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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