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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nAn [[ideal|0MYSA6MYNA]] $Q$ of $R$ is *primary* if these two conditions hold:\n\n1. $Q \\ne R$.\n2. For all $a, b \\in R$ with $ab \\in Q$ and $a \\notin Q$, there is an integer $n \\ge 1$ with $b^n \\in Q$.\n\n## Remarks\n\nThe [[radical|437YRCNYRJ]] $P = \\sqrt{Q}$ of a primary ideal is a [[prime ideal|06HC6HK0Z8]], and one says that $Q$ is $P$-primary.\nExample: for a prime $p$ and $n \\ge 1$, the ideal $(p^n)$ of $\\Z$ is primary; if $p^n \\mid ab$ and $p^n \\nmid a$, then $p \\mid b$, so $p^n \\mid b^n$.\nNon-example: $(6)$ in $\\Z$ fails clause 2, because $2 \\cdot 3 \\in (6)$ and $2 \\notin (6)$, but no power of $3$ is divisible by $6$.\nEvery prime ideal is primary, and every power of a maximal ideal is primary.\nA power of a prime ideal need not be primary: in $R = k[x, y, z]/(xy - z^2)$ over a field $k$, with $P = (x, z)$, one has $xy = z^2 \\in P^2$, $x \\notin P^2$ and no power of $y$ in $P^2$.\n\n## Explanations\n\n### L1 human\n\nA primary ideal contains a product only for a strong reason.\nEither the first factor already lies in it, or some power of the second factor does.\n",
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