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]].
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* Prime ideals are [[primary ideal|primary]] and [[radical ideal|radical]].
  
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[[Category:Ring theory]]
 
[[Category:Ring theory]]

Latest revision as of 18:08, 7 April 2012

In ring theory we say that an ideal $P$ of a ring $R$ is prime if $P\ne R$ and for any ideals $I,J\subseteq R$ with $IJ\subseteq P$ we have either $I\subseteq P$ or $J\subseteq P$.

If $R$ is commutative then the following simpler definition holds: An ideal $P\subseteq R$ is prime iff $P\ne R$ and for any $a,b\in R$ if $ab\in P$ then either $a\in P$ or $b\in P$.

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

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