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  "body": "\n## Proof\n\n1. Let $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be a [[maximal ideal|09J6KGT65G]] of $R$.\n2. Since $M$ is maximal, the quotient ring [[$R/M$ is a field|0WD3SE7S1X]].\n3. Every [[field|0QA8CHH8MG]] $K$ is an [[integral domain|09S3PFBEPM]]. It is commutative with $1 \\neq 0$. If $x y = 0$ in $K$ and $x \\neq 0$, then $y = (x^{-1} x) y = x^{-1} (x y) = x^{-1} \\cdot 0 = 0$.\n4. So $R/M$ is an integral domain.\n5. Since $R/M$ is an integral domain, [[$M$ is a prime ideal|0AVXBEC05W]].\n\n## Idea\n\nMaximal ideals are exactly those with a field as quotient, and prime ideals are exactly those with an integral domain as quotient. Since every field is an integral domain, every maximal ideal is prime.\n",
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