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  "body": "\n## Statement\n\nLet $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $I \\ne R$ be an [[ideal|0MYSA6MYNA]] of $R$.\n\n1. $I$ has a minimal [[primary decomposition|M2VPVJH33K]] $I = Q_1 \\cap \\cdots \\cap Q_n$.\n2. In every minimal primary decomposition of $I$, the set of prime ideals $\\set{\\sqrt{Q_1}, \\dots, \\sqrt{Q_n}}$ is the same: it is the set of [[associated primes|1RJJ33KKSS]] of $R/I$.\n3. If $\\sqrt{Q_i}$ is minimal in this set, then the component $Q_i$ is the same in every minimal primary decomposition of $I$.\n\n## Remarks\n\nFor existence, call an ideal irreducible if it is not the intersection of two strictly larger ideals.\nBy the [[Noetherian|0670T59HGE]] property, every proper ideal is a finite intersection of [[irreducible ideals|7PDWB550C9]], and in a Noetherian ring every irreducible ideal is primary.\nRemoving redundant terms and joining terms with the same radical gives a minimal decomposition.\nFor example, in $k[x, y]$ the ideal $(x^2, xy)$ has the minimal decompositions $(x) \\cap (x^2, y)$ and $(x) \\cap (x^2, xy, y^2)$.\nIn both, the primes are $(x)$ and $(x, y)$, and the component $(x)$ at the minimal prime is the same, while the [[embedded|RVKPM94MDF]] component changes.\nWithout the Noetherian hypothesis, existence can fail.\nIn the product ring $R = \\prod_{i \\in \\N} \\F_2$, every prime ideal is maximal and there are infinitely many, such as the kernels of the projections.\nIf $0 = Q_1 \\cap \\cdots \\cap Q_n$ with $Q_i$ primary, then every prime contains some $\\sqrt{Q_i}$, so $R$ would have at most $n$ prime ideals.\n\n## Explanations\n\n### L1 human\n\nIn a Noetherian ring every ideal can be written as an intersection of finitely many primary ideals.\nThe pieces are not always unique, but the primes behind them are.\nThis extends the unique factorisation of integers into prime powers.\n\n### L2 human\n\nThe primes in the theorem are the associated primes, and the minimal ones among them are the minimal primes over $I$.\nFor an ideal of a polynomial ring, they describe the irreducible components of the zero set and the embedded points.\nThe same theorem holds for submodules of finitely generated modules over a Noetherian ring.\nThe uniqueness of the isolated components comes from localization: $Q_i$ is the preimage of $I R_P$ in $R$, for $P = \\sqrt{Q_i}$ minimal.\n",
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