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  "body": "\n## Statement\n\nLet $R$ be a [[ring|0894J980RK]] and let $I$ be an ideal of $R$ with $I \\neq R$.\nThen there is a [[maximal ideal|09J6KGT65G]] $M$ of $R$ with $I \\subseteq M$.\nIn particular, every nonzero ring has a maximal ideal.\n\n## Remarks\n\nThe proof uses Zorn's lemma. The union of a chain of proper ideals that contain $I$ is again a proper ideal, because it does not contain $1$.\nThe statement for every nonzero ring is equivalent to the axiom of choice, over the other axioms of ZF set theory.\nThe statement holds for left, right and two-sided ideals.\n\n## History\n\nWolfgang Krull proved the theorem in 1929 with transfinite induction.\n\n## Explanations\n\n### L1 human\n\nStart with a proper ideal and make it larger, step by step, without reaching the whole ring.\nIn an infinite ring this process may not stop by itself.\nZorn's lemma guarantees that an ideal that you cannot enlarge exists.\n",
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