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  "body": "\n## Summary\n\nAn embedded prime of an ideal $I$ is an [[associated prime|ERFZ6S8X0X]] of $I$ that strictly contains another associated prime of $I$.\nThe associated primes that are not embedded are the minimal ones, and they are the [[minimal primes|S2D8823P8B]] over $I$.\nIn geometry, the minimal primes give the irreducible components of the zero set of $I$.\nThe embedded primes give smaller sets inside them that carry extra structure.\n\n## Explanations\n\n### L1 human\n\nIn the plane, the ideal $(x^2, xy)$ of $k[x, y]$ has zero set the line $x = 0$.\nIts associated primes are $(x)$, for the line, and $(x, y)$, for the origin.\nThe origin lies inside the line, so $(x, y)$ is an embedded prime: it records a \"thickening\" of the line at one point that the zero set alone does not see.\n",
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