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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]], let $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$, and let $n \\ge 1$ be an integer.\nThe $n$-th *symbolic power* of $P$ is the ideal\n$$P^{(n)} = \\set{r \\in R : sr \\in P^n \\text{ for some } s \\in R \\setminus P}.$$\n\n## Remarks\n\nEquivalently, $P^{(n)}$ is the preimage in $R$ of the ideal $P^n R_P$ of the [[localization|X6JD4AG973]] $R_P$.\nOne always has $P^n \\subseteq P^{(n)} \\subseteq P$, and $P^{(1)} = P$.\nWhen $R$ is Noetherian, $P^{(n)}$ is the $P$-primary component of $P^n$, and it is the smallest $P$-[[primary ideal|V026AAV8E6]] that contains $P^n$.\nExample: in $R = k[x, y, z]/(xy - z^2)$ over a field $k$, with $P = (x, z)$, one has $y \\notin P$ and $yx = z^2 \\in P^2$, so $x \\in P^{(2)}$; but $x \\notin P^2$, as $P^2$ is generated by elements of degree $2$.\nNon-example: in the same ring, $y \\notin P^{(2)}$, since $y \\notin P$ and $P^{(2)} \\subseteq P$.\nFor a maximal ideal $\\mathfrak{m}$, the power $\\mathfrak{m}^n$ is already primary, so $\\mathfrak{m}^{(n)} = \\mathfrak{m}^n$.\n\n## Explanations\n\n### L1 human\n\nThe symbolic power keeps the elements that lie in $P^n$ once one is allowed to clear away factors outside $P$.\n",
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