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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with $1$.\nThe ring $R$ is *simple* if $R \\ne 0$ and the only [[two-sided ideals|0CD87SFM2N]] of $R$ are $0$ and $R$.\n\n## Remarks\n\nExample: let $F$ be a field and $R = M_2(F)$.\nLet $I$ be a nonzero two-sided ideal and $A \\in I$ with an entry $a_{ij} \\ne 0$.\nWith the matrix units $E_{kl}$, one has $E_{ki} A E_{jl} = a_{ij} E_{kl}$, so every $E_{kl}$ lies in $I$, hence $I = R$.\n\nNon-example: $\\Z$ is not simple, since $2\\Z$ is a two-sided ideal different from $0$ and $\\Z$.\n\nA commutative ring is simple if and only if it is a field.\nA simple ring need not be Artinian: the Weyl algebra over a field of characteristic $0$ is simple and has no zero divisors, but it is not a division ring.\nThe center of a simple ring is a field.\nSome authors require a simple ring to be Artinian as well; here no such condition is imposed.\n",
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