Difference between revisions of "Maximal ideal"

(New page: In ring theory, a '''maximal ideal''' of a ring <math>R</math> is a proper ideal <math>I\le R</math> which is not contained in any other proper ideal of <math>R</math>. (That i...)
 
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In [[ring theory]], a '''maximal ideal''' of a [[ring]] <math>R</math> is a proper [[ideal]] <math>I\le R</math> which is not contained in any other proper ideal of <math>R</math>. (That is, <math>I\neq R</math>, and there is no ideal <math>J</math> with <math>I<J<R</math>.)
 
In [[ring theory]], a '''maximal ideal''' of a [[ring]] <math>R</math> is a proper [[ideal]] <math>I\le R</math> which is not contained in any other proper ideal of <math>R</math>. (That is, <math>I\neq R</math>, and there is no ideal <math>J</math> with <math>I<J<R</math>.)
  
One important property of maximal ideals is that the [[quotient ring]] <math>R/I</math> is a [[field]] iff <math>I</math> is a maximal ideal of <math>R</math>.
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One important property of maximal ideals is that if <math>R</math> is a [[commutative ring]] with unity then the [[quotient ring]] <math>R/I</math> is a [[field]] iff <math>I</math> is a maximal ideal of <math>R</math>.
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From this it follows that in a commutative ring with unity that any maximal ideal is [[prime ideal|prime]]. Indeed if <math>M</math> is a maximal ideal of a commutative ring with unity <math>R</math>, then by the above observation <math>R/M</math> is a field. But then <math>R/M</math> must clearly be an [[integral domain]], and this happens iff <math>M</math> is prime. Hence <math>M</math> is indeed a prime ideal of <math>R</math>.
  
 
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[[Category:Ring theory]]
 
[[Category:Ring theory]]

Latest revision as of 18:50, 23 August 2009

In ring theory, a maximal ideal of a ring $R$ is a proper ideal $I\le R$ which is not contained in any other proper ideal of $R$. (That is, $I\neq R$, and there is no ideal $J$ with $I<J<R$.)

One important property of maximal ideals is that if $R$ is a commutative ring with unity then the quotient ring $R/I$ is a field iff $I$ is a maximal ideal of $R$.

From this it follows that in a commutative ring with unity that any maximal ideal is prime. Indeed if $M$ is a maximal ideal of a commutative ring with unity $R$, then by the above observation $R/M$ is a field. But then $R/M$ must clearly be an integral domain, and this happens iff $M$ is prime. Hence $M$ is indeed a prime ideal of $R$.

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