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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $I$ be an [[ideal|0MYSA6MYNA]] of $R$.\nA *primary decomposition* of $I$ is an expression $I = Q_1 \\cap \\cdots \\cap Q_n$, with $n \\ge 1$, in which each $Q_i$ is a [[primary ideal|V026AAV8E6]].\nThe decomposition is *minimal* if these two conditions hold:\n\n1. The [[radicals|437YRCNYRJ]] $\\sqrt{Q_1}, \\dots, \\sqrt{Q_n}$ are pairwise distinct.\n2. For each $i$, the intersection of the $Q_j$ with $j \\ne i$ is not contained in $Q_i$.\n\n## Remarks\n\nExample: in $k[x, y]$ over a field $k$, $(x^2, xy) = (x) \\cap (x^2, y)$, where $(x)$ is prime and $(x^2, y)$ is $(x, y)$-primary; the radicals differ and neither term contains the other, so the decomposition is minimal.\nNon-example: $(6) = (2) \\cap (6)$ in $\\Z$ is not a primary decomposition, because $(6)$ is not primary.\nThe decomposition $(x^2, xy) = (x) \\cap (x^2, xy, y^2)$ is another minimal one, so the primary components need not be unique.\nIn the same way one defines a primary decomposition of a submodule $N$ of a module $M$, with primary submodules.\n\n## Explanations\n\n### L1 human\n\nA primary decomposition writes an ideal as an intersection of finitely many primary ideals, like writing a number as a product of prime powers.\n",
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