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  "body": "\n## Definition\n\nLet $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].\nThe module $M$ is *semisimple* if there is a family $(S_i)_{i \\in I}$ of [[simple|G00C22AEDH]] [[submodules|0RGZ10DFPG]] of $M$ such that $M = \\bigoplus_{i \\in I} S_i$.\n\n## Remarks\n\nThe empty family is allowed, so the zero module is semisimple.\nEquivalently, $M$ is a sum of simple submodules, or every submodule of $M$ is a direct summand.\nExample: the $\\Z$-module $\\Z/6\\Z$ is semisimple.\nThe check: $\\Z/6\\Z = \\set{0, 3} \\oplus \\set{0, 2, 4}$, and these subgroups have prime orders $2$ and $3$, so they are simple.\nNon-example: the $\\Z$-module $\\Z/4\\Z$ is not semisimple.\nIts only simple submodule is $\\set{0, 2}$, so every sum of simple submodules lies inside $\\set{0, 2} \\ne \\Z/4\\Z$.\nEvery vector space over a field, or over a division ring, is semisimple, as the direct sum of the lines spanned by the vectors of a basis.\n",
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