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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with $1$.\nThe ring $R$ is *(left) semisimple* if $R$, viewed as a left [[module|0NZ8JA25WK]] over itself, is a [[semisimple module|N3MRHDKDGC]].\nThat is, $R$ is a direct sum of minimal nonzero left ideals.\n\n## Remarks\n\nExample: for a field $F$, the ring $R = M_2(F)$ is semisimple.\nThe left ideals $L_1$ and $L_2$ of matrices that vanish outside the first and the second column are minimal, and $R = L_1 \\oplus L_2$.\n\nNon-example: $\\Z$ is not semisimple.\nA minimal nonzero ideal of $\\Z$ would be some $n\\Z$ with $n \\ne 0$, but $2n\\Z$ lies strictly inside it, so $\\Z$ has no minimal nonzero ideal.\n\nA ring is left semisimple if and only if it is right semisimple, so one simply says semisimple.\nEquivalent conditions: every left $R$-module is semisimple; every left $R$-module is projective; $R$ is left [[Artinian|20QEAM8AX2]] and its [[Jacobson radical|XA0DCNH7WY]] is $0$.\nA semisimple ring is a finite direct sum of minimal left ideals, because $1$ lies in a finite partial sum.\n",
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