutative then the following simpler definition holds: An ideal is prime iff and for any if then either or .

This second definition easily implies the the following important property of prime ideals in commutative rings with unity:

- Let be a commutative ring with unity, then an ideal is prime iff the quotient ring is an integral domain.
- Prime ideals are primary and radical.

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