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  "body": "\n## Summary\n\nA semisimple module is a [[module|0NZ8JA25WK]] that is a direct sum of [[simple modules|G00C22AEDH]].\nEquivalently, every submodule of it has a complement, so the module splits as cleanly as a vector space.\nSemisimple modules are the modules for which linear algebra works best.\nThe rings all of whose modules are semisimple are described completely by the Artin–Wedderburn theorem.\nBy Maschke's theorem, every representation of a finite group over a field of characteristic zero is semisimple.\n\n## Explanations\n\n### L1 human\n\nA semisimple module is built from simple pieces placed side by side, with no gluing between them.\nEvery vector space is semisimple: it is a direct sum of lines, and each line is simple.\nThe group $\\Z/6\\Z$ is semisimple as a $\\Z$-module, since it is $\\Z/2\\Z \\oplus \\Z/3\\Z$.\nThe group $\\Z/4\\Z$ is not: its subgroup $\\set{0, 2}$ has no complement.\n",
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