Maximal ideal

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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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