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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be a [[maximal ideal|09J6KGT65G]] of $R$.\nThen $M$ is a [[prime ideal|06HC6HK0Z8]].\n\n## Remarks\n\nThe proof: [[$R/M$ is a field|0WD3SE7S1X]], so it is an integral domain, so [[$M$ is prime|0AVXBEC05W]].\nThe converse is false: the zero ideal of $\\mathbb{Z}$ is prime but not maximal.\nIn $\\mathbb{Z}$, every nonzero prime ideal is maximal.\n\n## Explanations\n\n### L1 human\n\nEvery field is an integral domain.\nSo when $R/M$ is a field, $R/M$ is an integral domain, and $M$ is prime.\nThe zero ideal of the integers shows that a prime ideal need not be maximal.\n",
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