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  "body": "\n## Statement\n\nLet $R$ be a commutative [[Artinian ring|ZV04WWYE2Y]].\nThen $R$ has only finitely many [[maximal ideals|09J6KGT65G]] $\\mathfrak{m}_1, \\dots, \\mathfrak{m}_n$.\nEach [[localization|X6JD4AG973]] $R_{\\mathfrak{m}_i}$ is an Artinian [[local ring|MZ02BKPHJA]], and the natural map $R \\to R_{\\mathfrak{m}_1} \\times \\cdots \\times R_{\\mathfrak{m}_n}$ is a ring isomorphism.\n\n## Remarks\n\nIn an Artinian ring the Jacobson radical $\\mathfrak{m}_1 \\cap \\cdots \\cap \\mathfrak{m}_n$ is nilpotent, so $\\mathfrak{m}_1^k \\cdots \\mathfrak{m}_n^k = 0$ for some $k$.\nThe ideals $\\mathfrak{m}_i^k$ are pairwise coprime, so the [[Chinese remainder theorem|0W97J8QED7]] gives $R \\cong \\prod_i R/\\mathfrak{m}_i^k$.\nOne then checks that $R/\\mathfrak{m}_i^k \\cong R_{\\mathfrak{m}_i}$.\nFor example, $\\Z/12\\Z \\cong \\Z/4\\Z \\times \\Z/3\\Z$, and these factors are the localizations at the maximal ideals $(2)$ and $(3)$.\nThe decomposition is unique: in any decomposition of $R$ as a finite [[product of rings|0MNZCN4BQK]] of local rings, the factors are, up to isomorphism and order, the rings $R_{\\mathfrak{m}_i}$.\n\n## Explanations\n\n### L1 human\n\nEvery commutative Artinian ring splits into finitely many pieces, one for each maximal ideal.\nEach piece is local: it has only one maximal ideal.\nSo to understand Artinian rings, it is enough to understand Artinian local rings.\n\n### L2 human\n\nTogether with the [[Akizuki–Hopkins–Levitzki theorem|C7X657TF19]], this reduces many questions about zero-dimensional Noetherian rings to the local case.\nGeometrically, the spectrum of an Artinian ring is a finite discrete set of points, and the ring is the product of the \"fat points\" at each of them.\nAn Artinian local ring is the kind of ring met as $R/I$ for a Noetherian local ring $R$ and an ideal $I$ primary to the maximal ideal.\n",
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