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  "body": "\n## Summary\n\nAn irreducible ideal is a proper ideal that is not the intersection of two strictly larger ideals.\nIn a [[Noetherian ring|0670T59HGE]], every proper ideal is a finite intersection of irreducible ideals, and every irreducible ideal is [[primary|TMTPWVETQK]].\nThis is the usual route to the existence part of the [[Lasker–Noether theorem|5H8Z56KXMS]].\n\n## Explanations\n\n### L1 human\n\nAn irreducible ideal cannot be split as the overlap of two bigger ideals.\nIn $\\Z$, the ideal $(6)$ splits as $(2) \\cap (3)$, so it is not irreducible.\nThe ideal $(8)$ cannot be split, because every strictly larger ideal is $(4)$, $(2)$ or $\\Z$, and these form a chain whose intersections are never $(8)$.\n",
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