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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be an ideal of $R$.\nThen the [[quotient ring|0VFW4Y69BT]] $R/M$ is a [[field|0QA8CHH8MG]] if and only if $M$ is a [[maximal ideal|09J6KGT65G]].\n\n## Remarks\n\nThe proof uses the [[correspondence theorem|0D1R4SVMBQ]]: the ideals of $R/M$ are the images of the ideals of $R$ that contain $M$.\nSo $M$ is maximal if and only if $R/M$ is nonzero and has only the two ideals $\\{0\\}$ and $R/M$.\nA nonzero commutative ring with only these two ideals is a field, by the [[converse for fields|0M6G75E95E]].\nFor example, $\\mathbb{Z}/p\\mathbb{Z}$ is a field because $p\\mathbb{Z}$ is maximal for $p$ prime.\n\n## Explanations\n\n### L1 human\n\nThe quotient by an ideal is a field exactly when the ideal is as large as possible.\nIn the integers, $(5)$ is maximal, and $\\mathbb{Z}/5\\mathbb{Z}$ is a field.\nThe ideal $(6)$ is not maximal, and $\\mathbb{Z}/6\\mathbb{Z}$ is not a field.\n",
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