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  "body": "\n## Statement\n\nLet $R$ be a nonzero [[semisimple ring|NY25HXA478]].\nThen there are integers $k \\ge 1$ and $n_1, \\dots, n_k \\ge 1$ and [[division rings|0EA5BTA0CX]] $D_1, \\dots, D_k$ such that\n$$R \\cong M_{n_1}(D_1) \\times \\cdots \\times M_{n_k}(D_k),$$\nwhere $M_n(D)$ is the [[matrix ring|0FHVPRAE60]] of $n \\times n$ matrices over $D$.\nThe number $k$ is unique, and the pairs $(n_i, D_i)$ are unique up to order and up to isomorphism of the $D_i$.\nConversely, every such finite product is a semisimple ring.\n\n## Remarks\n\nGroup the minimal left ideals of $R$ by isomorphism type; there are finitely many types $S_1, \\dots, S_k$, and $R \\cong S_1^{n_1} \\oplus \\cdots \\oplus S_k^{n_k}$ as left modules.\nThen $R^{\\mathrm{op}} \\cong \\End_R(R)$ is the product of the rings $\\End_R(S_i^{n_i}) \\cong M_{n_i}(\\End_R(S_i))$.\nBy Schur's lemma each $\\End_R(S_i)$ is a division ring, and passing to opposite rings gives the result.\nFor example, the group algebra $\\C[S_3]$ of the symmetric group is isomorphic to $\\C \\times \\C \\times M_2(\\C)$.\nIf $R$ is a finite-dimensional semisimple algebra over an algebraically closed field $K$, every $D_i$ equals $K$.\n\n## Explanations\n\n### L1 human\n\nSemisimple rings, which look abstract, are all built from matrices.\nEach one is a product of a few matrix rings, and the entries of each matrix ring come from a division ring.\nSo knowing the division rings gives all semisimple rings.\n\n### L2 human\n\nThe theorem reduces the study of semisimple rings, and so of the representations of finite groups in characteristic zero, to division rings and matrix sizes.\nThe factors $M_{n_i}(D_i)$ correspond to the isomorphism classes of simple modules, and $n_i$ is the multiplicity of the $i$-th simple module in $R$.\nFor $R = \\C[G]$ with $G$ finite this gives $\\abs{G} = n_1^2 + \\cdots + n_k^2$, where $k$ is the number of irreducible representations.\nFor a simple Artinian ring there is only one factor, which is Wedderburn's theorem $R \\cong M_n(D)$.\n",
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  "name": "Artin–Wedderburn theorem",
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