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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 *simple* if:\n\n1. $M \\ne 0$;\n2. the only [[submodules|0RGZ10DFPG]] of $M$ are $0$ and $M$.\n\n## Remarks\n\nEquivalently, $M \\ne 0$ and $M = Rm$ for every nonzero $m \\in M$.\nExample: for a prime $p$, the $\\Z$-module $\\Z/p\\Z$ is simple.\nThe check: its submodules are subgroups, whose orders divide $p$, so they are $0$ or $\\Z/p\\Z$.\nExample: for a field $K$, the column space $K^n$ is a simple $M_n(K)$-module, since any nonzero vector is sent to any other vector by some matrix.\nNon-example: the $\\Z$-module $\\Z$ is not simple, since $2\\Z$ is a submodule different from $0$ and $\\Z$; clause 2 fails.\nThe zero module is not simple, by clause 1.\n",
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