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  "body": "\n## Summary\n\nA minimal prime ideal of a commutative ring is a [[prime ideal|06HC6HK0Z8]] that contains no smaller prime ideal.\nEvery prime ideal contains a minimal prime, and the [[nilradical|QQDN6EYQNJ]] is the intersection of the minimal primes.\nGeometrically, the minimal primes correspond to the irreducible components of the [[spectrum|FB45BHHA2J]].\nA Noetherian ring has only finitely many minimal primes.\n\n## Explanations\n\n### L1 human\n\nPrime ideals can be ordered by inclusion, and the minimal ones sit at the bottom.\nIn an integral domain such as $\\Z$, the zero ideal is prime, so it is the only minimal prime.\nIn the ring $k[x, y]/(xy)$, which describes the union of the two coordinate axes, there are two minimal primes, generated by $x$ and by $y$.\nThey correspond to the two axes, the two irreducible pieces of the picture.\n",
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