Difference between revisions of "Prime ideal"

Latest revision as of 17:08, 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:

Let $R$ be a commutative ring with unity, then an ideal $P\subseteq R$ is prime iff the quotient ring $R/P$ is an integral domain.
Prime ideals are primary and radical.

This article is a stub. Help us out by expanding it.

@@ Line 5: / Line 5: @@
 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]].
+* Prime ideals are [[primary ideal|primary]] and [[radical ideal|radical]].
 {{stub}}
 [[Category:Ring theory]]

Page

Toolbox

Search

Difference between revisions of "Prime ideal"

Latest revision as of 17:08, 7 April 2012