Maximal ideal

From AoPSWiki

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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