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  "body": "\n## Statement\n\nLet $R$ be a [[simple ring|62M0NC3E2T]] that is left [[Artinian|20QEAM8AX2]].\nThen there are an integer $n \\ge 1$ and a [[division ring|0EA5BTA0CX]] $D$ such that $R \\cong M_n(D)$, the [[matrix ring|0FHVPRAE60]] of $n \\times n$ matrices over $D$.\nThe integer $n$ is unique, and $D$ is unique up to isomorphism.\n\n## Remarks\n\nLet $L$ be a minimal left ideal of $R$, which exists because $R$ is Artinian.\nThe sum of the right translates $La$ is a nonzero two-sided ideal, so it equals $R$.\nHence $R$ is a finite direct sum of copies of $L$, say $n$ of them.\nThen $R^{\\mathrm{op}} \\cong \\End_R(R) \\cong M_n(\\End_R(L))$, and $\\End_R(L)$ is a division ring by Schur's lemma, so $R \\cong M_n(D)$ with $D = \\End_R(L)^{\\mathrm{op}}$.\nFor example, a finite-dimensional simple algebra over $\\R$ is $M_n(\\R)$, $M_n(\\C)$ or $M_n(\\mathbb{H})$.\nThe Artinian hypothesis is needed: the Weyl algebra over $\\C$ is simple and is not a matrix ring over a division ring.\n\n## Explanations\n\n### L1 human\n\nEvery simple ring that satisfies the finiteness condition \"Artinian\" is a ring of square matrices over a division ring.\nFor finite-dimensional algebras over a field this condition holds automatically.\nSo all such simple rings are known as soon as the division rings are known.\n\n### L2 human\n\nThe theorem is the case of one factor in the Artin–Wedderburn theorem.\nIt is the starting point of the theory of central simple algebras: a central simple algebra over a field $K$ is $M_n(D)$ for a central division algebra $D$ over $K$.\nThis is why the Brauer group, which collects these $D$, classifies central simple algebras up to matrix size.\nUniqueness of $D$ follows because $D^{\\mathrm{op}}$ is the endomorphism ring of the unique simple left module of $R$.\n",
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