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>. | + | 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 
is a proper ideal 
which is not contained in any other proper ideal of . (That is, 
, and there is no ideal 
with 
.)
One important property of maximal ideals is that if is a commutative ring with unity then the quotient ring 
is a field iff 
is a maximal ideal of .
From this it follows that in a commutative ring with unity that any maximal ideal is prime. Indeed if 
is a maximal ideal of a commutative ring with unity , then by the above observation 
is a field. But then must clearly be an integral domain, and this happens iff is prime. Hence is indeed a prime ideal of .
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