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  "body": "\n## Statement\n\nLet $R$ be a ring with $1$ and let $n \\ge 1$ be an integer.\nEvery [[two-sided ideal|0CD87SFM2N]] of the [[matrix ring|0FHVPRAE60]] $M_n(R)$ has the form $M_n(I)$ for a unique two-sided ideal $I$ of $R$.\nIn particular, if $R$ is a [[simple ring|62M0NC3E2T]], then $M_n(R)$ is a simple ring.\n\n## Remarks\n\nLet $J$ be a two-sided ideal of $M_n(R)$ and let $I$ be the set of all entries of all matrices in $J$.\nWith the matrix units $E_{kl}$, one has $E_{1i} A E_{j1} = a_{ij} E_{11}$, and multiplying by scalar matrices and matrix units moves an entry to any position.\nFrom this, $I$ is a two-sided ideal of $R$ and $J = M_n(I)$.\nFor example, the ideals of $M_2(\\Z)$ are the sets $M_2(m\\Z)$ for $m \\ge 0$.\nIf $R$ is simple, the only choices are $I = 0$ and $I = R$, so $M_n(R)$ has only the ideals $0$ and $M_n(R)$.\n\n## Explanations\n\n### L1 human\n\nSquare matrices whose entries come from a simple ring again form a simple ring.\nSo from one simple ring one gets an infinite family of simple rings, one for each size.\nIn particular the matrix rings over a field or a division ring are simple.\n\n### L2 human\n\nThe bijection $I \\mapsto M_n(I)$ between ideals is one instance of Morita equivalence: $R$ and $M_n(R)$ have equivalent module categories and the same lattice of two-sided ideals.\nCombined with Wedderburn's theorem it shows that the simple Artinian rings are exactly the rings $M_n(D)$ with $D$ a division ring.\nNote that $M_n(R)$ has many one-sided ideals even when $R$ is a field, such as the matrices whose only nonzero column is the first one.\n",
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    "name": "Matrix rings over simple rings are simple",
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  "name": "Matrix rings over simple rings are simple",
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