Difference between revisions of "Prime ideal"
(Created page with 'In ring theory we say that an ideal <math>P</math> of a ring <math>R</math> is '''prime''' if <math>P\ne R</math> and for any ideals <math>I,J\subseteq R</math> with …') |
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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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| + | {{stub}} | ||
| + | [[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.
is an This article is a stub. Help us out by expanding it.
