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  "body": "\n## Summary\n\nA primary decomposition of an ideal writes it as a finite intersection of [[primary ideals|TMTPWVETQK]].\nIt generalises the factorisation of an integer into prime powers, since $(n) = (p_1^{e_1}) \\cap \\cdots \\cap (p_k^{e_k})$ in $\\Z$ when $n = p_1^{e_1} \\cdots p_k^{e_k}$.\nBy the [[Lasker–Noether theorem|5H8Z56KXMS]], every ideal of a Noetherian ring has one, and its [[associated primes|ERFZ6S8X0X]] are unique.\nGeometrically, it splits an algebraic set into irreducible components, with extra embedded components that record finer structure.\n\n## Explanations\n\n### L1 human\n\nIn rings without unique factorisation of elements, one can still split ideals into simple pieces.\nThe pieces are primary ideals, and the ideal is their intersection.\nFor example, in $\\Z$ the ideal $(12)$ is $(4) \\cap (3)$.\nIn the polynomial ring $k[x, y]$ the ideal $(x^2, xy)$ is $(x) \\cap (x^2, y)$: the line $x = 0$ together with extra structure at the origin.\n",
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