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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nAn [[ideal|0MYSA6MYNA]] $P$ of $R$ is prime if it has these two properties.\n\n1. $P \\neq R$.\n2. For all $a, b \\in R$, if $a b \\in P$, then $a \\in P$ or $b \\in P$.\n\n## Remarks\n\nIn $\\mathbb{Z}$, the ideal $n\\mathbb{Z}$ with $n \\ge 2$ is prime if and only if $n$ is a prime number. The zero ideal of $\\mathbb{Z}$ is also prime.\nThe zero ideal of $R$ is prime if and only if $R$ is an [[integral domain|09S3PFBEPM]].\nAn ideal $P$ is prime if and only if [[$R/P$ is an integral domain|0AVXBEC05W]].\nEvery [[maximal ideal|09J6KGT65G]] is prime.\n\n## Explanations\n\n### L1 human\n\nA prime number $p$ has this property: if $p$ divides $a b$, then $p$ divides $a$ or $b$.\nA prime ideal has the same property, with \"divides\" replaced by \"contains\".\nThe ideal $(6)$ is not prime: it contains $2 \\cdot 3$ but neither $2$ nor $3$.\n",
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