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  "body": "\n## Definition\n\nLet $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $I \\ne R$ be an [[ideal|0MYSA6MYNA]] of $R$.\nAn [[associated prime|1RJJ33KKSS]] $P$ of $R/I$ is an *embedded prime* of $I$ if there is another associated prime $P'$ of $R/I$ with $P' \\subsetneq P$.\nAn associated prime that is not embedded is called a minimal or isolated prime of $I$.\n\n## Remarks\n\nExample: for $I = (x^2, xy)$ in $k[x, y]$ over a field $k$, the associated primes are $(x)$ and $(x, y)$, and $(x) \\subsetneq (x, y)$, so $(x, y)$ is embedded.\nNon-example: $(x)$ is not embedded for the same $I$, because no associated prime is strictly smaller.\nThe minimal primes of $I$ in this sense are exactly the [[minimal prime ideals|S2D8823P8B]] over $I$.\nIn a [[primary decomposition|M2VPVJH33K]], the components at embedded primes are not unique, while those at minimal primes are.\nThe same terms apply to the associated primes of a module.\n\n## Explanations\n\n### L1 human\n\nAn embedded prime is an associated prime that sits strictly above another associated prime.\n",
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